4.1 Extended polarization currents whose distribution patterns propagate faster than light in vacuum

The experimentally realized source distribution described in § 2 is a generic member of a wide class of rotating source distributions. Any electric polarization $\boldsymbol{P}$ whose distribution pattern rotates uniformly with the constant angular frequency $\unicode[STIX]{x1D714}$ gives rise to a charge density $\unicode[STIX]{x1D70C}=-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{P}$ and a current density $\boldsymbol{j}=\unicode[STIX]{x2202}\boldsymbol{P}/\unicode[STIX]{x2202}t$ that, like $\boldsymbol{P}$ itself, depend on the azimuthal angle $\unicode[STIX]{x1D711}$ in only the combination

(4.1) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D711}}=\unicode[STIX]{x1D711}-\unicode[STIX]{x1D714}t,\end{eqnarray}$$

i.e. are of the forms

(4.2) $$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}P_{r,\unicode[STIX]{x1D711},z}(r,\unicode[STIX]{x1D711},z,t)\\ \unicode[STIX]{x1D70C}(r,\unicode[STIX]{x1D711},z,t)\\ j_{r,\unicode[STIX]{x1D711},z}(r,\unicode[STIX]{x1D711},z,t)\end{array}\right] & = & \displaystyle \left[\begin{array}{@{}c@{}}P_{r,\unicode[STIX]{x1D711},z}(r,\hat{\unicode[STIX]{x1D711}},z,t)\\ \unicode[STIX]{x1D70C}(r,\hat{\unicode[STIX]{x1D711}},z,t)\\ j_{r,\unicode[STIX]{x1D711},z}(r,\hat{\unicode[STIX]{x1D711}},z,t)\end{array}\right],\end{eqnarray}$$

where $(r,\unicode[STIX]{x1D711},z)$ are, as in § 2, the cylindrical polar coordinates based on the axis of rotation, $t$ (assumed to be ${\geqslant}0$ ) is time and $P_{r,\unicode[STIX]{x1D711},z}$ and $j_{r,\unicode[STIX]{x1D711},z}$ are the cylindrical components of $\boldsymbol{P}$ and $\boldsymbol{j}$ , respectively.

In (4.1) and (4.2) the coordinates $t$ and $\unicode[STIX]{x1D711}$ both range over $(0,\infty )$ but the coordinate $\hat{\unicode[STIX]{x1D711}}$ has a limited range of length $2\unicode[STIX]{x03C0}$ , e.g.

(4.3) $$\begin{eqnarray}0\leqslant \hat{\unicode[STIX]{x1D711}}<2\unicode[STIX]{x03C0}.\end{eqnarray}$$

As can be seen from the alternative form $\unicode[STIX]{x1D711}=\hat{\unicode[STIX]{x1D711}}+\unicode[STIX]{x1D714}t$ of (4.1), $\hat{\unicode[STIX]{x1D711}}$ is a Lagrangian coordinate that labels the rotating volume elements of the current distribution on each circle $r=\text{const}.$ , $z=\text{const}.$ , by their azimuthal positions at the time $t=0$ . This coordinate cannot range over a wider interval because the aggregate of volume elements that constitute a rotating source in its entirety can at most occupy an azimuthal interval of length $2\unicode[STIX]{x03C0}$ at any given time (e.g. at $t=0$ ). The polarization distribution $P_{r,\unicode[STIX]{x1D711},z}(r,\unicode[STIX]{x1D711},z,t)=s_{r,\unicode[STIX]{x1D711},z}(r,z)\cos (m\hat{\unicode[STIX]{x1D711}})$ given in (2.1), on which the analysis in the following sections will be based, is an example of this class of sources in which the range of $\hat{\unicode[STIX]{x1D711}}$ is likewise subject to the constraint (4.3).

Note that beyond $r=c/\unicode[STIX]{x1D714}$ (which I will refer to as the light cylinder) the distribution patterns of the above charge-current densities move with linear speeds $r\unicode[STIX]{x1D714}$ exceeding the speed of light in vacuum, $c$ . This is not inconsistent with the requirements of special relativity because the superluminally moving pattern is created by the coordinated motion of aggregates of subluminally moving particles (Bolotovskii & Ginzburg Reference Bolotovskii and Ginzburg1972; Ginzburg Reference Ginzburg1972; Bolotovskii & Bykov Reference Bolotovskii and Bykov1990). Not only is a superluminal current distribution of this type already generated in the laboratory (see Ardavan et al. Reference Ardavan, Hayes, Singleton, Ardavan, Fopma and Halliday2004b ; Bolotovskii & Serov Reference Bolotovskii and Serov2005, and § 2), but it also occurs in the magnetospheres of astrophysical objects containing rapidly rotating neutron stars such as pulsars (see Ardavan Reference Ardavan1981; Spitkovsky Reference Spitkovsky2006; Ardavan et al. Reference Ardavan, Ardavan, Singleton and Perez2008c ; Kalapotharakos et al. Reference Kalapotharakos, Contopoulos and Kazanas2012; Tchekhovskoy, Philippov & Spitkovsky Reference Tchekhovskoy, Philippov and Spitkovsky2016, and § 12).

4.2 Radiation field of a superluminally rotating charge-current distribution

According to (3.2), the radiation fields associated with the retarded potential (3.7) are given by

(4.4) $$\begin{eqnarray}\left[\begin{array}{@{}c@{}}\boldsymbol{E}\\ \boldsymbol{B}\end{array}\right]=\frac{1}{c^{2}}\int \text{d}^{3}\boldsymbol{x}\,\text{d}t\,\frac{\unicode[STIX]{x1D6FF}^{\prime }(t-t_{P}+R/c)}{R}\left[\begin{array}{@{}c@{}}\boldsymbol{j}-\unicode[STIX]{x1D70C}c\,\hat{\boldsymbol{n}}\\ \hat{\boldsymbol{n}}\times \boldsymbol{ j}\end{array}\right],\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}^{\prime }$ denotes the derivative of $\unicode[STIX]{x1D6FF}$ with respect to its argument and $\hat{\boldsymbol{n}}=\unicode[STIX]{x1D735}_{P}R=\boldsymbol{R}/R$ . The terms arising from the differentiation of $R^{-1}$ which describe static fields (terms that are non-zero even when the charge-current distribution is time independent) have been discarded here: in addition to decaying faster with distance, these terms are negligibly smaller than the retained terms in cases where the radiation frequency is appreciably larger than the rotation frequency (i.e. the integer $m$ in (2.1) appreciably exceeds unity). For an observation point that is located at infinity, the unit vector $\hat{\boldsymbol{n}}$ is independent of the integration variables $(\boldsymbol{x},t)$ and can be taken outside the above integrals to obtain $\boldsymbol{B}=\hat{\boldsymbol{n}}\times \boldsymbol{E}$ in the limit $|\boldsymbol{x}_{P}|\rightarrow \infty$ . Since we will be concerned also with observation points that lie at finite distances from the source, however, I will take the dependence of $\hat{\boldsymbol{n}}$ on $\boldsymbol{x}$ and $\boldsymbol{x}_{P}$ into account and treat $\boldsymbol{E}$ and $\boldsymbol{B}$ as two independent vectors in this paper.

For the purposes of calculating the fields generated by the sources in (4.2) and (4.3), the space–time of source points may be marked either with $(\boldsymbol{x},t)=(r,\unicode[STIX]{x1D711},z,t)$ or with the coordinates $(r,\hat{\unicode[STIX]{x1D711}},z,t)$ that naturally appear in the description of such rotating sources. Once $\hat{\unicode[STIX]{x1D711}}$ , with the range $(0,2\unicode[STIX]{x03C0})$ , is adopted as one of the coordinates, either $t$ or $\unicode[STIX]{x1D711}$ (which have unlimited ranges) could be used to track the time evolution of the rotating source point $(r,\hat{\unicode[STIX]{x1D711}},z)$ .

Changing the variables of integration in (4.4) from $(\boldsymbol{x},t)=(r,\unicode[STIX]{x1D711},z,t)$ to $(r,\hat{\unicode[STIX]{x1D711}},z,\unicode[STIX]{x1D711})$ and introducing the dimensionless coordinates $\hat{r}=r\unicode[STIX]{x1D714}/c$ and $\hat{z}=z\unicode[STIX]{x1D714}/c$ , we obtain

(4.5) $$\begin{eqnarray}\left[\begin{array}{@{}c@{}}\boldsymbol{E}\\ \boldsymbol{B}\end{array}\right]=\frac{1}{\unicode[STIX]{x1D714}}\mathop{\sum }_{k=1}^{\infty }\int _{{\mathcal{S}}}\hat{r}\,\text{d}\hat{r}\,\text{d}\hat{\unicode[STIX]{x1D711}}\,\text{d}\hat{z}\,\int _{\hat{\unicode[STIX]{x1D711}}+2(k-1)\unicode[STIX]{x03C0}}^{\hat{\unicode[STIX]{x1D711}}+2k\unicode[STIX]{x03C0}}\text{d}\unicode[STIX]{x1D711}\,\frac{\unicode[STIX]{x1D6FF}^{\prime }(g-\unicode[STIX]{x1D719})}{\hat{R}}\left[\begin{array}{@{}c@{}}\boldsymbol{j}-\unicode[STIX]{x1D70C}c\,\hat{\boldsymbol{n}}\\ \hat{\boldsymbol{n}}\times \boldsymbol{ j}\end{array}\right],\end{eqnarray}$$

where

(4.6) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{R}=[(\hat{z}-\hat{z}_{P})^{2}+{\hat{r}_{P}}^{2}+\hat{r}^{2}-2\hat{r}_{P}\hat{r}\cos (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P})]^{1/2}, & \displaystyle\end{eqnarray}$$

(4.7) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{\boldsymbol{n}}=\{[\hat{r}_{P}-\hat{r}\cos (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P})]\hat{\boldsymbol{e}}_{r_{P}}-\hat{r}\sin (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P})\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D711}_{P}}-(\hat{z}-\hat{z}_{P})\hat{\boldsymbol{e}}_{z_{P}}\}/\hat{R}, & \displaystyle\end{eqnarray}$$

the function $g(\hat{r},\unicode[STIX]{x1D711},\hat{z};\hat{r}_{P},\unicode[STIX]{x1D711}_{P},\hat{z}_{P})$ is defined by

(4.8) $$\begin{eqnarray}g\equiv \unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P}+\hat{R},\end{eqnarray}$$

the variable $\unicode[STIX]{x1D719}$ in the argument of the delta function stands for

(4.9a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D719}\equiv \hat{\unicode[STIX]{x1D711}}-\hat{\unicode[STIX]{x1D711}}_{P}\quad \text{with}\quad \hat{\unicode[STIX]{x1D711}}_{P}\equiv \unicode[STIX]{x1D711}_{P}-\unicode[STIX]{x1D714}t_{P},\end{eqnarray}$$

and $(\hat{\boldsymbol{e}}_{r_{P}},\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D711}_{P}},\hat{\boldsymbol{e}}_{z_{P}})$ are the cylindrical base vectors at the observation point $(r_{P},\unicode[STIX]{x1D711}_{P},z_{P})$ . In (4.5), the domain of integration over the $(\hat{r},\hat{\unicode[STIX]{x1D711}},\hat{z})$ space consists of the support ${\mathcal{S}}$ of the source density $j^{\unicode[STIX]{x1D707}}$ and the range of integration with respect to $\unicode[STIX]{x1D711}$ is given by the extended interval of azimuthal angle traversed by the source in the course of its rotations prior to the observation time $t_{P}$ .

I have expressed the range of $\unicode[STIX]{x1D711}$ integration as a sum of the intervals of length $2\unicode[STIX]{x03C0}$ that the element initially located at $\hat{\unicode[STIX]{x1D711}}$ traverses during each of its individual rotations: $k$ is a positive integer enumerating successive rotation periods (the first rotation period being designated by $k=1$ ) and the summation extends over the set of rotations executed by the source over its lifetime. Given $(\hat{r},\hat{\unicode[STIX]{x1D711}},\hat{z})$ and $(r_{P},\unicode[STIX]{x1D711}_{P},z_{P},t_{P})$ , there are a limited number of values of $k$ for which $g-\unicode[STIX]{x1D719}$ vanishes and so the integral in (4.5) is non-zero. In other words, the contribution received from the source point $(\hat{r},\hat{\unicode[STIX]{x1D711}},\hat{z})$ at the space–time observation point $(r_{P},\unicode[STIX]{x1D711}_{P},z_{P},t_{P})$ is made during a limited number of its (earlier) rotation periods (see appendix B).

4.3 The Green’s function for the problem and its loci of singularities

To put the current density $\boldsymbol{j}=j_{r}\hat{\boldsymbol{e}}_{r}+j_{\unicode[STIX]{x1D711}}\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D711}}+j_{z}\hat{\boldsymbol{e}}_{z}$ into a form suitable for performing the integration with respect to $\unicode[STIX]{x1D711}$ , we need to express the $\unicode[STIX]{x1D711}$ -dependent base vectors $(\hat{\boldsymbol{e}}_{r},\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D711}},\hat{\boldsymbol{e}}_{z})$ associated with the source point $(r,\unicode[STIX]{x1D711},z)$ in terms of the constant base vectors $(\hat{\boldsymbol{e}}_{r_{P}},\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D711}_{P}},\hat{\boldsymbol{e}}_{z_{P}})$ at the observation point $(r_{P},\unicode[STIX]{x1D711}_{P},z_{P})$ :

(4.10) $$\begin{eqnarray}\left[\begin{array}{@{}c@{}}\hat{\boldsymbol{e}}_{r}\\ \hat{\boldsymbol{e}}_{\unicode[STIX]{x1D711}}\\ \hat{\boldsymbol{e}}_{z}\end{array}\right]=\left[\begin{array}{@{}ccc@{}}\cos (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P}) & \sin (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P}) & 0\\ -\text{sin}(\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P}) & \cos (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P}) & 0\\ 0 & 0 & 1\end{array}\right]\left[\begin{array}{@{}c@{}}\hat{\boldsymbol{e}}_{r_{P}}\\ \hat{\boldsymbol{e}}_{\unicode[STIX]{x1D711}_{P}}\\ \hat{\boldsymbol{e}}_{z_{P}}\end{array}\right].\end{eqnarray}$$

Once the resulting expression,

(4.11) $$\begin{eqnarray}\boldsymbol{j}=[j_{r}\cos (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P})-j_{\unicode[STIX]{x1D711}}\sin (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P})]\hat{\boldsymbol{e}}_{r_{P}}+[j_{r}\sin (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P})+j_{\unicode[STIX]{x1D711}}\cos (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P})]\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D711}_{P}}+j_{z}\hat{\boldsymbol{e}}_{z_{P}},\end{eqnarray}$$

and the expression in (4.7) for $\hat{\boldsymbol{n}}$ are inserted in (4.5) and $\unicode[STIX]{x1D6FF}^{\prime }(g-\unicode[STIX]{x1D719})$ is written as $-\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}(g-\unicode[STIX]{x1D719})/\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D711}}$ (see (4.9)), we arrive at

(4.12) $$\begin{eqnarray}\left[\begin{array}{@{}c@{}}\boldsymbol{E}\\ \boldsymbol{B}\end{array}\right]=-\frac{1}{\unicode[STIX]{x1D714}}\mathop{\sum }_{n=1}^{2}\mathop{\sum }_{j=1}^{3}\int _{{\mathcal{S}}}\hat{r}\,\text{d}\hat{r}\,\text{d}\hat{\unicode[STIX]{x1D711}}\,\text{d}\hat{z}\,\frac{\unicode[STIX]{x2202}G_{nj}}{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D711}}}\left[\begin{array}{@{}c@{}}\boldsymbol{u}_{nj}\\ \boldsymbol{ v}_{nj}\end{array}\right],\end{eqnarray}$$

with

(4.13) $$\begin{eqnarray}\displaystyle & \displaystyle \left[\begin{array}{@{}c@{}}\boldsymbol{u}_{11}\\ \boldsymbol{ u}_{12}\\ \boldsymbol{ u}_{13}\end{array}\right]=\left[\begin{array}{@{}c@{}}j_{r}\hat{\boldsymbol{e}}_{r_{P}}+j_{\unicode[STIX]{x1D711}}\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D711}_{P}}\\ -j_{\unicode[STIX]{x1D711}}\hat{\boldsymbol{e}}_{r_{P}}+j_{r}\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D711}_{P}}\\ j_{z}\hat{\boldsymbol{e}}_{z_{P}}\end{array}\right], & \displaystyle\end{eqnarray}$$

(4.14) $$\begin{eqnarray}\displaystyle & \displaystyle \left[\begin{array}{@{}c@{}}\boldsymbol{u}_{21}\\ \boldsymbol{ u}_{22}\\ \boldsymbol{ u}_{23}\end{array}\right]=\unicode[STIX]{x1D70C}c\left[\begin{array}{@{}c@{}}\hat{r}\hat{\boldsymbol{e}}_{r_{P}}\\ \hat{r}\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D711}_{P}}\\ -\hat{r}_{P}\hat{\boldsymbol{e}}_{r_{P}}+(\hat{z}-\hat{z}_{P})\hat{\boldsymbol{e}}_{z_{P}}\end{array}\right], & \displaystyle\end{eqnarray}$$

(4.15) $$\begin{eqnarray}\displaystyle & \displaystyle \left[\begin{array}{@{}ccc@{}}\boldsymbol{v}_{11} & \boldsymbol{v}_{12} & \boldsymbol{v}_{13}\end{array}\right]=\left[\begin{array}{@{}ccc@{}}0 & 0 & 0\end{array}\right], & \displaystyle\end{eqnarray}$$

and

(4.16) $$\begin{eqnarray}\displaystyle & \left[\begin{array}{@{}c@{}}\boldsymbol{v}_{21}\\ \boldsymbol{ v}_{22}\\ \boldsymbol{ v}_{23}\end{array}\right]=\left[\begin{array}{@{}c@{}}-(\hat{z}-\hat{z}_{P})\boldsymbol{u}_{12}+\hat{r}j_{z}\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D711}_{P}}+\hat{r}_{P}j_{\unicode[STIX]{x1D711}}\hat{\boldsymbol{e}}_{z_{P}}\\ \hat{\boldsymbol{e}}_{z_{P}}\times \boldsymbol{v}_{21}+\hat{r}_{P}j_{r}\hat{\boldsymbol{e}}_{z_{P}}\\ -\hat{r}_{P}j_{z}\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D711}_{P}}-\hat{r}j_{\unicode[STIX]{x1D711}}\hat{\boldsymbol{e}}_{z_{P}}\end{array}\right], & \displaystyle\end{eqnarray}$$

in which

(4.17) $$\begin{eqnarray}\left[\begin{array}{@{}c@{}}G_{n1}\\ G_{n2}\\ G_{n3}\end{array}\right]=\mathop{\sum }_{k=1}^{\infty }\int _{\hat{\unicode[STIX]{x1D711}}+2(k-1)\unicode[STIX]{x03C0}}^{\hat{\unicode[STIX]{x1D711}}+2k\unicode[STIX]{x03C0}}\text{d}\unicode[STIX]{x1D711}\,\frac{\unicode[STIX]{x1D6FF}(g-\unicode[STIX]{x1D719})}{\hat{R}^{n}}\left[\begin{array}{@{}c@{}}\cos (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P})\\ \sin (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P})\\ 1\end{array}\right]\end{eqnarray}$$

denotes the outcome of the remaining integration with respect to $\unicode[STIX]{x1D711}$ . Note that the dependence on $\hat{\unicode[STIX]{x1D711}}$ of the limits of integration in (4.17) does not contribute toward the values of the derivatives of $G_{nj}$ with respect to $\hat{\unicode[STIX]{x1D711}}$ (see (B 7)).

The function $G_{nj}(\hat{r},\hat{\unicode[STIX]{x1D711}},\hat{z};\hat{r}_{P},\hat{\unicode[STIX]{x1D711}}_{P},\hat{z}_{P})$ here acts as the Green’s function for the present problem. It describes the Liénard–Wiechert field that arises from an individual volume element of the rotating distribution pattern of the source. If we specialize the current distribution to a rotating point charge $q$ , i.e. let $j_{r}=j_{z}=0$ and $j_{\unicode[STIX]{x1D711}}=r_{s}\unicode[STIX]{x1D714}q\unicode[STIX]{x1D6FF}(r-r_{s})\unicode[STIX]{x1D6FF}(\hat{\unicode[STIX]{x1D711}})\unicode[STIX]{x1D6FF}(z)$ with a constant $r_{s}$ , then (4.12) at an observation point in the far zone would describe the familiar field of synchrotron radiation when $r_{s}<c/\unicode[STIX]{x1D714}$ and a synergic field combining attributes of both synchrotron and Čerenkov emissions when $r_{s}>c/\unicode[STIX]{x1D714}$ (see, e.g. Ardavan et al. Reference Ardavan, Ardavan and Singleton2004c ).

Depending on the value of

(4.18) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}=({\hat{r}_{P}}^{2}-1)(\hat{r}^{2}-1)-(\hat{z}-\hat{z}_{P})^{2}\end{eqnarray}$$

for a given source point $(r,\hat{\unicode[STIX]{x1D711}},z)$ with $r\unicode[STIX]{x1D714}>c$ , the $\unicode[STIX]{x1D711}$ -dependence of the function $g$ that appears in the definition of the Green’s function $G_{nj}$ in (4.17) has one of the generic forms shown in figure 4. As can be seen from the curve labelled $\unicode[STIX]{x1D6E5}>0$ in this figure, there are values,

(4.19) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{\pm }=\unicode[STIX]{x1D711}_{P}+2k\unicode[STIX]{x03C0}-\arccos \left(\frac{1\mp \unicode[STIX]{x1D6E5}^{1/2}}{\hat{r}\hat{r}_{P}}\right),\end{eqnarray}$$

of the retarded position of the source point at which

(4.20) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}g}{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}}=1+\frac{\hat{r}\hat{r}_{P}\sin (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P})}{\hat{R}}\end{eqnarray}$$

vanishes and so $G_{nj}$ diverges. These turning points of $g$ occur at source points for which $\unicode[STIX]{x2202}(R|_{\unicode[STIX]{x1D711}=\hat{\unicode[STIX]{x1D711}}+\unicode[STIX]{x1D714}t})/\unicode[STIX]{x2202}t=-c$ , i.e. the source points that approach the observer, along the radiation direction $\hat{\boldsymbol{n}}$ , with the speed of light at the retarded time. The inflection point of $g$ (see the curve labelled $\unicode[STIX]{x1D6E5}=0$ in figure 4), at which

(4.21) $$\begin{eqnarray}\left.\frac{\unicode[STIX]{x2202}^{2}g}{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}^{2}}\right|_{\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}_{\pm }}=\mp \frac{\unicode[STIX]{x1D6E5}^{1/2}}{\hat{R}_{\pm }}\end{eqnarray}$$

in addition vanishes, occurs at source points that approach the observer not only with the wave speed but also with zero acceleration at the retarded time, i.e. for which both $\unicode[STIX]{x2202}(R|_{\unicode[STIX]{x1D711}=\hat{\unicode[STIX]{x1D711}}+\unicode[STIX]{x1D714}t})/\unicode[STIX]{x2202}t=-c$ and $\unicode[STIX]{x2202}^{2}(R|_{\unicode[STIX]{x1D711}=\hat{\unicode[STIX]{x1D711}}+\unicode[STIX]{x1D714}t})/\unicode[STIX]{x2202}t^{2}=0$ at the time when $g|_{\unicode[STIX]{x1D711}=\hat{\unicode[STIX]{x1D711}}+\unicode[STIX]{x1D714}t}=\unicode[STIX]{x1D719}$ and $\unicode[STIX]{x2202}g/\unicode[STIX]{x2202}\unicode[STIX]{x1D711}=\unicode[STIX]{x2202}^{2}g/\unicode[STIX]{x2202}\unicode[STIX]{x1D711}^{2}=0$ . In (4.21),

(4.22) $$\begin{eqnarray}\hat{R}_{\pm }=[(\hat{z}-\hat{z}_{P})^{2}+\hat{r}^{2}+{\hat{r}_{P}}^{2}-2(1\mp \unicode[STIX]{x1D6E5}^{1/2})]^{1/2}\end{eqnarray}$$

is the value of $\hat{R}$ at the extrema $\unicode[STIX]{x1D711}_{\pm }$ of $g$ .

The envelope of the wave fronts emanating from a given rotating source element $(\hat{r},\hat{\unicode[STIX]{x1D711}},\hat{z})$ , on which $\unicode[STIX]{x2202}g/\unicode[STIX]{x2202}\unicode[STIX]{x1D711}$ vanishes, consists of the rigidly rotating two-sheeted surface $\hat{\unicode[STIX]{x1D711}}-\hat{\unicode[STIX]{x1D711}}_{P}=g(\unicode[STIX]{x1D711}_{\pm })$ in the space $(\hat{r}_{P},\hat{\unicode[STIX]{x1D711}}_{P},\hat{z}_{P})$ of observation points. This surface, which is shown in figures 5 and 6, is described by

(4.23) $$\begin{eqnarray}\unicode[STIX]{x1D719}_{\pm }\equiv \hat{\unicode[STIX]{x1D711}}_{\pm }-\hat{\unicode[STIX]{x1D711}}_{P}=\unicode[STIX]{x1D711}_{\pm }-\unicode[STIX]{x1D711}_{P}+\hat{R}_{\pm }\end{eqnarray}$$

(see (4.8), (4.9), (4.19) and (4.22)). The two sheets of this surface tangentially meet along a cusp on which $\unicode[STIX]{x2202}^{2}g/\unicode[STIX]{x2202}\unicode[STIX]{x1D711}^{2}$ as well as $\unicode[STIX]{x2202}g/\unicode[STIX]{x2202}\unicode[STIX]{x1D711}$ vanishes (see figures 6 and 7). Three distinct wave fronts, emitted at three differing values of the retarded time, pass through any given observation point inside the envelope. At an observation point located on the envelope or its cusp, respectively two or all three of these waves coalesce and interfere constructively (see figure 4).

4.4 Bifurcation surface of an observation point

Reciprocally, the locus in the space of source points $(\hat{r},\hat{\unicode[STIX]{x1D711}},\hat{z})$ on which $\unicode[STIX]{x2202}g/\unicode[STIX]{x2202}\unicode[STIX]{x1D711}$ vanishes is a two-sheeted cusped surface issuing from the fixed observation point $P$ (see figure 8). I refer to this locus, which is described by (4.23) for fixed values of $(\hat{r}_{P},\hat{\unicode[STIX]{x1D711}}_{P},\hat{z}_{P})$ rather than fixed values of $(\hat{r},\hat{\unicode[STIX]{x1D711}},\hat{z})$ , as the bifurcation surface of the observation point $P$ . The two sheets $\unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{+}$ and $\unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{-}$ of this surface, respectively referred to as the regular and singular sheets, meet along the following cusp:

(4.24) $$\begin{eqnarray}\displaystyle C:\left\{\begin{array}{@{}l@{}}\hat{r}=\hat{r}_{C}(\hat{z})=[1+(\hat{z}-\hat{z}_{P})^{2}/({\hat{r}_{P}}^{2}-1)]^{1/2},\quad \\ \unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}_{C}(\hat{z})=\unicode[STIX]{x1D711}_{P}+2k\unicode[STIX]{x03C0}-\arccos [1/(\hat{r}\hat{r}_{P})],\quad \end{array}\right. & & \displaystyle\end{eqnarray}$$

where $k$ is the same integer as that appearing in (4.5). I refer to both $C$ and its projection onto the $(r,z)$ plane as the cusp locus of the bifurcation surface; whether it is $C$ itself or its projection that is referred to will be clear from the context.

The source points inside the bifurcation surface, close to its cusp, make their contributions toward the observed value of the field at three distinct retarded positions in their trajectory (where a horizontal line $g=\unicode[STIX]{x1D719}$ in figure 4 intersects the curve $\unicode[STIX]{x1D6E5}>0$ between its extrema), while those outside the bifurcation surface make their contributions at a single retarded position (where the curve $\unicode[STIX]{x1D6E5}<0$ is intersected by $g=\unicode[STIX]{x1D719}$ in figure 4). For the source points on the bifurcation surface (i.e. those for which $g=\unicode[STIX]{x1D719}_{\pm }$ in figure 4), two of the contributing retarded positions coalesce at the extrema of the curve $\unicode[STIX]{x1D6E5}>0$ in figure 4 giving rise to a divergent value of the Green’s function at $P$ . For the source points located on the cusp locus $C$ of the bifurcation surface (i.e. those for which $\unicode[STIX]{x1D6E5}=0$ in figure 4), all three of the contributing retarded positions coalesce at the inflection point of the curve $\unicode[STIX]{x1D6E5}=0$ in figure 4 giving rise to a higher-order singularity in $G_{nj}$ . In the following section, I use the time-domain version (Burridge Reference Burridge1995) of the method of Chester et al. (Reference Chester, Friedman and Ursell1957) to derive a uniform asymptotic approximation to the value of $G_{nj}$ for the source points close to the cusp $C$ of the bifurcation surface.

4.5 A uniform asymptotic approximation to the value of the Green’s function near the cusp locus of the bifurcation surface

As long as the observation point does not coincide with the source point, the function $g(\unicode[STIX]{x1D711})$ is analytic and the following transformation of the integration variable in (4.17) from $\unicode[STIX]{x1D711}$ to $\unicode[STIX]{x1D708}$ is permissible

(4.25) $$\begin{eqnarray}g(\unicode[STIX]{x1D711})={\textstyle \frac{1}{3}}\unicode[STIX]{x1D708}^{3}-{c_{1}}^{2}\unicode[STIX]{x1D708}+c_{2},\end{eqnarray}$$

in which

(4.26a,b ) $$\begin{eqnarray}c_{1}=[{\textstyle \frac{3}{4}}(\unicode[STIX]{x1D719}_{+}-\unicode[STIX]{x1D719}_{-})]^{1/3}\quad \text{and}\quad c_{2}={\textstyle \frac{1}{2}}(\unicode[STIX]{x1D719}_{+}+\unicode[STIX]{x1D719}_{-}),\end{eqnarray}$$

are chosen such that the values of the two functions on opposite sides of (4.25) coincide at their extrema when $\unicode[STIX]{x1D6E5}$ is positive. Thus an alternative exact expression for $G_{nj}$ is

(4.27) $$\begin{eqnarray}G_{nj}=\mathop{\sum }_{k=1}^{\infty }{\mathcal{H}}\int _{-\infty }^{\infty }\text{d}\unicode[STIX]{x1D708}\,f_{nj}\unicode[STIX]{x1D6FF}\left(\frac{1}{3}\unicode[STIX]{x1D708}^{3}-{c_{1}}^{2}\unicode[STIX]{x1D708}+c_{2}-\unicode[STIX]{x1D719}\right),\end{eqnarray}$$

where

(4.28) $$\begin{eqnarray}\left[\begin{array}{@{}c@{}}f_{n1}\\ f_{n2}\\ f_{n3}\end{array}\right]=\frac{1}{\hat{R}^{n}}\frac{\text{d}\unicode[STIX]{x1D711}}{\text{d}\unicode[STIX]{x1D708}}\left[\begin{array}{@{}c@{}}\cos (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P})\\ \sin (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P})\\ 1\end{array}\right],\end{eqnarray}$$

and the step function ${\mathcal{H}}$ is non-zero only if the argument of the delta function in (4.17) vanishes within the original domain of integration $\hat{\unicode[STIX]{x1D711}}+2(k-1)\unicode[STIX]{x03C0}\leqslant \unicode[STIX]{x1D711}\leqslant \hat{\unicode[STIX]{x1D711}}+2k\unicode[STIX]{x03C0}$ , i.e. if $g|_{\unicode[STIX]{x1D711}=\hat{\unicode[STIX]{x1D711}}+2k\unicode[STIX]{x03C0}}-\unicode[STIX]{x1D719}\geqslant 0$ but $g|_{\unicode[STIX]{x1D711}=\hat{\unicode[STIX]{x1D711}}+2(k-1)\unicode[STIX]{x03C0}}-\unicode[STIX]{x1D719}\leqslant 0$ . Inserting $\unicode[STIX]{x1D711}=\hat{\unicode[STIX]{x1D711}}+2k\unicode[STIX]{x03C0}$ and $\unicode[STIX]{x1D711}=\hat{\unicode[STIX]{x1D711}}+2(k-1)\unicode[STIX]{x03C0}$ in the definition of $g$ in (4.8) and simplifying the resulting expressions by means of (4.9), we can write this step function as

(4.29) $$\begin{eqnarray}{\mathcal{H}}=\text{H}[\hat{R}|_{\unicode[STIX]{x1D711}=\hat{\unicode[STIX]{x1D711}}}-\unicode[STIX]{x1D714}t_{P}+2k\unicode[STIX]{x03C0}]-\text{H}[\hat{R}|_{\unicode[STIX]{x1D711}=\hat{\unicode[STIX]{x1D711}}}-\unicode[STIX]{x1D714}t_{P}+2(k-1)\unicode[STIX]{x03C0}],\end{eqnarray}$$

in which $\text{H}(x)$ denotes the Heaviside step function.

In cases where the distance $\hat{R}_{P}=(\hat{r}_{P}^{2}+\hat{z}_{P}^{2})^{1/2}$ of the observation point from the origin of the reference frame is much larger than the coordinates $\hat{r}$ and $\hat{z}$ of the source point, equation (4.29) reduces to

(4.30) $$\begin{eqnarray}{\mathcal{H}}_{\infty }=\text{H}[\hat{R}_{P}-\unicode[STIX]{x1D714}t_{P}+2k\unicode[STIX]{x03C0}]-\text{H}[\hat{R}_{P}-\unicode[STIX]{x1D714}t_{P}+2(k-1)\unicode[STIX]{x03C0}],\qquad \hat{R}_{P}\gg 1.\end{eqnarray}$$

The step function ${\mathcal{H}}$ picks out the particular rotation cycle (or cycles) during which the signal that reaches the observation point $(\hat{r}_{P},\unicode[STIX]{x1D711}_{P},\hat{z}_{P})$ at the observation time $t_{P}$ is emitted by the source point $(\hat{r},\hat{\unicode[STIX]{x1D711}},\hat{z})$ .

Note that $c_{1}(\hat{r},\hat{z};\hat{r}_{P},\hat{z}_{P})$ in the expression for $G_{nj}$ vanishes on the cusp locus of the bifurcation surface where $\unicode[STIX]{x1D6E5}$ equals zero and $\unicode[STIX]{x1D719}_{-}=\unicode[STIX]{x1D719}_{+}$ . The leading term in the asymptotic expansion of the integral in (4.27) in the vicinity of the cusp locus of the bifurcation surface, i.e. for small $c_{1}$ , can be found by replacing $f_{j}$ by $p_{j}+q_{j}\unicode[STIX]{x1D708}$ ,

(4.31) $$\begin{eqnarray}G_{nj}\simeq \mathop{\sum }_{k=1}^{\infty }{\mathcal{H}}\int _{-\infty }^{\infty }\text{d}\unicode[STIX]{x1D708}\,(p_{nj}+q_{nj}\unicode[STIX]{x1D708})\unicode[STIX]{x1D6FF}\left(\frac{1}{3}\unicode[STIX]{x1D708}^{3}-{c_{1}}^{2}\unicode[STIX]{x1D708}+c_{2}-\unicode[STIX]{x1D719}\right),\quad c_{1}\ll 1,\end{eqnarray}$$

where

(4.32) $$\begin{eqnarray}p_{nj}={\textstyle \frac{1}{2}}(f_{nj}|_{\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}_{-}}+f_{nj}|_{\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}_{+}}),\end{eqnarray}$$

(4.33) $$\begin{eqnarray}q_{nj}={\textstyle \frac{1}{2}}{c_{1}}^{-1}(f_{nj}|_{\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}_{-}}-f_{nj}|_{\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}_{+}})\end{eqnarray}$$

(see Chester et al. (Reference Chester, Friedman and Ursell1957) and note that $\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}_{-}$ maps onto $\unicode[STIX]{x1D708}=c_{1}$ and $\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}_{+}$ onto $\unicode[STIX]{x1D708}=-c_{1}$ ). To evaluate the integral in (4.31) we need to know the roots of the cubic function that appears in the argument of the Dirac $\unicode[STIX]{x1D6FF}$ function in this expression. Depending on whether the source point is located inside or outside the bifurcation surface, the roots of

(4.34) $$\begin{eqnarray}{\textstyle \frac{1}{3}}\unicode[STIX]{x1D708}^{3}-{c_{1}}^{2}\unicode[STIX]{x1D708}+c_{2}-\unicode[STIX]{x1D719}=0\end{eqnarray}$$

for $\unicode[STIX]{x1D6E5}>0$ are given, respectively, by

(4.35) $$\begin{eqnarray}\unicode[STIX]{x1D708}=\unicode[STIX]{x1D708}_{\ell }=2c_{1}\cos \left({\textstyle \frac{2}{3}}\ell \unicode[STIX]{x03C0}+{\textstyle \frac{1}{3}}\arccos \unicode[STIX]{x1D712}\right),\quad |\unicode[STIX]{x1D712}|<1,\end{eqnarray}$$

with $\ell =0,1$ and $2$ , or by

(4.36) $$\begin{eqnarray}\unicode[STIX]{x1D708}=\unicode[STIX]{x1D708}_{\text{out}}=2c_{1}\text{sgn}(\unicode[STIX]{x1D712})\cosh \left({\textstyle \frac{1}{3}}\text{arccosh}|\unicode[STIX]{x1D712}|\right),\quad |\unicode[STIX]{x1D712}|>1,\end{eqnarray}$$

where

(4.37) $$\begin{eqnarray}\unicode[STIX]{x1D712}=\frac{3(\unicode[STIX]{x1D719}-c_{2})}{2{c_{1}}^{3}}.\end{eqnarray}$$

Note that $\unicode[STIX]{x1D712}$ equals $1$ on the sheet $\unicode[STIX]{x1D719}_{+}$ of the bifurcation surface and $-1$ on the sheet  $\unicode[STIX]{x1D719}_{-}$ .

The integral multiplying $p_{j}$ in (4.31) therefore has the following value when the source point lies inside the bifurcation surface ( $|\unicode[STIX]{x1D712}|<1$ ) and (4.34) has the three roots given in (4.35),

(4.38) $$\begin{eqnarray}\displaystyle & & \displaystyle \int _{-\infty }^{\infty }\text{d}\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FF}\left(\frac{1}{3}\unicode[STIX]{x1D708}^{3}-{c_{1}}^{2}\unicode[STIX]{x1D708}+c_{2}-\unicode[STIX]{x1D719}\right)=\mathop{\sum }_{\ell =0}^{2}|\unicode[STIX]{x1D708}_{\ell }^{2}-c_{1}^{2}|^{-1}\nonumber\\ \displaystyle & & \displaystyle \qquad =\mathop{\sum }_{\ell =0}^{2}c_{1}^{-2}\left|4\cos ^{2}\left(\frac{2}{3}\ell \unicode[STIX]{x03C0}+\frac{1}{3}\arccos \unicode[STIX]{x1D712}\right)-1\right|^{-1},\quad |\unicode[STIX]{x1D712}|<1.\end{eqnarray}$$

Using the trigonometric identity $4\cos ^{2}\unicode[STIX]{x1D6FC}-1=\sin 3\unicode[STIX]{x1D6FC}/\sin \unicode[STIX]{x1D6FC}$ , we can write this as

(4.39) $$\begin{eqnarray}\displaystyle & & \displaystyle \int _{-\infty }^{\infty }\text{d}\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FF}\left(\frac{1}{3}\unicode[STIX]{x1D708}^{3}-{c_{1}}^{2}\unicode[STIX]{x1D708}+c_{2}-\unicode[STIX]{x1D719}\right)=c_{1}^{-2}(1-\unicode[STIX]{x1D712}^{2})^{-1/2}\mathop{\sum }_{\ell =0}^{2}\left|\sin \left(\frac{2}{3}\ell \unicode[STIX]{x03C0}+\frac{1}{3}\arccos \unicode[STIX]{x1D712}\right)\right|\nonumber\\ \displaystyle & & \displaystyle \qquad =2c_{1}^{-2}(1-\unicode[STIX]{x1D712}^{2})^{-1/2}\cos \left(\frac{1}{3}\arcsin \unicode[STIX]{x1D712}\right),\quad |\unicode[STIX]{x1D712}|<1,\end{eqnarray}$$

in which I have evaluated the sum by adding the sine functions two at a time. When the source point lies outside the bifurcation surface ( $|\unicode[STIX]{x1D712}|>1$ ), the above integral receives a contribution only from the single value of $\unicode[STIX]{x1D708}$ given in (4.36) and we obtain

(4.40) $$\begin{eqnarray}\int _{-\infty }^{\infty }\text{d}\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FF}\left(\frac{1}{3}\unicode[STIX]{x1D708}^{3}-{c_{1}}^{2}\unicode[STIX]{x1D708}+c_{2}-\unicode[STIX]{x1D719}\right)=c_{1}^{-2}(\unicode[STIX]{x1D712}^{2}-1)^{-1/2}\sinh \left(\frac{1}{3}\text{arccosh}|\unicode[STIX]{x1D712}|\right),\quad |\unicode[STIX]{x1D712}|>1,\end{eqnarray}$$

where this time I have used the identity $4\cosh ^{2}\unicode[STIX]{x1D6FC}-1=\sinh (3\unicode[STIX]{x1D6FC})/\sinh \unicode[STIX]{x1D6FC}$ .

The second part of the integral in (4.31) can be evaluated in exactly the same way. It has the value

(4.41) $$\begin{eqnarray}\int _{-\infty }^{\infty }\text{d}\unicode[STIX]{x1D708}\,\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FF}\left(\frac{1}{3}\unicode[STIX]{x1D708}^{3}-{c_{1}}^{2}\unicode[STIX]{x1D708}+c_{2}-\unicode[STIX]{x1D719}\right)=-2c_{1}^{-1}(1-\unicode[STIX]{x1D712}^{2})^{-1/2}\sin \left(\frac{2}{3}\arcsin \unicode[STIX]{x1D712}\right),\quad |\unicode[STIX]{x1D712}|<1,\end{eqnarray}$$

when the source point lies inside the bifurcation surface ( $|\unicode[STIX]{x1D712}|<1$ ) and the value

(4.42) $$\begin{eqnarray}\displaystyle & & \displaystyle \int _{-\infty }^{\infty }\text{d}\unicode[STIX]{x1D708}\,\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FF}\left(\frac{1}{3}\unicode[STIX]{x1D708}^{3}-{c_{1}}^{2}\unicode[STIX]{x1D708}+c_{2}-\unicode[STIX]{x1D719}\right)\nonumber\\ \displaystyle & & \displaystyle \qquad =c_{1}^{-1}(\unicode[STIX]{x1D712}^{2}-1)^{-1/2}\text{sgn}(\unicode[STIX]{x1D712})\sinh \left(\frac{2}{3}\text{arccosh}|\unicode[STIX]{x1D712}|\right),\quad |\unicode[STIX]{x1D712}|>1,\end{eqnarray}$$

when the source point lies outside the bifurcation surface ( $|\unicode[STIX]{x1D712}|>1$ ). Inserting (4.39)–(4.42) in (4.31), we obtain

(4.43) $$\begin{eqnarray}G_{nj}^{\text{in}}\simeq \mathop{\sum }_{k=1}^{\infty }2{\mathcal{H}}c_{1}^{-2}(1-\unicode[STIX]{x1D712}^{2})^{-1/2}\!\left[p_{nj}\cos \left(\frac{1}{3}\arcsin \unicode[STIX]{x1D712}\!\right)-c_{1}q_{nj}\sin \left(\frac{2}{3}\arcsin \unicode[STIX]{x1D712}\!\right)\!\right]\!,\quad |\unicode[STIX]{x1D712}|<1,\end{eqnarray}$$

and

(4.44) $$\begin{eqnarray}\displaystyle G_{nj}^{\text{out}} & \simeq & \displaystyle \mathop{\sum }_{k=1}^{\infty }{\mathcal{H}}c_{1}^{-2}(\unicode[STIX]{x1D712}^{2}-1)^{-1/2}\left[p_{nj}\sinh \left(\frac{1}{3}\text{arccosh}|\unicode[STIX]{x1D712}|\right)\right.\nonumber\\ \displaystyle & & \displaystyle \left.+\,c_{1}q_{nj}\text{sgn}(\unicode[STIX]{x1D712})\sinh \left(\frac{2}{3}\text{arccosh}|\unicode[STIX]{x1D712}|\right)\right],\quad |\unicode[STIX]{x1D712}|>1,\end{eqnarray}$$

where $G_{nj}^{\text{in}}$ and $G_{nj}^{\text{out}}$ denote the values of $G_{nj}$ over $\unicode[STIX]{x1D6E5}\geqslant 0$ inside and outside the bifurcation surface, respectively.

For the source points in $\unicode[STIX]{x1D6E5}<0$ , the functions $\unicode[STIX]{x1D719}_{-}$ and $\unicode[STIX]{x1D719}_{+}$ are complex conjugate of one another so that the coefficient $c_{2}$ is still real but $c_{1}$ is pure imaginary: the relevant cube root of $\unicode[STIX]{x1D719}_{+}-\unicode[STIX]{x1D719}_{-}$ in the expression for $c_{1}$ is in this case given by $-\text{i}|\unicode[STIX]{x1D719}_{+}-\unicode[STIX]{x1D719}_{-}|^{1/3}$ , which casts (4.26a ) into the form $c_{1}=-\text{i}[3/4|\unicode[STIX]{x1D719}_{+}-\unicode[STIX]{x1D719}_{-}|]^{1/3}$ . Since neither $g(\unicode[STIX]{x1D711})$ nor the cubic expression to which $g$ is transformed have any extrema in this case, there is only one real solution to (4.34) when $\unicode[STIX]{x1D6E5}<0$ and $c_{1}$ is pure imaginary. This solution, which can be found by writing the coefficient ${c_{1}}^{2}$ of $\unicode[STIX]{x1D708}$ (in which $c_{1}$ is pure imaginary) as $-|c_{1}|^{2}$ prior to solving the cubic, is given by

(4.45) $$\begin{eqnarray}\unicode[STIX]{x1D708}=2c_{1}\sinh \left({\textstyle \frac{1}{3}}\text{arcsinh}\unicode[STIX]{x1D712}^{\prime }\right),\end{eqnarray}$$

with

(4.46) $$\begin{eqnarray}\unicode[STIX]{x1D712}^{\prime }=\frac{3(\unicode[STIX]{x1D719}-c_{2})}{2|c_{1}|^{3}}.\end{eqnarray}$$

Following the same procedure as that employed in deriving (4.40) and (4.42), we obtain

(4.47) $$\begin{eqnarray}\displaystyle G_{nj}^{\text{sub}} & \simeq & \displaystyle \mathop{\sum }_{k=1}^{\infty }{\mathcal{H}}c_{1}^{-2}({\unicode[STIX]{x1D712}^{\prime }}^{2}+1)^{-1/2}\left[p_{nj}\cosh \left(\frac{1}{3}\text{arcsinh}\unicode[STIX]{x1D712}^{\prime }\right)\right.\nonumber\\ \displaystyle & & \displaystyle \left.+\,|c_{1}|q_{nj}\sinh \left(\frac{2}{3}\text{arcsinh}\unicode[STIX]{x1D712}^{\prime }\right)\right],\quad |\unicode[STIX]{x1D712}^{\prime }|>1,\end{eqnarray}$$

for the value $G_{nj}^{\text{sub}}$ of the Green’s function in $\unicode[STIX]{x1D6E5}<0$ where the source points approach the observer with subluminal speeds.

To complete the derivation of $G_{nj}$ , we need to evaluate the coefficients $p_{j}$ and $q_{j}$ which are defined by (4.32), (4.33) and (4.28). The indeterminate quantities $\text{d}\unicode[STIX]{x1D711}/\text{d}\unicode[STIX]{x1D708}|_{\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}_{\pm }}$ that appear in these definitions have to be found by repeated differentiation of (4.25) with respect to $\unicode[STIX]{x1D708}$ , and the evaluation of the resulting relations

(4.48) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}g}{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}}\frac{\text{d}\unicode[STIX]{x1D711}}{\text{d}\unicode[STIX]{x1D708}}=\unicode[STIX]{x1D708}^{2}-{c_{1}}^{2},\end{eqnarray}$$

and

(4.49) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}^{2}g}{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}^{2}}\left(\frac{\text{d}\unicode[STIX]{x1D711}}{\text{d}\unicode[STIX]{x1D708}}\right)^{2}+\frac{\unicode[STIX]{x2202}g}{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}}\frac{\text{d}^{2}\unicode[STIX]{x1D711}}{\text{d}\unicode[STIX]{x1D708}^{2}}=2\unicode[STIX]{x1D708},\end{eqnarray}$$

at $\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}_{\pm }$ . This procedure, which amounts to applying the l’Hôpital rule, yields

(4.50) $$\begin{eqnarray}\left.\frac{\text{d}\unicode[STIX]{x1D711}}{\text{d}\unicode[STIX]{x1D708}}\right|_{\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}_{\pm }}=\left(\frac{2c_{1}\hat{R}_{\pm }}{\unicode[STIX]{x1D6E5}^{1/2}}\right)^{1/2}.\end{eqnarray}$$

Equation (4.50) together with (4.28), (4.19) and (4.22), now yield the following values of

(4.51) $$\begin{eqnarray}\left[\begin{array}{@{}c@{}}f_{n1}\\ f_{n2}\\ f_{n3}\end{array}\right]_{\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}_{\pm }}=\frac{1}{\hat{r}\hat{r}_{P}\hat{R}_{\pm }^{n-1/2}}\left(\frac{2c_{1}}{\unicode[STIX]{x1D6E5}^{1/2}}\right)^{1/2}\left[\begin{array}{@{}c@{}}1\mp \unicode[STIX]{x1D6E5}^{1/2}\\ -\hat{R}_{\pm }\\ \hat{r}\hat{r}_{P}\end{array}\right],\end{eqnarray}$$

and hence the following values of

(4.52) $$\begin{eqnarray}\left[\begin{array}{@{}c@{}}p_{n1}\\ p_{n2}\\ p_{n3}\end{array}\right]=\frac{1}{\hat{r}\hat{r}_{P}}\left(\frac{c_{1}}{2\unicode[STIX]{x1D6E5}^{1/2}}\right)^{1/2}\left[\begin{array}{@{}c@{}}\hat{R}_{+}^{-n+1/2}+\hat{R}_{-}^{-n+1/2}+\unicode[STIX]{x1D6E5}^{1/2}(\hat{R}_{-}^{-n+1/2}-\hat{R}_{+}^{-n+1/2})\\ -(\hat{R}_{-}^{-n+3/2}+\hat{R}_{+}^{-n+3/2})\\ \hat{r}\hat{r}_{P}(\hat{R}_{-}^{-n+1/2}+\hat{R}_{+}^{-n+1/2})\end{array}\right],\end{eqnarray}$$

and

(4.53) $$\begin{eqnarray}\left[\begin{array}{@{}c@{}}q_{n1}\\ q_{n2}\\ q_{n3}\end{array}\right]=\frac{1}{\hat{r}\hat{r}_{P}(2c_{1}\unicode[STIX]{x1D6E5}^{1/2})^{1/2}}\left[\begin{array}{@{}c@{}}\hat{R}_{-}^{-n+1/2}-\hat{R}_{+}^{-n+1/2}+\unicode[STIX]{x1D6E5}^{1/2}(\hat{R}_{-}^{-n+1/2}+\hat{R}_{+}^{-n+1/2})\\ \hat{R}_{+}^{-n+3/2}-\hat{R}_{-}^{-n+3/2}\\ \hat{r}\hat{r}_{P}(\hat{R}_{-}^{-n+1/2}-\hat{R}_{+}^{-n+1/2})\end{array}\right]\end{eqnarray}$$

(see (4.32) and (4.33)). For $\unicode[STIX]{x1D6E5}<0$ , the functions $c_{1}$ and $\unicode[STIX]{x1D6E5}^{1/2}$ in (4.52) and (4.53) are respectively given by $-\text{i}|\unicode[STIX]{x1D719}_{+}-\unicode[STIX]{x1D719}_{-}|^{1/3}$ and $-\text{i}|\unicode[STIX]{x1D6E5}|^{1/2}$ , so that $p_{j}$ and $q_{j}$ are real also in this case: $\hat{R}_{-}$ is the complex conjugate of $\hat{R}_{+}$ (see (4.22)).

The two-dimensional loci $\unicode[STIX]{x1D712}=\pm 1$ across which the resulting expression

(4.54) $$\begin{eqnarray}G_{nj}=\left\{\begin{array}{@{}ll@{}}G_{nj}^{\text{in}}\quad & \unicode[STIX]{x1D6E5}>0,|\unicode[STIX]{x1D712}|<1\\ G_{nj}^{\text{out}}\quad & \unicode[STIX]{x1D6E5}\geqslant 0,|\unicode[STIX]{x1D712}|\geqslant 1\\ G_{nj}^{\text{sub}}\quad & \unicode[STIX]{x1D6E5}<0,|\unicode[STIX]{x1D712}^{\prime }|>1\end{array}\right.\end{eqnarray}$$

for the Green’s function changes form correspond to the two sheets $\unicode[STIX]{x1D719}_{\pm }$ of the bifurcation surface, respectively. As a source point $(r,\hat{\unicode[STIX]{x1D711}},z)$ in the vicinity of the cusp $C$ approaches the bifurcation surface from inside, i.e. as $\unicode[STIX]{x1D712}\rightarrow 1-$ or $\unicode[STIX]{x1D712}\rightarrow -1+$ , $G_{nj}^{\text{in}}$ diverges. However, as a source point approaches either one of the sheets of the bifurcation surface from outside, the numerator and the denominator in (4.44) vanish simultaneously and $G_{nj}^{\text{out}}$ tends to a finite limit,

(4.55) $$\begin{eqnarray}G_{nj}^{\text{out}}|_{\unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{\pm }}=G_{nj}^{\text{out}}|_{\unicode[STIX]{x1D712}=\pm 1}={\textstyle \frac{1}{3}}c_{1}^{-2}\left(p_{nj}\pm 2c_{1}q_{nj}\right).\end{eqnarray}$$

Note that $c_{1}$ , and hence $p_{nj}$ and $q_{nj}$ , are independent of $k$ (see (4.19), (4.23) and (4.26)). The only $k$ -dependent functions appearing in the expressions for $G_{nj}^{\text{out}}|_{\unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{\pm }}$ are the step functions ${\mathcal{H}}|_{\unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{\pm }}$ which can be summed over $k$ to obtain unity (see (4.29)). Thus the Green’s function $G_{nj}$ is singular only on the inner side of the bifurcation surface (see figures 9 and 10).

4.6 Hadamard’s finite part of the divergent integral representing the field

It follows from (4.43) and (4.54) that the factor $\unicode[STIX]{x2202}G_{nj}/\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D711}}$ in the integrand of the integral (4.12) diverges as $(1-\unicode[STIX]{x1D712}^{2})^{-3/2}$ and so has a non-integrable singularity on the bifurcation surface where $\unicode[STIX]{x1D712}^{2}$ equals 1. This singularity has arisen because we differentiated the retarded potential (3.7) under the integral sign when calculating the field. Had we evaluated the integral in (3.7) prior to differentiating it we would have found a singularity-free expression. Interchanging the orders of integration and differentiation is mathematically permissible when the integrand is discontinuous only if one treats the resulting integral as a generalized function and so one handles any non-integrable singularities that consequently arise by means of Hadamard’s regularization technique (see Ardavan Reference Ardavan1999; Hadamard Reference Hadamard2003; Hoskins Reference Hoskins2009, and the illustrative example in appendix A).

Hadamard’s procedure consists of performing an integration by parts and discarding the divergent (integrated) term in the resulting expression. The remaining finite part is the value that Hadamard’s regularization assigns to the integral; in the present case, it is the value we would have obtained if we had first evaluated the finite integral representing the retarded potential and had differentiated the result $A^{\unicode[STIX]{x1D707}}(\boldsymbol{x}_{P},t_{P})$ of that evaluation subsequently. (The more direct approach, in which the potential is first evaluated and then differentiated, cannot of course be carried out for any realistic source distribution analytically.)

The $\hat{\unicode[STIX]{x1D711}}$ coordinates $\hat{\unicode[STIX]{x1D711}}_{\pm }$ of the two sheets of the bifurcation surface depend on the observation time $t_{P}$ (see (4.23) and (4.9)), so that these two sheets move across the $\hat{\unicode[STIX]{x1D711}}$ extent of the source distribution as $t_{P}$ elapses. If the position of the observation point is such that the cusp locus of the bifurcation surface intersects the source distribution, the two sheets of this surface (which tangentially meet at the cusp) will divide the volume of the source into a part that lies inside and a part that lies outside the bifurcation surface. The Lagrangian coordinates $\hat{\unicode[STIX]{x1D711}}$ designating the initial azimuthal positions of the constituent volume elements of a source that fully occupies an annular region range over the interval $0\leqslant \hat{\unicode[STIX]{x1D711}}<2\unicode[STIX]{x03C0}$ . The $(\hat{r},\hat{z})$ coordinates of these source elements either fall in $\unicode[STIX]{x1D6E5}\geqslant 0$ or in $\unicode[STIX]{x1D6E5}<0$ . The elements in $\unicode[STIX]{x1D6E5}\geqslant 0$ are always divided into two sets: a set inside the bifurcation surface for which $\hat{\unicode[STIX]{x1D711}}_{-}\leqslant \hat{\unicode[STIX]{x1D711}}\leqslant \hat{\unicode[STIX]{x1D711}}_{+}$ and so the Green’s function $G_{nj}$ has the form $G_{nj}^{\text{in}}$ and a set outside for which $\hat{\unicode[STIX]{x1D711}}$ lies either in $(0,\hat{\unicode[STIX]{x1D711}}_{-})$ or in $(\hat{\unicode[STIX]{x1D711}}_{+},2\unicode[STIX]{x03C0})$ and so $G_{nj}$ has the form $G_{nj}^{\text{out}}$ (see (4.54)). On the other hand, if the position of the observation point is such that $\unicode[STIX]{x1D6E5}<0$ for all values of $(\hat{r},\hat{z})$ in ${\mathcal{S}}^{\prime }$ (see (2.7)), then the source lies entirely outside the bifurcation surface and $G_{nj}$ has the form $G_{nj}^{\text{sub}}$ . Note that, for certain space–time coordinates of the observation point $P$ , the values of $\hat{\unicode[STIX]{x1D711}}_{-}$ and $\hat{\unicode[STIX]{x1D711}}_{+}$ that lie in the interval $(0,2\unicode[STIX]{x03C0})$ could correspond to different rotation periods, i.e. to different values of $k$ (see (4.19), (4.22) and (4.23)). To simplify the notation, here I adopt an observation time $t_{P}$ at which the values of $\hat{\unicode[STIX]{x1D711}}_{-}$ and $\hat{\unicode[STIX]{x1D711}}_{+}$ that lie in the interval $(0,2\unicode[STIX]{x03C0})$ correspond to the same rotation period $k$ .

Breaking up the volume of integration in the expression for one of the radiation fields, e.g. $\boldsymbol{E}$ , into the domains of validity of $G_{nj}^{\text{in}}$ , $G_{nj}^{\text{out}}$ and $G_{nj}^{\text{sub}}$ , we can therefore write the $\hat{\unicode[STIX]{x1D711}}$ -integral over $\boldsymbol{u}_{nj}$ in (4.12) as

(4.56) $$\begin{eqnarray}\displaystyle \boldsymbol{I}_{\hat{\unicode[STIX]{x1D711}}} & \equiv & \displaystyle \int _{0}^{2\unicode[STIX]{x03C0}}\text{d}\hat{\unicode[STIX]{x1D711}}\,\boldsymbol{u}_{nj}\frac{\unicode[STIX]{x2202}G_{nj}}{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D711}}}\nonumber\\ \displaystyle & = & \displaystyle \text{H}(\unicode[STIX]{x1D6E5})\left[\left(\int _{0}^{\hat{\unicode[STIX]{x1D711}}_{-}}+\int _{\hat{\unicode[STIX]{x1D711}}_{+}}^{2\unicode[STIX]{x03C0}}\right)\text{d}\hat{\unicode[STIX]{x1D711}}\,\boldsymbol{u}_{nj}\frac{\unicode[STIX]{x2202}G_{nj}^{\text{out}}}{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D711}}}+\int _{\hat{\unicode[STIX]{x1D711}}_{-}}^{\hat{\unicode[STIX]{x1D711}}_{+}}\text{d}\hat{\unicode[STIX]{x1D711}}\,\boldsymbol{u}_{nj}\frac{\unicode[STIX]{x2202}G_{nj}^{\text{in}}}{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D711}}}\right]\nonumber\\ \displaystyle & & \displaystyle +\text{H}(-\unicode[STIX]{x1D6E5})\int _{0}^{2\unicode[STIX]{x03C0}}\text{d}\hat{\unicode[STIX]{x1D711}}\,\boldsymbol{u}_{nj}\frac{\unicode[STIX]{x2202}G_{nj}^{\text{sub}}}{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D711}}}.\end{eqnarray}$$

If we now integrate every term of the above expression by parts, recall that $\hat{\unicode[STIX]{x1D711}}=0$ labels the same source point as does $\hat{\unicode[STIX]{x1D711}}=2\unicode[STIX]{x03C0}$ , and use the fact that the exact version of $G_{nj}$ given in (4.17) is periodic in $\hat{\unicode[STIX]{x1D711}}$ as well as in $\unicode[STIX]{x1D711}$ (with the same period $2\unicode[STIX]{x03C0}$ ), we arrive at

(4.57) $$\begin{eqnarray}\displaystyle \boldsymbol{I}_{\hat{\unicode[STIX]{x1D711}}} & = & \displaystyle \text{H}(\unicode[STIX]{x1D6E5})\left\{\left[\boldsymbol{u}_{nj}\left(G_{nj}^{\text{in}}-G_{nj}^{\text{out}}\right)\right]_{\hat{\unicode[STIX]{x1D711}}=\hat{\unicode[STIX]{x1D711}}_{-}}^{\hat{\unicode[STIX]{x1D711}}=\hat{\unicode[STIX]{x1D711}}_{+}}-\left(\int _{0}^{\hat{\unicode[STIX]{x1D711}}_{-}}+\int _{\hat{\unicode[STIX]{x1D711}}_{+}}^{2\unicode[STIX]{x03C0}}\right)\text{d}\hat{\unicode[STIX]{x1D711}}\,\frac{\unicode[STIX]{x2202}\boldsymbol{u}_{nj}}{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D711}}}G_{nj}^{\text{out}}\right.\nonumber\\ \displaystyle & & \displaystyle \left.\vphantom{\left[\left(G_{nj}^{\text{in}}\right)\right]_{\hat{\unicode[STIX]{x1D711}}=\hat{\unicode[STIX]{x1D711}}_{-}}^{\hat{\unicode[STIX]{x1D711}}=\hat{\unicode[STIX]{x1D711}}_{+}}}-\,\int _{\hat{\unicode[STIX]{x1D711}}_{-}}^{\hat{\unicode[STIX]{x1D711}}_{+}}\text{d}\hat{\unicode[STIX]{x1D711}}\,\frac{\unicode[STIX]{x2202}\boldsymbol{u}_{nj}}{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D711}}}G_{nj}^{\text{in}}\right\}-\text{H}(-\unicode[STIX]{x1D6E5})\int _{0}^{2\unicode[STIX]{x03C0}}\text{d}\hat{\unicode[STIX]{x1D711}}\,\frac{\unicode[STIX]{x2202}\boldsymbol{u}_{nj}}{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D711}}}G_{nj}^{\text{sub}},\end{eqnarray}$$

an expression that reduces to

(4.58) $$\begin{eqnarray}\displaystyle \boldsymbol{I}_{\hat{\unicode[STIX]{x1D711}}}=\text{H}(\unicode[STIX]{x1D6E5})[\boldsymbol{u}_{nj}(G_{nj}^{\text{in}}-G_{nj}^{\text{out}})]_{\hat{\unicode[STIX]{x1D711}}=\hat{\unicode[STIX]{x1D711}}_{-}}^{\hat{\unicode[STIX]{x1D711}}=\hat{\unicode[STIX]{x1D711}}_{+}}-\int _{0}^{2\unicode[STIX]{x03C0}}\text{d}\hat{\unicode[STIX]{x1D711}}\,\frac{\unicode[STIX]{x2202}\boldsymbol{u}_{nj}}{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D711}}}G_{nj}, & & \displaystyle\end{eqnarray}$$

once the integrals over $G_{nj}^{\text{in}}$ , $G_{nj}^{\text{out}}$ and $G_{nj}^{\text{sub}}$ are combined in the light of (4.54).

We have seen in the last paragraph of § 4.5 that the value of $G_{nj}^{\text{in}}$ at $\hat{\unicode[STIX]{x1D711}}=\hat{\unicode[STIX]{x1D711}}_{\pm }$ diverges (figures 9 and 10). The Hadamard finite part of $\boldsymbol{I}_{\hat{\unicode[STIX]{x1D711}}}$ is therefore given by the right-hand side of (4.58) without the divergent terms involving $G_{nj}^{\text{in}}|_{\hat{\unicode[STIX]{x1D711}}=\hat{\unicode[STIX]{x1D711}}_{-}}$ and $G_{nj}^{\text{in}}|_{\hat{\unicode[STIX]{x1D711}}=\hat{\unicode[STIX]{x1D711}}_{+}}$ ,

(4.59) $$\begin{eqnarray}\text{Fp}\{\boldsymbol{I}_{\hat{\unicode[STIX]{x1D711}}}\}=-\text{H}(\unicode[STIX]{x1D6E5})\boldsymbol{u}_{j}G_{nj}^{\text{out}}|_{\hat{\unicode[STIX]{x1D711}}=\hat{\unicode[STIX]{x1D711}}_{-}}^{\hat{\unicode[STIX]{x1D711}}=\hat{\unicode[STIX]{x1D711}}_{+}}-\int _{0}^{2\unicode[STIX]{x03C0}}\text{d}\hat{\unicode[STIX]{x1D711}}\,\frac{\unicode[STIX]{x2202}\boldsymbol{u}_{nj}}{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D711}}}G_{nj},\end{eqnarray}$$

where $\text{Fp}\{\boldsymbol{I}_{\hat{\unicode[STIX]{x1D711}}}\}$ denotes the Hadamard finite part of the divergent integral $\boldsymbol{I}_{\hat{\unicode[STIX]{x1D711}}}$ (see Hadamard Reference Hadamard2003; Hoskins Reference Hoskins2009). This procedure applies also to the expression for the radiation field $\boldsymbol{B}$ in (4.12) except that $\boldsymbol{u}_{nj}$ in (4.56)–(4.59) is everywhere replaced by  $\boldsymbol{v}_{nj}$ .

Once the integrals with respect to $\hat{\unicode[STIX]{x1D711}}$ in (4.12) are equated to the expression on the right-hand side of (4.59) and its counterpart for $\boldsymbol{B}$ , we find that

(4.60) $$\begin{eqnarray}\left[\begin{array}{@{}c@{}}\boldsymbol{E}\\ \boldsymbol{ B}\end{array}\right]=\left[\begin{array}{@{}c@{}}\boldsymbol{E}^{\text{v}}\\ \boldsymbol{ B}^{\text{v}}\end{array}\right]+\left[\begin{array}{@{}c@{}}\boldsymbol{E}_{+}^{\text{b}}\\ \boldsymbol{ B}_{+}^{\text{b}}\end{array}\right]-\left[\begin{array}{@{}c@{}}\boldsymbol{E}_{-}^{\text{b}}\\ \boldsymbol{ B}_{-}^{\text{b}}\end{array}\right]\end{eqnarray}$$

with

(4.61) $$\begin{eqnarray}\left[\begin{array}{@{}c@{}}\boldsymbol{E}^{\text{v}}\\ \boldsymbol{B}^{\text{v}}\end{array}\right]=\frac{1}{\unicode[STIX]{x1D714}}\mathop{\sum }_{n=1}^{2}\mathop{\sum }_{j=1}^{3}\int _{{\mathcal{S}}}\hat{r}\,\text{d}\hat{r}\,\text{d}\hat{\unicode[STIX]{x1D711}}\,\text{d}\hat{z}\,G_{nj}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D711}}}\left[\begin{array}{@{}c@{}}\boldsymbol{u}_{nj}\\ \boldsymbol{ v}_{nj}\end{array}\right],\end{eqnarray}$$

and

(4.62) $$\begin{eqnarray}\left[\begin{array}{@{}c@{}}\boldsymbol{E}_{\pm }^{\text{b}}\\ \boldsymbol{B}_{\pm }^{\text{b}}\end{array}\right]=\frac{1}{\unicode[STIX]{x1D714}}\mathop{\sum }_{n=1}^{2}\mathop{\sum }_{j=1}^{3}\int _{{\mathcal{S}}^{\prime }}\hat{r}\,\text{d}\hat{r}\,\text{d}\hat{z}\,\text{H}(\unicode[STIX]{x1D6E5})\,G_{nj}^{\text{out}}\left.\left[\begin{array}{@{}c@{}}\boldsymbol{u}_{nj}\\ \boldsymbol{ v}_{nj}\end{array}\right]\right|_{\hat{\unicode[STIX]{x1D711}}=\hat{\unicode[STIX]{x1D711}}_{\pm }},\end{eqnarray}$$

where ${\mathcal{S}}^{\prime }$ is the projection of the support ${\mathcal{S}}$ of the source distribution onto the $(r,z)$ plane (see (2.7)). The term $[\begin{array}{@{}cc@{}}\boldsymbol{E}^{\text{v}} & \boldsymbol{B}^{\text{v}}\end{array}]$ constitutes the contribution from the entire volume of the source while the terms $[\begin{array}{@{}cc@{}}\boldsymbol{E}_{\pm }^{\text{b}} & \boldsymbol{B}_{\pm }^{\text{b}}\end{array}]$ denote the contributions from the discontinuities of the Green’s function on the two sheets $\unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{\pm }$ of the bifurcation surface, respectively. We will see that the terms $[\begin{array}{@{}cc@{}}\boldsymbol{E}_{\pm }^{\text{b}} & \boldsymbol{B}_{\pm }^{\text{b}}\end{array}]$ describe unconventional radiation fields with characteristics that turn out to differ from any previously known radiation fields.

5.1 Cusp locus $C$ and its dual role in the spaces of source points and observation points

The boundary terms $[\begin{array}{@{}cc@{}}\boldsymbol{E}_{\pm }^{\text{b}} & \boldsymbol{B}_{\pm }^{\text{b}}\end{array}]$ receive contributions only from those source elements whose $(\hat{r},\hat{z})$ coordinates fall within the region $\unicode[STIX]{x1D6E5}\geqslant 0$ shown in figure 11, i.e. for which $\hat{r}\geqslant \hat{r}_{C}(\hat{z})$ (see (4.24)). In other words, $[\begin{array}{@{}cc@{}}\boldsymbol{E}_{\pm }^{\text{b}} & \boldsymbol{B}_{\pm }^{\text{b}}\end{array}]$ are non-zero either when the projection of the cusp locus of the bifurcation surface $C$ onto the $(\hat{r},\hat{z})$ plane intersects the domain ${\mathcal{S}}^{\prime }$ described by (2.7), in which case $\hat{r}_{L}\leqslant \hat{r}_{C}\leqslant \hat{r}_{U}$ for $-\hat{z}_{0}\leqslant \hat{z}\leqslant \hat{z}_{0}$ as shown in figure 11, or when $\hat{r}_{C}\leqslant \hat{r}_{L}$ for these values of $\hat{z}$ and the entire radial extent of the source lies inside the bifurcation surface. The intersection of ${\mathcal{S}}^{\prime }$ and $\unicode[STIX]{x1D6E5}\geqslant 0$ which constitutes the domain of integration in (5.8) thus changes as the location of the cusp locus $C$ (which depends on the position of the observation point) changes.

The parametric equation $\hat{r}=\hat{r}(\hat{z})$ , $\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}(\hat{z})$ , of the cusp locus of the bifurcation surface associated with a given observation point $(\hat{r}_{P},\hat{\unicode[STIX]{x1D711}}_{P},\hat{z}_{P})$ at the observation time $t_{P}$ was derived in (4.24). If we rewrite the two members of (4.24) in terms of the dimensionless polar coordinates $\hat{R}_{P}=(\hat{r}_{P}^{2}+\hat{z}_{P}^{2})^{1/2}$ , $\unicode[STIX]{x1D703}_{P}=\arccos (\hat{z}_{P}/\hat{R}_{P})$ , of the observation point $P$ and solve them for $\unicode[STIX]{x1D703}_{P}$ and $\unicode[STIX]{x1D711}_{P}$ as functions of $(\hat{r},\unicode[STIX]{x1D711},\hat{z})$ and $\hat{R}_{P}$ , we obtain

(5.9) $$\begin{eqnarray}\displaystyle & C:\left\{\begin{array}{@{}l@{}}\unicode[STIX]{x1D703}_{P}=\unicode[STIX]{x1D703}_{P}^{\text{c}}(\hat{r},\hat{z})\equiv \arccos \left\{{\displaystyle \frac{1}{\hat{R}_{P}\hat{r}}}\left[{\displaystyle \frac{\hat{z}}{\hat{r}}}\pm \left(\hat{r}^{2}-1\right)^{1/2}\left({\hat{R}_{P}}^{2}-1-{\displaystyle \frac{\hat{z}^{2}}{\hat{r}^{2}}}\right)^{1/2}\right]\right\},\quad \\ \unicode[STIX]{x1D711}_{P}=\unicode[STIX]{x1D711}_{P}^{\text{c}}(\hat{r},\unicode[STIX]{x1D711},\hat{z})\equiv \unicode[STIX]{x1D711}-2k\unicode[STIX]{x03C0}+\arccos \left({\displaystyle \frac{1}{\hat{r}\hat{R}_{P}\sin \unicode[STIX]{x1D703}_{P}}}\right),\quad \end{array}\right. & \displaystyle\end{eqnarray}$$

where the $\pm$ correspond to the two halves of the cusp curve below and above the plane $\hat{z}=\hat{z}_{P}$ , respectively, and $k$ is the positive integer enumerating successive rotation periods (see (4.5) and (4.19)).

The angle between the asymptotes to the hyperbola representing the projection of the cusp locus $C$ onto the $(\hat{r},\hat{z})$ -plane, as well as the radial coordinate of the point of intersection of this hyperbola with the plane $\hat{z}=0$ , depend on the coordinate $\unicode[STIX]{x1D703}_{P}$ of the observation point (see figure 11). As the observation point $P$ at a given distance $\hat{R}_{P}$ moves from the upper half of the rotation axis ( $\unicode[STIX]{x1D703}_{P}=0$ ) towards the plane of rotation ( $\unicode[STIX]{x1D703}_{P}=\unicode[STIX]{x03C0}/2$ ), the point of intersection of the cusp locus $C$ with the plane $\hat{z}=0$ gradually shifts across this plane from a large value ( $\lim _{\hat{R}_{P}\rightarrow \infty }\hat{r}_{C}=\csc \unicode[STIX]{x1D703}_{P}$ ) of the radial coordinate $\hat{r}$ towards the upper boundary $\hat{r}_{U}$ of the source distribution, across the radial extent of the source distribution towards its lower boundary $\hat{r}_{L}$ and eventually towards the light cylinder $\hat{r}=1$ . The cusp $C$ will thus lie to the right of the source distribution shown in figure 11 when $0<\unicode[STIX]{x1D703}_{P}\leqslant \unicode[STIX]{x1D703}_{L}^{\text{c}}$ , where

(5.10) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{L}^{\text{c}}=\unicode[STIX]{x1D703}_{P}^{\text{c}}|_{\hat{r}=\hat{r}_{U},\hat{z}=\hat{z}_{0}},\end{eqnarray}$$

intersects this source distribution while $\unicode[STIX]{x1D703}_{L}^{\text{c}}\leqslant \unicode[STIX]{x1D703}_{P}\leqslant \unicode[STIX]{x1D703}_{U}^{\text{c}}$ , where

(5.11) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{U}^{\text{c}}=\unicode[STIX]{x1D703}_{P}^{\text{c}}|_{\hat{r}=\hat{r}_{L},\hat{z}=-\hat{z}_{0}},\end{eqnarray}$$

and lies in $1<\hat{r}<\hat{r}_{L}$ (to the left of this source distribution) when $\unicode[STIX]{x1D703}_{U}^{\text{c}}\leqslant \unicode[STIX]{x1D703}_{P}\leqslant \unicode[STIX]{x03C0}/2$ . (Recall that $\hat{r}_{L}$ and $\hat{r}_{U}$ designate the inner and outer radial boundaries of the source distribution (see figure 11).)

Thus the cusp locus $C$ would intersect the support (2.7) of the source distribution only if the observation point lies within one of the following conical shells,

(5.12) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D703}_{L}^{\text{c}}\leqslant \unicode[STIX]{x1D703}_{P}\leqslant \unicode[STIX]{x1D703}_{U}^{\text{c}},\\ \displaystyle \unicode[STIX]{x03C0}-\unicode[STIX]{x1D703}_{U}^{\text{c}}\leqslant \unicode[STIX]{x1D703}_{P}\leqslant \unicode[STIX]{x03C0}-\unicode[STIX]{x1D703}_{L}^{\text{c}}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

The first member of (5.9), on which the definitions in (5.10) and (5.11) are based, reduces to

(5.13) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{P}^{\text{c}}=\arcsin \left(\frac{1}{\hat{r}}\right)-\left(\frac{\hat{z}}{\hat{r}}\right){\hat{R}_{P}}^{-1}\pm \frac{1}{2}\left(\hat{r}^{2}-1\right)^{1/2}\hat{R}_{P}^{-2}+\cdots\end{eqnarray}$$

in the far zone where $\hat{R}_{P}\gg 1$ . The leading term in this expansion (in powers of $\hat{R}_{P}^{-1}$ ) together with (5.10) and (5.11) shows that, in the limit $\hat{R}_{P}\rightarrow \infty$ , the angles $\unicode[STIX]{x1D703}_{L}^{\text{c}}$ and $\unicode[STIX]{x1D703}_{U}^{\text{c}}$ reduce to $\arcsin (1/\hat{r}_{U})$ and $\arcsin (1/\hat{r}_{L})$ , respectively. We shall see below that the radiation field $[\begin{array}{@{}cc@{}}\boldsymbol{E} & \boldsymbol{B}\end{array}]$ has radically differing characteristics in each of the three disjoint regions of space separated by these two cones (see figure 12).

6.1 Evaluation of $[\boldsymbol{E}^{\text{v}}\quad \boldsymbol{B}^{\text{v}}]$ at observation points for which $\unicode[STIX]{x1D703}_{L}^{\text{c}}\leqslant \unicode[STIX]{x1D703}_{P}\leqslant \unicode[STIX]{x03C0}-\unicode[STIX]{x1D703}_{L}^{\text{c}}$

When the polar coordinate $\unicode[STIX]{x1D703}_{P}$ of the observation point lies in $\unicode[STIX]{x1D703}_{L}^{\text{c}}\leqslant \unicode[STIX]{x1D703}_{P}\leqslant \unicode[STIX]{x03C0}-\unicode[STIX]{x1D703}_{L}^{\text{c}}$ , there are volume elements within the source distribution ${\mathcal{S}}$ that approach $P$ along the radiation direction with a speed exceeding $c$ at the retarded time. For such source elements $\unicode[STIX]{x1D6E5}$ is positive. Depending on whether the $\hat{\unicode[STIX]{x1D711}}$ coordinates of these elements lie inside or outside the bifurcation surface associated with the observation point $P$ , the Green’s function $G_{nj}$ that appears in (5.7) has either the value $G_{nj}^{\text{in}}$ or the value $G_{nj}^{\text{out}}$ (see (4.54)). There are also source elements lying in $\unicode[STIX]{x1D6E5}<0$ for which $G_{nj}$ has the value $G_{nj}^{\text{sub}}$ (see figure 11). The expression in (5.7) can therefore be written as

(6.1) $$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}\boldsymbol{E}^{\text{v}}\\ \boldsymbol{ B}^{\text{v}}\end{array}\right] & = & \displaystyle m^{2}\mathop{\sum }_{n=1}^{2}\mathop{\sum }_{j=1}^{3}\int _{{\mathcal{S}}^{\prime }}\hat{r}\,\text{d}\hat{r}\,\text{d}\hat{z}\,\left[\begin{array}{@{}c@{}}\tilde{\boldsymbol{u}}_{nj}\\ \tilde{\boldsymbol{v}}_{nj}\end{array}\right]\left\{\text{H}(\unicode[STIX]{x1D6E5})\left[\int _{\hat{\unicode[STIX]{x1D711}}_{-}}^{\hat{\unicode[STIX]{x1D711}}_{+}}\text{d}\hat{\unicode[STIX]{x1D711}}\exp (-\text{i}m\hat{\unicode[STIX]{x1D711}})G_{nj}^{\text{in}}\right.\right.\nonumber\\ \displaystyle & & \displaystyle \left.\left.+\left(\int _{0}^{\hat{\unicode[STIX]{x1D711}}_{-}}+\int _{\hat{\unicode[STIX]{x1D711}}_{+}}^{2\unicode[STIX]{x03C0}}\right)\,\text{d}\hat{\unicode[STIX]{x1D711}}\exp (-\text{i}m\hat{\unicode[STIX]{x1D711}})G_{nj}^{\text{out}}\right]+\text{H}(-\unicode[STIX]{x1D6E5})\int _{0}^{2\unicode[STIX]{x03C0}}\text{d}\hat{\unicode[STIX]{x1D711}}\exp (-\text{i}m\hat{\unicode[STIX]{x1D711}})G_{nj}^{\text{sub}}\right\},\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

in which $G_{nj}^{\text{in}}$ , $G_{nj}^{\text{out}}$ and $G_{nj}^{\text{sub}}$ are given by (4.43), (4.44) and (4.47).

If we change the variable of integration in the integral over $\hat{\unicode[STIX]{x1D711}}_{-}\leqslant \hat{\unicode[STIX]{x1D711}}\leqslant \hat{\unicode[STIX]{x1D711}}_{+}$ in (6.1) from $\hat{\unicode[STIX]{x1D711}}$ to

(6.2) $$\begin{eqnarray}\unicode[STIX]{x1D713}=-2c_{1}\sin \left({\textstyle \frac{1}{3}}\arcsin \unicode[STIX]{x1D712}\right),\end{eqnarray}$$

in which $\unicode[STIX]{x1D712}$ depends on $\hat{\unicode[STIX]{x1D711}}$ as in (4.37), and note that the inversion of (6.2) yields

(6.3) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D711}}={\textstyle \frac{1}{3}}\unicode[STIX]{x1D713}^{3}-c_{1}^{2}\unicode[STIX]{x1D713}+c_{2}+\hat{\unicode[STIX]{x1D711}}_{P},\end{eqnarray}$$

we obtain

(6.4) $$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\hat{\unicode[STIX]{x1D711}}_{-}}^{\hat{\unicode[STIX]{x1D711}}_{+}}\text{d}\hat{\unicode[STIX]{x1D711}}\exp (-\text{i}m\hat{\unicode[STIX]{x1D711}})G_{nj}^{\text{in}}\nonumber\\ \displaystyle & & \displaystyle \qquad =2\mathop{\sum }_{k=1}^{\infty }\int _{-c_{1}}^{c_{1}}\text{d}\unicode[STIX]{x1D713}\,{\mathcal{H}}(p_{nj}+q_{nj}\unicode[STIX]{x1D713})\exp \left[-\text{i}m\left(\frac{1}{3}\unicode[STIX]{x1D713}^{3}-c_{1}^{2}\unicode[STIX]{x1D713}+\hat{\unicode[STIX]{x1D711}}_{P}+c_{2}\right)\right]\end{eqnarray}$$

(see (4.43)). The variables of integration in the remaining two integrals inside the square bracket in (6.1) can be similarly transformed from $\hat{\unicode[STIX]{x1D711}}$ to

(6.5) $$\begin{eqnarray}\unicode[STIX]{x1D6F9}=2c_{1}\text{sgn}(\unicode[STIX]{x1D712})\cosh \left({\textstyle \frac{1}{3}}\text{arccosh}|\unicode[STIX]{x1D712}|\right).\end{eqnarray}$$

This transformation and its inverse

(6.6) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D711}}=\text{sgn}(\unicode[STIX]{x1D712})\left({\textstyle \frac{1}{3}}\unicode[STIX]{x1D6F9}^{3}-c_{1}^{2}\unicode[STIX]{x1D6F9}\right)+c_{2}+\hat{\unicode[STIX]{x1D711}}_{P},\end{eqnarray}$$

then result in

(6.7) $$\begin{eqnarray}\displaystyle \left(\int _{0}^{\hat{\unicode[STIX]{x1D711}}_{-}}+\int _{\hat{\unicode[STIX]{x1D711}}_{+}}^{2\unicode[STIX]{x03C0}}\right)\text{d}\hat{\unicode[STIX]{x1D711}}G_{nj}^{\text{out}} & = & \displaystyle \mathop{\sum }_{k=1}^{\infty }\left(\int _{\unicode[STIX]{x1D6F9}_{L}}^{-2c_{1}}+\int _{2c_{1}}^{\unicode[STIX]{x1D6F9}_{U}}\right)\text{d}\unicode[STIX]{x1D6F9}{\mathcal{H}}(p_{nj}+q_{nj}\unicode[STIX]{x1D6F9})\nonumber\\ \displaystyle & & \displaystyle \times \,\exp \left[-\text{i}m\left(\frac{1}{3}\unicode[STIX]{x1D6F9}^{3}-c_{1}^{2}\unicode[STIX]{x1D6F9}+c_{2}+\hat{\unicode[STIX]{x1D711}}_{P}\right)\right],\end{eqnarray}$$

where

(6.8) $$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{L}=-2c_{1}\cosh \left[\frac{1}{3}\text{arccosh}\left(\frac{3}{2}\frac{\hat{\unicode[STIX]{x1D711}}_{P}+c_{2}}{c_{1}^{3}}\right)\right]\end{eqnarray}$$

and

(6.9) $$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{U}=2c_{1}\cosh \left[\frac{1}{3}\text{arccosh}\left(\frac{3}{2}\frac{2\unicode[STIX]{x03C0}-\hat{\unicode[STIX]{x1D711}}_{P}-c_{2}}{c_{1}^{3}}\right)\right]\end{eqnarray}$$

are the values of $\unicode[STIX]{x1D6F9}$ corresponding to $\hat{\unicode[STIX]{x1D711}}=0$ and $\hat{\unicode[STIX]{x1D711}}=2\unicode[STIX]{x03C0}$ , respectively (see (4.37) and (4.44)). For any given observation point with the space–time coordinates ( $\hat{R}_{P}$ , $\unicode[STIX]{x1D703}_{L}^{\text{c}}\leqslant \unicode[STIX]{x1D703}_{P}\leqslant \unicode[STIX]{x03C0}-\unicode[STIX]{x1D703}_{L}^{\text{c}}$ , $\unicode[STIX]{x1D711}_{P}$ , $t_{P}$ ) the value of $k$ (in $c_{2}$ ) that is selected by the step function ${\mathcal{H}}$ (or its far-field version ${\mathcal{H}}_{\infty }$ ) will automatically render the arguments of the $\text{arccosh}$ functions in (6.8) and (6.9) positive and yield a positive $\unicode[STIX]{x1D6F9}_{U}$ and a negative $\unicode[STIX]{x1D6F9}_{L}$ .

Rather than evaluating the remaining integral in (6.1) by substituting the expression for $G_{nj}^{\text{sub}}$ in its integrand, here I replace $G_{nj}$ in (5.7) by its original representation (4.17) to write this integral as

(6.10) $$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{j=1}^{3}\int _{{\mathcal{S}}^{\prime }}\hat{r}\,\text{d}\hat{r}\,\text{d}\hat{z}\text{H}(-\unicode[STIX]{x1D6E5})\left[\begin{array}{@{}c@{}}\tilde{\boldsymbol{u}}_{nj}\\ \tilde{\boldsymbol{v}}_{nj}\end{array}\right]\int _{0}^{2\unicode[STIX]{x03C0}}\text{d}\hat{\unicode[STIX]{x1D711}}\exp (-\text{i}m\hat{\unicode[STIX]{x1D711}})G_{nj}^{\text{sub}}=\mathop{\sum }_{k=1}^{\infty }\int _{{\mathcal{S}}^{\prime }}\hat{r}\,\text{d}\hat{r}\,\text{d}\hat{z}\,\text{H}(-\unicode[STIX]{x1D6E5})\nonumber\\ \displaystyle & & \displaystyle \qquad \quad \times \int _{0}^{2\unicode[STIX]{x03C0}}\text{d}\hat{\unicode[STIX]{x1D711}}\,\exp (-\text{i}m\hat{\unicode[STIX]{x1D711}})\int _{\hat{\unicode[STIX]{x1D711}}+2(k-1)\unicode[STIX]{x03C0}}^{\hat{\unicode[STIX]{x1D711}}+2k\unicode[STIX]{x03C0}}\text{d}\unicode[STIX]{x1D711}\frac{\unicode[STIX]{x1D6FF}(g-\unicode[STIX]{x1D719})}{\hat{R}^{n}}\nonumber\\ \displaystyle & & \displaystyle \qquad \quad \times \left\{\cos (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P})\left[\begin{array}{@{}c@{}}\tilde{\boldsymbol{u}}_{n1}\\ \tilde{\boldsymbol{v}}_{n1}\end{array}\right]+\sin (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P})\left[\begin{array}{@{}c@{}}\tilde{\boldsymbol{u}}_{n2}\\ \tilde{\boldsymbol{v}}_{n2}\end{array}\right]+\left[\begin{array}{@{}c@{}}\tilde{\boldsymbol{u}}_{n3}\\ \tilde{\boldsymbol{v}}_{n3}\end{array}\right]\right\}.\end{eqnarray}$$

Given that $g$ is a monotonic function of $\unicode[STIX]{x1D711}$ in $\unicode[STIX]{x1D6E5}<0$ and that the integrand in (6.10) is periodic in $\unicode[STIX]{x1D711}$ with the period $2\unicode[STIX]{x03C0}$ , it makes no difference which period, i.e. which value of $k$ , makes the contribution received at the observation time. We can therefore replace the range of the $\unicode[STIX]{x1D711}$ -integral by $(0,2\unicode[STIX]{x03C0})$ (omitting the summation over $k$ ) and perform the trivial integration with respect to $\hat{\unicode[STIX]{x1D711}}$ to obtain

(6.11) $$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{j=1}^{3}\int _{{\mathcal{S}}^{\prime }}\hat{r}\,\text{d}\hat{r}\,\text{d}\hat{z}\,\text{H}(-\unicode[STIX]{x1D6E5})\left[\begin{array}{@{}c@{}}\tilde{\boldsymbol{u}}_{nj}\\ \tilde{\boldsymbol{v}}_{nj}\end{array}\right]\int _{0}^{2\unicode[STIX]{x03C0}}\text{d}\hat{\unicode[STIX]{x1D711}}\exp (-\text{i}m\hat{\unicode[STIX]{x1D711}})G_{nj}^{\text{sub}}=\exp (-\text{i}m\hat{\unicode[STIX]{x1D711}}_{P})\int _{{\mathcal{S}}^{\prime }}\hat{r}\,\text{d}\hat{r}\,\text{d}\hat{z}\,\text{H}(-\unicode[STIX]{x1D6E5})\nonumber\\ \displaystyle & & \displaystyle \qquad \times \int _{0}^{2\unicode[STIX]{x03C0}}\text{d}\unicode[STIX]{x1D711}\,\frac{\exp (-\text{i}mg)}{\hat{R}^{n}}\left\{\cos (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P})\left[\begin{array}{@{}c@{}}\tilde{\boldsymbol{u}}_{n1}\\ \tilde{\boldsymbol{v}}_{n1}\end{array}\right]+\sin (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P})\left[\begin{array}{@{}c@{}}\tilde{\boldsymbol{u}}_{n2}\\ \tilde{\boldsymbol{v}}_{n2}\end{array}\right]+\left[\begin{array}{@{}c@{}}\tilde{\boldsymbol{u}}_{n3}\\ \tilde{\boldsymbol{v}}_{n3}\end{array}\right]\right\}\end{eqnarray}$$

for the value of the last integral in (6.1).

Inserting (6.4), (6.7) and (6.11) in (6.1), we arrive at

(6.12) $$\begin{eqnarray}\displaystyle & & \displaystyle \left[\begin{array}{@{}c@{}}\boldsymbol{E}^{\text{v}}\\ \boldsymbol{ B}^{\text{v}}\end{array}\right]=m^{2}\exp (-\text{i}m\hat{\unicode[STIX]{x1D711}}_{P})\mathop{\sum }_{n=1}^{2}\int _{{\mathcal{S}}^{\prime }}\hat{r}\,\text{d}\hat{r}\,\text{d}\hat{z}\,\left\{\text{H}(\unicode[STIX]{x1D6E5})\mathop{\sum }_{j=1}^{3}\mathop{\sum }_{k=1}^{\infty }\exp (-\text{i}mc_{2})\left[\begin{array}{@{}c@{}}\tilde{\boldsymbol{u}}_{nj}\\ \tilde{\boldsymbol{v}}_{nj}\end{array}\right]\right.\nonumber\\ \displaystyle & & \displaystyle \quad \times \left(\int _{\unicode[STIX]{x1D6F9}_{L}}^{\unicode[STIX]{x1D6F9}_{U}}-\int _{-2c_{1}}^{2c_{1}}+2\int _{-c_{1}}^{c_{1}}\right)\,\text{d}\unicode[STIX]{x1D6F9}(p_{nj}+q_{nj}\unicode[STIX]{x1D6F9}){\mathcal{H}}\exp \left[-\text{i}m\left(\frac{1}{3}\unicode[STIX]{x1D6F9}^{3}-c_{1}^{2}\unicode[STIX]{x1D6F9}\right)\right]+\text{H}(-\unicode[STIX]{x1D6E5})\nonumber\\ \displaystyle & & \displaystyle \quad \left.\vphantom{\mathop{\sum }_{j=1}^{3}}\times \,\int _{0}^{2\unicode[STIX]{x03C0}}\text{d}\unicode[STIX]{x1D711}\frac{\exp (-\text{i}mg)}{\hat{R}^{n}}\left(\cos (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P})\left[\begin{array}{@{}c@{}}\tilde{\boldsymbol{u}}_{n1}\\ \tilde{\boldsymbol{v}}_{n1}\end{array}\right]+\sin (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P})\left[\begin{array}{@{}c@{}}\tilde{\boldsymbol{u}}_{n2}\\ \tilde{\boldsymbol{v}}_{n2}\end{array}\right]+\left[\begin{array}{@{}c@{}}\tilde{\boldsymbol{u}}_{n3}\\ \tilde{\boldsymbol{v}}_{n3}\end{array}\right]\right)\right\},\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where the integration variable $\unicode[STIX]{x1D713}$ in (6.4) has been renamed $\unicode[STIX]{x1D6F9}$ and the limits of integration in (6.7) have been placed in alternative positions. For small $c_{1}$ , the difference between the values of the two integrals over $-2c_{1}\leqslant \unicode[STIX]{x1D6F9}\leqslant 2c_{1}$ and $-c_{1}\leqslant \unicode[STIX]{x1D6F9}\leqslant c_{1}$ is negligibly small compared to that of the first integral inside the parentheses. Once these two integrals are ignored, the remaining $k$ -dependent function in the resulting expression can be summed,

(6.13) $$\begin{eqnarray}\mathop{\sum }_{k=1}^{\infty }{\mathcal{H}}=1,\end{eqnarray}$$

since the $k$ -dependence of $c_{2}$ does not influence the value of $\exp (-\text{i}mc_{2})$ (see (4.19), (4.23) and (4.26)). This enables us to obtain the asymptotic value of the integral over $\unicode[STIX]{x1D6F9}_{L}\leqslant \unicode[STIX]{x1D6F9}\leqslant \unicode[STIX]{x1D6F9}_{U}$ by simply extending its range,

(6.14) $$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\unicode[STIX]{x1D6F9}_{L}}^{\unicode[STIX]{x1D6F9}_{U}}\text{d}\unicode[STIX]{x1D6F9}(p_{nj}+q_{nj}\unicode[STIX]{x1D6F9})\exp \left[-\text{i}m\left(\frac{1}{3}\unicode[STIX]{x1D6F9}^{3}-c_{1}^{2}\unicode[STIX]{x1D6F9}\right)\right]\nonumber\\ \displaystyle & & \displaystyle \qquad \simeq 2\int _{0}^{\infty }\text{d}\unicode[STIX]{x1D6F9}\,\left\{p_{nj}\cos \left[m\left(\frac{1}{3}\unicode[STIX]{x1D6F9}^{3}-c_{1}^{2}\unicode[STIX]{x1D6F9}\right)\right]\right.\nonumber\\ \displaystyle & & \displaystyle \qquad \quad \left.-\,\text{i}q_{nj}\unicode[STIX]{x1D6F9}\sin \left[m\left(\frac{1}{3}\unicode[STIX]{x1D6F9}^{3}-c_{1}^{2}\unicode[STIX]{x1D6F9}\right)\right]\right\},\quad m\gg 1,\end{eqnarray}$$

because the phase of the exponential factor in its integrand is in a canonical form as it stands.

The imaginary part of the $\unicode[STIX]{x1D6F9}$ -integral in (6.14) can be obtained by differentiating the real part of this integral with respect to $c_{1}^{2}$ and dividing the resulting expression by $m$ . From (9.5.1) of Olver et al. (Reference Olver, Lozier, Boisvert and Clark2010) and (6.13) and (6.14) it follows, therefore, that

(6.15) $$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}\boldsymbol{E}^{\text{v}}\\ \boldsymbol{ B}^{\text{v}}\end{array}\right] & \simeq & \displaystyle m^{2}\exp (-\text{i}m\hat{\unicode[STIX]{x1D711}}_{P})\mathop{\sum }_{n=1}^{2}\int _{{\mathcal{S}}^{\prime }}\hat{r}\,\text{d}\hat{r}\,\text{d}\hat{z}\,\left\{\text{H}(\unicode[STIX]{x1D6E5})\mathop{\sum }_{j=1}^{3}2\unicode[STIX]{x03C0}m^{-1/3}\exp (-\text{i}mc_{2})\left[\begin{array}{@{}c@{}}\tilde{\boldsymbol{u}}_{nj}\\ \tilde{\boldsymbol{v}}_{nj}\end{array}\right]\right.\nonumber\\ \displaystyle & & \displaystyle \times \left[p_{nj}\text{Ai}\left(-m^{2/3}c_{1}^{2}\right)+\text{i}m^{-1/3}q_{nj}\text{Ai}^{\prime }\left(-m^{2/3}c_{1}^{2}\right)\right]+\text{H}(-\unicode[STIX]{x1D6E5})\nonumber\\ \displaystyle & & \displaystyle \left.\vphantom{\mathop{\sum }_{j=1}^{3}}\times \int _{0}^{2\unicode[STIX]{x03C0}}\text{d}\unicode[STIX]{x1D711}\frac{\exp (-\text{i}mg)}{\hat{R}^{n}}\left(\cos (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P})\left[\begin{array}{@{}c@{}}\tilde{\boldsymbol{u}}_{n1}\\ \tilde{\boldsymbol{v}}_{n1}\end{array}\right]+\sin (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P})\left[\begin{array}{@{}c@{}}\tilde{\boldsymbol{u}}_{n2}\\ \tilde{\boldsymbol{v}}_{n2}\end{array}\right]+\left[\begin{array}{@{}c@{}}\tilde{\boldsymbol{u}}_{n3}\\ \tilde{\boldsymbol{v}}_{n3}\end{array}\right]\right)\right\},\nonumber\\ \displaystyle & & \displaystyle \qquad \unicode[STIX]{x1D703}_{L}^{\text{c}}\leqslant \unicode[STIX]{x1D703}_{P}\leqslant \unicode[STIX]{x03C0}-\unicode[STIX]{x1D703}_{L}^{\text{c}},\quad m\gg 1,\end{eqnarray}$$

in which $\text{Ai}$ and $\text{Ai}^{\prime }$ are the Airy function and the derivative of the Airy function with respect to its argument, respectively.

On the other hand, evaluation of the leading term in the asymptotic expansion of the $\unicode[STIX]{x1D711}$ -integral in

(6.16) $$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}\boldsymbol{E}^{\text{v}}\\ \boldsymbol{ B}^{\text{v}}\end{array}\right] & = & \displaystyle m^{2}\mathop{\sum }_{n=1}^{2}\int _{{\mathcal{S}}^{\prime }}\hat{r}\,\text{d}\hat{r}\,\text{d}\hat{z}\int _{0}^{2\unicode[STIX]{x03C0}}\text{d}\unicode[STIX]{x1D711}\frac{\exp [-\text{i}m(g+\hat{\unicode[STIX]{x1D711}}_{P})]}{\hat{R}^{n}}\nonumber\\ \displaystyle & & \displaystyle \times \left(\cos (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P})\left[\begin{array}{@{}c@{}}\tilde{\boldsymbol{u}}_{n1}\\ \tilde{\boldsymbol{v}}_{n1}\end{array}\right]+\sin (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{P})\left[\begin{array}{@{}c@{}}\tilde{\boldsymbol{u}}_{n2}\\ \tilde{\boldsymbol{v}}_{n2}\end{array}\right]+\left[\begin{array}{@{}c@{}}\tilde{\boldsymbol{u}}_{n3}\\ \tilde{\boldsymbol{v}}_{n3}\end{array}\right]\right)\end{eqnarray}$$

by the method of Chester et al. (Reference Chester, Friedman and Ursell1957) results in exactly the same expression as that which multiplies $\text{H}(\unicode[STIX]{x1D6E5})$ in (6.15). Hence, the expression in (6.16) gives the combined contributions of the source elements in both $\unicode[STIX]{x1D6E5}<0$ and $\unicode[STIX]{x1D6E5}>0$ . It turns out that this expression could have been directly obtained in the present case by performing the integration with respect to $t$ in (4.4), even though such a procedure is not generally applicable to cases in which the retarded time is multi-valued (see appendix B).

6.2 Evaluation of $[\boldsymbol{E}^{\text{v}}\quad \boldsymbol{B}^{\text{v}}]$ at observation points for which $0<\unicode[STIX]{x1D703}_{P}\leqslant \unicode[STIX]{x1D703}_{L}^{\text{c}}$ or $\unicode[STIX]{x03C0}-\unicode[STIX]{x1D703}_{L}^{\text{c}}\leqslant \unicode[STIX]{x1D703}_{P}<\unicode[STIX]{x03C0}$

Equation (6.16) applies also to an observation point for which the entire source lies outside the bifurcation surface, i.e. for which $\hat{r}_{U}<\hat{r}_{C}$ (see figure 11). None of the source elements in ${\mathcal{S}}^{\prime }$ can approach observers that are located in $0<\unicode[STIX]{x1D703}_{P}\leqslant \unicode[STIX]{x1D703}_{L}^{\text{c}}$ or $\unicode[STIX]{x03C0}-\unicode[STIX]{x1D703}_{L}^{\text{c}}\leqslant \unicode[STIX]{x1D703}_{P}<\unicode[STIX]{x03C0}$ with a superluminal speed along the radiation direction. As a result, $\unicode[STIX]{x1D6E5}$ is negative throughout the source distribution (2.7) and the field $[\begin{array}{@{}cc@{}}\boldsymbol{E}^{\text{v}} & \boldsymbol{B}^{\text{v}}\end{array}]$ that is generated outside these two cones (i.e. outside the coloured regions in figure 12) is the same as any other conventional radiation field.

It is customary, when deriving (6.16) from (4.4), to replace the term $\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}/\unicode[STIX]{x2202}t$ that results from the integration with respect to $t$ by $-c\,\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{j}$ (from the equation of continuity) and to apply a subsequent integration by parts with respect to $\boldsymbol{x}$ (by means of the divergence theorem) to write the conventional radiation field as

(6.17) $$\begin{eqnarray}\left[\begin{array}{@{}c@{}}\boldsymbol{E}^{\text{v}}\\ \boldsymbol{B}^{\text{v}}\end{array}\right]=\frac{1}{c^{2}}\int \text{d}^{3}\boldsymbol{x}\,\text{d}t\,\frac{\unicode[STIX]{x1D6FF}(t-t_{P}+R/c)}{R}\,\hat{\boldsymbol{n}}\times \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\left[\begin{array}{@{}c@{}}\hat{\boldsymbol{n}}\times \boldsymbol{j}\\ -\boldsymbol{j}\end{array}\right].\end{eqnarray}$$

However, of the two equivalent formulations given by (6.16) and (6.17), I will be using the former which can be more easily combined with the expressions I will derive for $[\begin{array}{@{}cc@{}}\boldsymbol{E}_{\pm }^{\text{b}} & \boldsymbol{E}_{\pm }^{\text{b}}\end{array}]$ .

7.1 Locus of stationary points, $S$ , of the phase of the exponential factor in the expression for $[\boldsymbol{E}_{-}^{\text{b}}\quad \boldsymbol{B}_{-}^{\text{b}}]$

Despite the apparent symmetry between the two terms $G_{nj}^{\text{out}}|_{\hat{\unicode[STIX]{x1D711}}_{\pm }}\exp (-\text{i}m\hat{\unicode[STIX]{x1D711}}_{\pm })$ in the expressions for $[\begin{array}{@{}cc@{}}\boldsymbol{E}_{\pm }^{\text{b}} & \boldsymbol{B}_{\pm }^{\text{b}}\end{array}]$ in (5.8), these two contributions toward the value of the unconventional radiation field differ radically: the phase $\hat{\unicode[STIX]{x1D711}}_{-}$ of the first exponential factor is stationary, as a function of $\hat{r}$ , along a curve in the $(r,\unicode[STIX]{x1D711},z)$ space while the phase $\hat{\unicode[STIX]{x1D711}}_{+}$ of the second exponential has no stationary points.

Differentiation of the functions $\unicode[STIX]{x1D719}_{\pm }=\hat{\unicode[STIX]{x1D711}}_{\pm }-\hat{\unicode[STIX]{x1D711}}_{P}=g(\unicode[STIX]{x1D711}_{\pm })$ with respect to $\hat{r}$ yields

(7.1) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{\pm }}{\unicode[STIX]{x2202}\hat{r}}=\frac{\hat{r}^{2}-1\pm \unicode[STIX]{x1D6E5}^{1/2}}{\hat{r}\hat{R}_{\pm }}\end{eqnarray}$$

(see (4.23)). Hence $\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{-}/\unicode[STIX]{x2202}\hat{r}$ vanishes for all $\hat{z}$ , along the projection $\hat{r}=\hat{r}_{S}(\hat{z})$ of the following three-dimensional curve onto the $(r,z)$ plane,

(7.2) $$\begin{eqnarray}\displaystyle & S:\left\{\begin{array}{@{}l@{}}\hat{r}=\hat{r}_{S}(\hat{z})=\{\frac{1}{2}({\hat{r}_{P}}^{2}+1)-{[\frac{1}{4}({\hat{r}_{P}}^{2}-1)^{2}-(\hat{z}-\hat{z}_{P})^{2}]^{1/2}\}}^{1/2},\quad \\ \unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}_{S}(\hat{z})=\unicode[STIX]{x1D711}_{P}+2k\unicode[STIX]{x03C0}-\arccos (\hat{r}_{S}/\hat{r}_{P}).\quad \end{array}\right. & \displaystyle\end{eqnarray}$$

Along this curve, which is depicted in figure 11, the two derivatives $\unicode[STIX]{x2202}g/\unicode[STIX]{x2202}\unicode[STIX]{x1D711}$ and $\unicode[STIX]{x2202}g/\unicode[STIX]{x2202}\hat{r}$ of the argument of the Dirac delta function in (4.17) vanish simultaneously and $\unicode[STIX]{x1D6E5}^{1/2}=\hat{r}^{2}-1$ (see (4.19) and (4.20)). At the point $\hat{z}=\hat{z}_{P}$ on $S$ the derivative $\unicode[STIX]{x2202}g/\unicode[STIX]{x2202}\hat{z}$ also vanishes (since $g$ depends on $\hat{z}$ only through $(\hat{z}-\hat{z}_{P})^{2}$ ), so that the derivatives of the phase of the Green’s function (4.17) with respect to all three of the integration variables in the expression for the field $[\begin{array}{@{}cc@{}}\boldsymbol{E} & \boldsymbol{B}\end{array}]$ (i.e. $\unicode[STIX]{x2202}g/\unicode[STIX]{x2202}\unicode[STIX]{x1D711}$ , $\unicode[STIX]{x2202}g/\unicode[STIX]{x2202}\hat{r}$ and $\unicode[STIX]{x2202}g/\unicode[STIX]{x2202}\hat{z}$ ) vanish simultaneously. I refer to both $S$ and its projection onto the $(r,z)$ plane as the locus of stationary points of $\unicode[STIX]{x1D719}_{-}$ ; whether it is $S$ itself or its projection that is referred to will be clear from the context.

Note that the coordinates of a far-field observation point need to satisfy $\hat{R}_{P}\geqslant 2\cot \unicode[STIX]{x1D703}_{P}\csc \unicode[STIX]{x1D703}_{P}$ for $\hat{r}_{S}$ to be real; otherwise, the expression inside the curly brackets in (7.2) would be negative. This constraint reflects the fact that the projection of locus $S$ onto the $(\hat{r},\hat{z})$ plane curves away from the rotation axis (see figure 11). The locus $S$ becomes more parallel to the rotation axis as the observation point moves into the far zone. But, no matter how large $\hat{R}_{P}$ may be, there are always ranges of values of the polar angle $\unicode[STIX]{x1D703}_{P}$ (close to $0$ and to $\unicode[STIX]{x03C0}$ ) for which the function $\unicode[STIX]{x1D719}_{-}$ has no stationary points.

By comparing (4.24) and (7.2), we can see that the radial separation between the curves $C$ and $S$ in figure 11 is exceedingly small when the observation point lies either in the far zone, $\hat{R}_{P}\gg 1$ , or close to the plane of rotation $\unicode[STIX]{x1D703}_{P}=\unicode[STIX]{x03C0}/2$ , so that $|\hat{z}-\hat{z}_{P}|\ll 1$ throughout the localized source distribution (2.7),

(7.3) $$\begin{eqnarray}\displaystyle & \hat{r}_{S}-\hat{r}_{C}\simeq \left\{\begin{array}{@{}ll@{}}\cot ^{4}\unicode[STIX]{x1D703}_{P}/(2\hat{R}_{P}^{2}\sin \unicode[STIX]{x1D703}_{P}),\quad & \hat{R}_{P}\gg 1\\ \frac{1}{2}(\hat{z}-\hat{z}_{P})^{4}/(\hat{r}_{P}^{2}-1)^{3},\quad & |\hat{z}-\hat{z}_{P}|\ll 1.\end{array}\right. & \displaystyle\end{eqnarray}$$

In other words, the locus $S$ of the stationary points of $\unicode[STIX]{x1D719}_{-}$ is essentially coincident with the cusp locus of the bifurcation surface (i.e. with the locus $C$ of the source elements that approach the observation point with the speed of light and zero acceleration at the retarded time) in these cases. This notwithstanding, (4.18) and (7.1) show that the value of the function $\unicode[STIX]{x1D6E5}$ undergoes a large change over a small interval in $\hat{r}$ : it vanishes on $C$ , equals $(\hat{r}_{S}^{2}-1)^{2}$ on $S$ and rapidly rises to as large a value as that of $\hat{R}_{P}^{2}$ (for $\hat{R}_{P}\gg 1$ ) a short distance away from $C$ . We will see that the sharp change in the value of $G_{j}^{\text{out}}|_{\unicode[STIX]{x1D719}_{-}}$ resulting from the proximity of the loci $C$ and $S$ renders the numerical evaluation of $[\begin{array}{@{}cc@{}}\boldsymbol{E}_{-}^{\text{b}} & \boldsymbol{B}_{-}^{\text{b}}\end{array}]$ in the far zone particularly challenging (§ 11).

If, in analogy with (5.9), we rewrite the two members of (7.2) in terms of the dimensionless polar coordinates $(\hat{R}_{P},\unicode[STIX]{x1D703}_{P})$ of the observation point $P$ and solve them for $\unicode[STIX]{x1D703}_{P}$ and $\unicode[STIX]{x1D711}_{P}$ as functions of $(\hat{r},\hat{z})$ and $\hat{R}_{P}$ , we obtain

(7.4) $$\begin{eqnarray}\displaystyle S:\left\{\begin{array}{@{}l@{}}\displaystyle \unicode[STIX]{x1D703}_{P}=\unicode[STIX]{x1D703}_{P}^{\text{s}}(\hat{r},\hat{z})\equiv \arccos \left\{\frac{1}{\hat{R}_{P}\hat{r}}\left[\frac{\hat{z}}{\hat{r}}+\left(\hat{r}^{2}-1\right)^{1/2}\left({\hat{R}_{P}}^{2}-\hat{r}^{2}-\frac{\hat{z}^{2}}{\hat{r}^{2}}\right)^{1/2}\right]\right\},\quad \\ \displaystyle \unicode[STIX]{x1D711}_{P}=\unicode[STIX]{x1D711}_{P}^{\text{s}}(\hat{r},\hat{z})\equiv \unicode[STIX]{x1D711}-2k\unicode[STIX]{x03C0}+\arccos \left(\frac{\hat{r}}{\hat{R}_{P}\sin \unicode[STIX]{x1D703}_{P}}\right),\quad \end{array}\right. & & \displaystyle\end{eqnarray}$$

where $k$ is the positive integer enumerating successive rotation periods. Hence, the projection of the segment $\hat{z}_{P}\geqslant \hat{z}$ of the locus $S$ onto the $(\hat{r},\hat{z})$ plane intersects the source distribution (2.7) at a given $-\hat{z}_{0}\leqslant \hat{z}\leqslant \hat{z}_{0}$ if the colatitude of the observation point $P$ lies in the interval $\unicode[STIX]{x1D703}_{L}^{\text{s}}\leqslant \unicode[STIX]{x1D703}_{P}\leqslant \unicode[STIX]{x1D703}_{U}^{\text{s}}$ , where

(7.5) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{L}^{\text{s}}=\unicode[STIX]{x1D703}_{P}^{\text{s}}|_{\hat{r}=\hat{r}_{U},\hat{z}=\hat{z}_{0}}\end{eqnarray}$$

and

(7.6) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{U}^{\text{s}}=\unicode[STIX]{x1D703}_{P}^{\text{s}}|_{\hat{r}=\hat{r}_{L},\hat{z}=-\hat{z}_{0}}.\end{eqnarray}$$

The radial coordinates of all source elements would exceed $\hat{r}_{S}$ , on the other hand, if $\unicode[STIX]{x1D703}_{U}^{\text{s}}\leqslant \unicode[STIX]{x1D703}_{P}\leqslant \unicode[STIX]{x03C0}-\unicode[STIX]{x1D703}_{U}^{\text{s}}$ (see (2.7) and figure 11).

The loci $C$ and $S$ would both lie within the source distribution (2.7), at every value of $\hat{z}$ in $-\hat{z}_{0}\leqslant \hat{z}\leqslant \hat{z}_{0}$ , if $\unicode[STIX]{x1D703}_{L}\leqslant \unicode[STIX]{x1D703}_{P}\leqslant \unicode[STIX]{x1D703}_{U}$ , where

(7.7) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{L}=\unicode[STIX]{x1D703}_{P}^{\text{s}}|_{\hat{r}=\hat{r}_{U},\hat{z}=-\hat{z}_{0}}\end{eqnarray}$$

and

(7.8) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{U}=\unicode[STIX]{x1D703}_{P}^{\text{c}}|_{\hat{r}=\hat{r}_{L},\hat{z}=\hat{z}_{0}}.\end{eqnarray}$$

This can be seen by noting that as the polar coordinate of the observation point increases (at a given $\hat{R}_{P}$ ) away from the rotation axis $\unicode[STIX]{x1D703}_{P}=0$ toward the equatorial plane $\unicode[STIX]{x1D703}_{P}=\unicode[STIX]{x03C0}/2$ , the curve $S$ intersects the entire $\hat{z}$ -extent of the source once it passes the corner $\hat{r}=\hat{r}_{U}$ , $\hat{z}=-\hat{z}_{0}$ of the rectangular support ${\mathcal{S}}^{\prime }$ of the source distribution (2.7) shown in figure 11. It is curve $C$ , on the other hand, that starts leaving the source at the corner $\hat{r}=\hat{r}_{L}$ , $\hat{z}=\hat{z}_{0}$ of ${\mathcal{S}}^{\prime }$ after $S$ and $C$ have swept across the $\hat{r}$ -extent of the source. If $\unicode[STIX]{x1D703}_{P}$ continues to increase past the value $\unicode[STIX]{x03C0}/2$ , then $S$ and $C$ (in that order) start entering ${\mathcal{S}}^{\prime }$ from the corner $\hat{r}=\hat{r}_{L}$ , $\hat{z}=\hat{z}_{0}$ when $\unicode[STIX]{x1D703}_{P}=\unicode[STIX]{x03C0}-\unicode[STIX]{x1D703}_{U}$ and will both intersect the entire $\hat{z}$ -extent of the source again when $\unicode[STIX]{x03C0}-\unicode[STIX]{x1D703}_{U}\leqslant \unicode[STIX]{x1D703}_{P}\leqslant \unicode[STIX]{x03C0}-\unicode[STIX]{x1D703}_{L}$ .

There are intervals of $\unicode[STIX]{x1D703}_{P}$ near $\unicode[STIX]{x1D703}_{L}$ or $\unicode[STIX]{x1D703}_{U}$ for which only one of the curves $C$ and $S$ intersects the source distribution, mostly over a limited section of $-\hat{z}_{0}\leqslant \hat{z}\leqslant \hat{z}_{0}$ (see figure 11). Evaluation of the radiation field in these transitional intervals whose widths rapidly shrink with increasing distance – as $\hat{R}_{P}^{-2}$ for $\hat{R}_{P}\gg 1$ (see (7.3)) – will be dealt with separately in § 9.

Note that the stationary point $\hat{r}=1$ , $\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}_{P}+2\unicode[STIX]{x03C0}k-\arccos (1/\hat{r})$ , $\hat{z}=\hat{z}_{P}$ , at which all three derivatives of $g$ vanish and the curves $C$ and $S$ meet tangentially (see figure 11) does not fall within the range of integration in (5.8) unless (i) there are source elements whose speeds equal the speed of light $c$ , and (ii) the observation point lies sufficiently close to the plane of rotation $\unicode[STIX]{x1D703}_{P}=\unicode[STIX]{x03C0}/2$ for its coordinate $\hat{z}_{P}=\hat{R}_{P}\cos \unicode[STIX]{x1D703}_{P}$ to match the coordinate $\hat{z}$ of some source elements. For the source distribution described in § 2, these requirements are met only if $\hat{r}_{L}\leqslant 1$ , i.e. the source elements at the inner radius of the dielectric move with a speed that is smaller than or equal to $c$ and the observation point lies within the following angular interval

(7.9) $$\begin{eqnarray}\frac{\unicode[STIX]{x03C0}}{2}-\arcsin \left(\frac{\hat{z}_{0}}{\hat{R}_{P}}\right)\leqslant \unicode[STIX]{x1D703}_{P}\leqslant \frac{\unicode[STIX]{x03C0}}{2}+\arcsin \left(\frac{\hat{z}_{0}}{\hat{R}_{P}}\right)\end{eqnarray}$$

encompassing the equatorial plane. Note also that the width of this interval decreases with increasing distance as $\hat{R}_{P}^{-1}$ in the far zone.

7.2 Paths of steepest descent for the exponential kernels $\exp (-\text{i}m\unicode[STIX]{x1D719}_{\pm })$

Owing to the proximity of the loci $C$ and $S$ and the rapid variation of $G_{nj}^{\text{out}}|_{\unicode[STIX]{x1D719}_{-}}$ in their vicinity, the contributions of the critical points discussed in the preceding section toward the value of the integral in (5.8) cannot be taken into account properly without resorting to a technique more discerning than a direct numerical integration. Here I evaluate the $\hat{r}$ -integral in (5.8) by the method of steepest descent (see, e.g. Bender & Orszag Reference Bender and Orszag1999). I regard the variable of integration $\hat{r}$ as complex, i.e. write

(7.10) $$\begin{eqnarray}\hat{r}=u+\text{i}v,\end{eqnarray}$$

in which $u$ and $v$ are real, and invoke Cauchy’s integral theorem to deform the original path of integration into the contours of steepest descent in the complex $(u,v)$ -plane that pass through the critical points of the phases $\unicode[STIX]{x1D719}_{\pm }(\hat{r},\hat{z})$ at a given $\hat{z}$ , i.e. through the stationary point $\hat{r}=\hat{r}_{S}$ and the boundary points $\hat{r}=\hat{r}_{C}$ or $\hat{r}_{L}$ and $\hat{r}=\hat{r}_{U}$ . We will see that the constant $m$ in the argument of the exponential factor in (5.8) need not be particularly large for the main contributions to the values of $[\begin{array}{@{}cc@{}}\boldsymbol{E}_{\pm }^{\text{b}} & \boldsymbol{B}_{\pm }^{\text{b}}\end{array}]$ to come from a limited segment of each path next to the critical point from which it issues.

The first step is to write $\unicode[STIX]{x1D719}_{\pm }$ as functions of $(u,v,\hat{z};\hat{r}_{P},\hat{z}_{P})$ . Inserting (7.10) in (4.18) and adopting the square root of the resulting complex expression which is positive on the real axis $v=0$ , we obtain

(7.11) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}^{1/2}=d\,\exp (\text{i}\unicode[STIX]{x1D6FF}),\end{eqnarray}$$

with

(7.12) $$\begin{eqnarray}d=\{[(\hat{r}_{P}^{2}-1)(u^{2}-v^{2}-1)-(\hat{z}-\hat{z}_{P})^{2}]^{2}+4(\hat{r}_{P}^{2}-1)^{2}u^{2}{v^{2}\}}^{1/4},\end{eqnarray}$$

and

(7.13) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}=\frac{1}{2}\arctan \frac{2(\hat{r}_{P}^{2}-1)uv}{(\hat{r}_{P}^{2}-1)(u^{2}-v^{2}-1)-(\hat{z}-\hat{z}_{P})^{2}}.\end{eqnarray}$$

Equation (4.22) together with (7.10) and (7.11) then yields

(7.14) $$\begin{eqnarray}\hat{R}_{\pm }=L_{\pm }\exp (\text{i}\unicode[STIX]{x1D70E}_{\pm }),\end{eqnarray}$$

in which

(7.15) $$\begin{eqnarray}L_{\pm }=\{[(\hat{z}-\hat{z}_{P})^{2}+\hat{r}_{P}^{2}+u^{2}-v^{2}-2(1\mp d\cos \unicode[STIX]{x1D6FF})]^{2}+4{(uv\pm d\sin \unicode[STIX]{x1D6FF})^{2}\}}^{1/4},\end{eqnarray}$$

and

(7.16) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{\pm }=\frac{1}{2}\arctan \frac{2(uv\pm d\sin \unicode[STIX]{x1D6FF})}{(\hat{z}-\hat{z}_{P})^{2}+\hat{r}_{P}^{2}+u^{2}-v^{2}-2(1\mp d\cos \unicode[STIX]{x1D6FF})}.\end{eqnarray}$$

From Euler’s formula $\exp (\text{i}x)=\cos x+\text{i}\sin x$ and (4.19), we have

(7.17) $$\begin{eqnarray}\exp [\text{i}(\unicode[STIX]{x1D711}_{\pm }-\unicode[STIX]{x1D711}_{P})]=\frac{1\mp \unicode[STIX]{x1D6E5}^{1/2}-\text{i}\hat{R}_{\pm }}{\hat{r}\hat{r}_{P}},\end{eqnarray}$$

so that the corresponding expression for $\unicode[STIX]{x1D711}_{\pm }$ can be written as

(7.18) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}_{\pm } & = & \displaystyle \unicode[STIX]{x1D711}_{P}+2k\unicode[STIX]{x03C0}+\unicode[STIX]{x1D709}_{\pm }-\unicode[STIX]{x1D702}+\text{i}\ln [\hat{r}_{P}(u^{2}+v^{2})^{1/2}]\nonumber\\ \displaystyle & & \displaystyle -\,(\text{i}/2)\ln \{1+d^{2}+L_{\pm }^{2}+2L_{\pm }\sin \unicode[STIX]{x1D70E}_{\pm }\mp 2d[L_{\pm }\sin (\unicode[STIX]{x1D70E}_{\pm }-\unicode[STIX]{x1D6FF})+\cos \unicode[STIX]{x1D6FF}]\},\quad\end{eqnarray}$$

with

(7.19) $$\begin{eqnarray}\unicode[STIX]{x1D709}_{\pm }=\arctan \frac{-L_{\pm }\cos \unicode[STIX]{x1D70E}_{\pm }\mp d\sin \unicode[STIX]{x1D6FF}}{1\mp d\cos \unicode[STIX]{x1D6FF}+L_{\pm }\sin \unicode[STIX]{x1D70E}_{\pm }},\end{eqnarray}$$

and

(7.20) $$\begin{eqnarray}\unicode[STIX]{x1D702}=\arctan (v/u),\end{eqnarray}$$

in the light of (7.10), (7.11), (7.14) and (7.17).

Inserting the above expressions for $\unicode[STIX]{x1D6E5}^{1/2}$ , $\hat{R}_{\pm }$ and $\unicode[STIX]{x1D711}_{\pm }$ in (4.23), we finally arrive at the following expressions for the real and imaginary parts of $\unicode[STIX]{x1D719}_{\pm }=\hat{\unicode[STIX]{x1D711}}_{\pm }-\hat{\unicode[STIX]{x1D711}}_{P}$ as functions of $(u,v,\hat{z},\hat{r}_{P},\hat{z}_{P})$ ,

(7.21) $$\begin{eqnarray}\text{Re}[\unicode[STIX]{x1D719}_{\pm }(u,v)]=L_{\pm }\cos \unicode[STIX]{x1D70E}_{\pm }+\unicode[STIX]{x1D709}_{\pm }-\unicode[STIX]{x1D702}+2k\unicode[STIX]{x03C0},\end{eqnarray}$$

and

(7.22) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{Im}[\unicode[STIX]{x1D719}_{\pm }(u,v)]=L_{\pm }\sin \unicode[STIX]{x1D70E}_{\pm }+\ln [\hat{r}_{P}(u^{2}+v^{2})^{1/2}]\nonumber\\ \displaystyle & & \displaystyle \qquad -\,(1/2)\ln \{1+d^{2}+L_{\pm }^{2}+2L_{\pm }\sin \unicode[STIX]{x1D70E}_{\pm }\mp 2d[L_{\pm }\sin (\unicode[STIX]{x1D70E}_{\pm }-\unicode[STIX]{x1D6FF})+\cos \unicode[STIX]{x1D6FF}]\}.\qquad\end{eqnarray}$$

The path of steepest descent through a given critical point of $\unicode[STIX]{x1D719}_{-}$ or $\unicode[STIX]{x1D719}_{+}$ is the curve in the complex $(u,v)$ -plane along which the corresponding phase $-\text{i}m\hat{\unicode[STIX]{x1D711}}_{-}$ or $-\text{i}m\hat{\unicode[STIX]{x1D711}}_{+}$ of the relevant exponential factor in (5.8) has a constant imaginary part and a negative real part (Bender & Orszag Reference Bender and Orszag1999).

To use Cauchy’s theorem to express the integrals over $\hat{r}$ in (5.8) as integrals over such steepest-descent paths, we also need the Jacobians of the transformations that map the real axis onto these paths. Along each path of steepest descent through a critical point of either $\exp (-\text{i}m\unicode[STIX]{x1D719}_{-})$ or $\exp (-\text{i}m\unicode[STIX]{x1D719}_{+})$ , the real part of the relevant $\unicode[STIX]{x1D719}_{\pm }$ is constant. So, setting the total derivative of $\text{Re}[\unicode[STIX]{x1D719}_{\pm }(u,v)]$ equal to zero, we find that the slope of a steepest-descent path is given by

(7.23) $$\begin{eqnarray}\frac{\text{d}v}{\text{d}u}=-\frac{{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}u}}[\text{Re}(\unicode[STIX]{x1D719}_{\pm })]}{{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v}}[\text{Re}(\unicode[STIX]{x1D719}_{\pm })]}=\frac{{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}u}}[\text{Re}(\unicode[STIX]{x1D719}_{\pm })]}{{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}u}}[\text{Im}(\unicode[STIX]{x1D719}_{\pm })]}=\frac{\text{Re}\left[{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{\pm }}{\unicode[STIX]{x2202}\hat{r}}}\right]}{\text{Im}\left[{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{\pm }}{\unicode[STIX]{x2202}\hat{r}}}\right]},\end{eqnarray}$$

where I have used the Cauchy–Riemann relation

(7.24) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v}\{\text{Re}[\unicode[STIX]{x1D719}_{\pm }(u,v)]\}=-\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}u}\{\text{Im}[\unicode[STIX]{x1D719}_{\pm }(u,v)]\},\end{eqnarray}$$

and the following expression for the derivatives of the complex functions $\unicode[STIX]{x1D719}_{\pm }(u,v)$ with respect to the complex variable $\hat{r}=u+\text{i}v$ :

(7.25) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{\pm }}{\unicode[STIX]{x2202}\hat{r}}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}u}\{\text{Re}[\unicode[STIX]{x1D719}_{\pm }(u,v)]\}+\text{i}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}u}\{\text{Im}[\unicode[STIX]{x1D719}_{\pm }(u,v)]\}.\end{eqnarray}$$

Equation (7.23) applies to a steepest-descent path through any critical point of either $\unicode[STIX]{x1D719}_{-}$ or $\unicode[STIX]{x1D719}_{+}$ , irrespective of whether it is parametrized by $u$ or $v$ (i.e. whether its slope is given by the last expression on the right-hand side of (7.23) or by the inverse of this expression).

The functions $\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{\pm }/\unicode[STIX]{x2202}\hat{r}$ have the values derived in (7.1). Writing $\hat{r}$ everywhere in the right-hand side of (7.1) as $u+\text{i}v$ and making use of (7.11) and (7.14), we arrive at

(7.26) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{\pm }}{\unicode[STIX]{x2202}\hat{r}}=\frac{u^{2}-v^{2}-1\pm d\cos \unicode[STIX]{x1D6FF}+\text{i}(2uv\pm d\sin \unicode[STIX]{x1D6FF})}{(u^{2}+v^{2})^{1/2}L_{\pm }\exp [\text{i}(\unicode[STIX]{x1D70E}_{\pm }+\unicode[STIX]{x1D702})]},\end{eqnarray}$$

so that insertion of the real and imaginary parts of the above expression in the last member of (7.23) yields

(7.27) $$\begin{eqnarray}\frac{\text{d}v}{\text{d}u}=\cot (\unicode[STIX]{x1D701}_{\pm }-\unicode[STIX]{x1D70E}_{\pm }-\unicode[STIX]{x1D702}),\end{eqnarray}$$

where

(7.28) $$\begin{eqnarray}\unicode[STIX]{x1D701}_{\pm }=\arctan \frac{2uv\pm d\sin \unicode[STIX]{x1D6FF}}{u^{2}-v^{2}-1\pm d\cos \unicode[STIX]{x1D6FF}}.\end{eqnarray}$$

The Jacobian of the transformation from $\hat{r}$ along the real axis to $u$ or $v$ along a steepest-descent path through a critical point of $\unicode[STIX]{x1D719}_{\pm }$ is therefore given by

(7.29) $$\begin{eqnarray}J_{u}^{\pm }=\frac{\text{d}\hat{r}}{\text{d}u}=1+\text{i}\cot (\unicode[STIX]{x1D701}_{\pm }-\unicode[STIX]{x1D70E}_{\pm }-\unicode[STIX]{x1D702}),\end{eqnarray}$$

or

(7.30) $$\begin{eqnarray}J_{v}^{\pm }=\frac{\text{d}\hat{r}}{\text{d}v}=\tan (\unicode[STIX]{x1D701}_{\pm }-\unicode[STIX]{x1D70E}_{\pm }-\unicode[STIX]{x1D702})+\text{i},\end{eqnarray}$$

depending, respectively, on whether the path in question is parametrized by $u$ or by $v$ .

We shall see below that a variable, more suitable than either $u$ or $v$ , for parametrizing the path of steepest descent through the critical point $u=\hat{r}_{C}$ , $v=0$ , is the radial distance $w=[(u-\hat{r}_{C})^{2}+v^{2}]^{1/2}$ from this point. If we mark the complex $\hat{r}$ plane by a set of polar coordinates centred on the point $u=\hat{r}_{C}$ , $v=0$ , i.e. write

(7.31) $$\begin{eqnarray}\hat{r}=u+\text{i}v=\hat{r}_{C}+w\exp (\text{i}\unicode[STIX]{x1D706}),\end{eqnarray}$$

and express the Cartesian coordinates $u$ and $v$ in (7.27) in terms of the polar coordinates $w$ and $\unicode[STIX]{x1D706}$ , we find that

(7.32) $$\begin{eqnarray}\frac{\text{d}\unicode[STIX]{x1D706}}{\text{d}w}=\frac{1}{w}\cot (\unicode[STIX]{x1D701}_{\pm }-\unicode[STIX]{x1D70E}_{\pm }-\unicode[STIX]{x1D702}+\unicode[STIX]{x1D706}).\end{eqnarray}$$

This together with (7.31) yields

(7.33) $$\begin{eqnarray}J_{w}^{\pm }=\frac{\text{d}\hat{r}}{\text{d}w}=\exp (\text{i}\unicode[STIX]{x1D706})[1+\text{i}\cot (\unicode[STIX]{x1D701}_{\pm }-\unicode[STIX]{x1D70E}_{\pm }-\unicode[STIX]{x1D702}+\unicode[STIX]{x1D706})]\end{eqnarray}$$

for the Jacobian of the transformation from $\hat{r}$ along the real axis to $w$ along a steepest-descent path through the point $\hat{r}=\hat{r}_{C}$ (i.e. through $w=0$ ).

It should be noted (for purposes of numerical evaluation of the above expressions) that the real functions that are defined in terms of an $\arctan$ , i.e. $\unicode[STIX]{x1D6FF}$ , $\unicode[STIX]{x1D70E}_{\pm }$ , $\unicode[STIX]{x1D709}_{\pm }$ , $\unicode[STIX]{x1D701}_{\pm }$ and $\unicode[STIX]{x1D706}$ , are continuous along each steepest-descent path. Any discontinuities arising from the multi-valuedness of $\arctan$ should be removed by adopting an appropriate branch of $\arctan$ . The multi-valued complex function $c_{1}(u+\text{i}v,\hat{z},\hat{r}_{P},\hat{z}_{P})$ is rendered continuous along every one of the paths described in this section by choosing the cubic root of  $\unicode[STIX]{x1D719}_{+}-\unicode[STIX]{x1D719}_{-}$ in (4.26) as follows: along the lower branch of the path ${\mathcal{L}}_{S}$ in figures 16 and 19, for which $u<0$ ,

(7.34) $$\begin{eqnarray}\displaystyle c_{1}|_{{\mathcal{L}}_{S},u<0}=\left\{\begin{array}{@{}ll@{}}[\frac{3}{4}(\unicode[STIX]{x1D719}_{+}-\unicode[STIX]{x1D719}_{-})]^{1/3},\quad & \text{arg}(\unicode[STIX]{x1D719}_{+}-\unicode[STIX]{x1D719}_{-})>0\\ \exp (2\text{i}\unicode[STIX]{x03C0}/3)[\frac{3}{4}(\unicode[STIX]{x1D719}_{+}-\unicode[STIX]{x1D719}_{-})]^{1/3},\quad & \text{arg}(\unicode[STIX]{x1D719}_{+}-\unicode[STIX]{x1D719}_{-})\leqslant 0,\end{array}\right. & & \displaystyle\end{eqnarray}$$

along the paths ${\mathcal{L}}_{C}$ in figures 16 and 18,

(7.35) $$\begin{eqnarray}\displaystyle & c_{1}|_{{\mathcal{L}}_{C}}=\left\{\begin{array}{@{}ll@{}}\exp (-2\text{i}\unicode[STIX]{x03C0}/3)[\frac{3}{4}(\unicode[STIX]{x1D719}_{+}-\unicode[STIX]{x1D719}_{-})]^{1/3},\quad & \text{arg}(\unicode[STIX]{x1D719}_{+}-\unicode[STIX]{x1D719}_{-})>0\\ \text{}[\frac{3}{4}(\unicode[STIX]{x1D719}_{+}-\unicode[STIX]{x1D719}_{-})]^{1/3},\quad & \text{arg}(\unicode[STIX]{x1D719}_{+}-\unicode[STIX]{x1D719}_{-})\leqslant 0,\end{array}\right. & \displaystyle\end{eqnarray}$$

and along all other paths $c_{1}$ is given by $[3(\unicode[STIX]{x1D719}_{+}-\unicode[STIX]{x1D719}_{-})/4]^{1/3}$ irrespective of the sign of $\text{arg}(\unicode[STIX]{x1D719}_{+}-\unicode[STIX]{x1D719}_{-})$ . Along the path ${\mathcal{L}}_{S}$ , the phase of $c_{1}(u+\text{i}v,\hat{z},\hat{r}_{P},\hat{z}_{P})$ discontinuously changes by $2\unicode[STIX]{x03C0}/3$ across the saddle point at $\hat{r}=\hat{r}_{S}$ . This correspondingly shifts the phase of the Green’s function $G_{nj}^{\text{out}}|_{{\mathcal{L}}_{S}}$ by $\unicode[STIX]{x03C0}$ across $\hat{r}=\hat{r}_{S}$ , a phase shift similar to that encountered across a focal point in optics. The choice in (7.35) is dictated by the analytic expression for the approximate value of the phase of $c_{1}|_{{\mathcal{L}}_{C}}$ in the vicinity of $\hat{r}=\hat{r}_{C}$ (see (7.46) below). It is understood that the branch cuts for any complex multi-valued functions are selected to lie outside the closed areas in the $(u,v)$ plane around which the contour integrations are performed.

7.3 Paths of steepest descent through the critical points of the phase $\unicode[STIX]{x1D719}_{-}$ for observation points in $\unicode[STIX]{x1D703}_{L}\leqslant \unicode[STIX]{x1D703}_{P}\leqslant \unicode[STIX]{x1D703}_{U}$ or $\unicode[STIX]{x03C0}-\unicode[STIX]{x1D703}_{U}\leqslant \unicode[STIX]{x1D703}_{P}\leqslant \unicode[STIX]{x03C0}-\unicode[STIX]{x1D703}_{L}$

At observation points for which the cusp locus $C$ of the bifurcation surface intersects the source distribution ${\mathcal{S}}^{\prime }$ , the $\hat{r}$ -integral in the expression for $[\begin{array}{@{}cc@{}}\boldsymbol{E}_{-}^{\text{b}} & \boldsymbol{B}_{-}^{\text{b}}\end{array}]$ in (5.8) extends over $\hat{r}_{C}\leqslant \hat{r}\leqslant \hat{r}_{U}$ , since $\text{H}(\unicode[STIX]{x1D6E5})$ in its integrand would vanish over the remaining segment $\hat{r}_{L}\leqslant \hat{r}<\hat{r}_{C}$ of the range of integration with respect to $\hat{r}$ (see § 5.1 and figure 11). The critical points of $\unicode[STIX]{x1D719}_{-}$ for the asymptotic evaluation of the $\hat{r}$ -integral for large $m$ would consist, therefore, of the locus $\hat{r}=\hat{r}_{S}$ of stationary points of the phase $\hat{\unicode[STIX]{x1D711}}_{-}$ of the exponential and the boundaries $\hat{r}=\hat{r}_{C}$ and $\hat{r}=\hat{r}_{U}$ of the domain of integration.

The path of steepest descent for the exponential factor $\exp (-\text{i}m\hat{\unicode[STIX]{x1D711}}_{-})$ , in the complex $(u,v)$ plane, through the point $u=\hat{r}_{S}$ , $v=0$ , at which $\unicode[STIX]{x1D719}_{-}(u,v)$ has a saddle point is parametrically described by the solution $v=v_{S}(u)$ of the transcendental equation

(7.36) $$\begin{eqnarray}\text{Re}[\unicode[STIX]{x1D719}_{-}(u,v)]=\unicode[STIX]{x1D719}_{-}(\hat{r}_{S},0)\equiv \unicode[STIX]{x1D719}_{S}\end{eqnarray}$$

that satisfies $v_{S}(\hat{r}_{S})=0$ and the condition

(7.37) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{S}\equiv \text{Im}[\unicode[STIX]{x1D719}_{-}(u,v)]|_{v=v_{S}(u)}\leqslant 0\end{eqnarray}$$

at all relevant values of the curve parameter $u$ and the fixed parameters $(\hat{z},\hat{r}_{P},\hat{z}_{P})$ . I denote this path which consists of two segments, one in $v<0$ and one in $v>0$ , by ${\mathcal{L}}_{S}$ (see figures 13–15). From the plots of $\unicode[STIX]{x1D6FE}_{S}$ versus $u$ , it can be seen that for $v_{S}(u)\geqslant 0$ , the condition in (7.37) is satisfied by the segment $u\geqslant \hat{r}_{S}$ of the solution $v=v_{S}(u)$ (see figure 14), while for $v_{S}(u)\leqslant 0$ , the condition in (7.37) is satisfied by the segment $u\leqslant \hat{r}_{S}$ of this solution (see figure 15).

Figure 13.

The solutions $v=v_{S}(u)$ of (7.36) for $\hat{z}=0$ , $\hat{R}_{P}=100$ and $\unicode[STIX]{x1D703}_{P}=\unicode[STIX]{x03C0}/3$ in the vicinity of the saddle point $(\hat{r}_{S},0)$ of the function $\text{Re}[\unicode[STIX]{x1D719}_{-}(u,v)]$ . As shown by figures 14 and 15, the segments here designated by ${\mathcal{L}}_{S}$ satisfy the condition in (7.37) and so constitute the paths of steepest descent through the saddle point $(\hat{r}_{S},0)$ .

Figure 14.

The function $\unicode[STIX]{x1D6FE}_{S}(u)$ , here plotted for $\hat{z}=0$ , $\hat{R}_{P}=100$ and $\unicode[STIX]{x1D703}_{P}=\unicode[STIX]{x03C0}/3$ , shows that of the two segments of the solution to (7.36) for which $v_{S}(u)\geqslant 0$ (the upper segments in figure 13), only the segment $u\geqslant \hat{r}_{S}$ (on the right) satisfies the requirement in (7.37).

Figure 15.

The function $\unicode[STIX]{x1D6FE}_{S}(u)$ , here plotted for $\hat{z}=0$ , $\hat{R}_{P}=100$ and $\unicode[STIX]{x1D703}_{P}=\unicode[STIX]{x03C0}/3$ , shows that of the two segments of the solution to (7.36) for which $v_{S}(u)\leqslant 0$ (the lower segments in figure 13), only the segment $u\leqslant \hat{r}_{S}$ (on the left) satisfies the requirement in (7.37).

Figure 16.

The complex $\hat{r}=u+\text{i}v$ plane with a shift in the position of the imaginary axis which places the saddle point $(\hat{r}_{S},0)$ of $\unicode[STIX]{x1D719}_{-}(u,v,\hat{z},\hat{R}_{P},\unicode[STIX]{x1D703}_{P})$ at the origin. The curves ${\mathcal{L}}_{S}$ , ${\mathcal{L}}_{C}$ and ${\mathcal{L}}_{U}$ delineate the paths of steepest descent of $\exp (-\text{i}m\unicode[STIX]{x1D719}_{-})$ through the following critical points, respectively: the saddle point $(\hat{r}_{S},0)$ , the cusp point $(\hat{r}_{C},0)$ and the boundary point $(\hat{r}_{U},0)$ . Here, the cusp point lies between the lower and upper boundaries $(\hat{r}_{L},0)$ and $(\hat{r}_{U},0)$ of the source distribution (see figure 11). The segment $\hat{r}_{C}+\unicode[STIX]{x1D716}\leqslant u\leqslant \hat{r}_{U}$ of the real axis, together with ${\mathcal{L}}_{S}$ , ${\mathcal{L}}_{C}$ , ${\mathcal{L}}_{U}$ and the indentation ${\mathcal{L}}_{\unicode[STIX]{x1D716}}$ , surrounding the singularity of $G_{j}^{\text{out}}|_{\unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{-}}$ at the cusp point, constitute the contours of integration for the evaluation of the part $[\boldsymbol{E}_{-}^{\text{b}}\quad \boldsymbol{B}_{-}^{\text{b}}]$ of the field given by (7.59). The arrows show the adopted directions of integration along the various contours. This figure is plotted for the following set of values of the parameters: $\hat{R}_{P}=10^{2}$ , $\unicode[STIX]{x1D703}_{P}=\unicode[STIX]{x03C0}/3$ , $\hat{z}=0$ , $m=10$ and $\hat{r}_{U}=1.15474$ .

When the cusp curve of the bifurcation surface intersects the source distribution, as in figure 11, the point $\hat{r}=\hat{r}_{C}$ (rather than $\hat{r}=\hat{r}_{L}$ ) constitutes the lower boundary of the domain of integration with respect to $\hat{r}$ in (5.8). In terms of the polar coordinates introduced in (7.31), the path of steepest descent for $\exp [-\text{i}m\unicode[STIX]{x1D719}_{-}(w,\unicode[STIX]{x1D706})]$ through the boundary point $\hat{r}=\hat{r}_{C}$ is given by the solution $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{C}^{-}(w)$ of the transcendental equation

(7.38) $$\begin{eqnarray}\text{Re}[\unicode[STIX]{x1D719}_{-}(w,\unicode[STIX]{x1D706})]=\unicode[STIX]{x1D719}_{-}|_{\hat{r}=\hat{r}_{C}}\equiv \unicode[STIX]{x1D719}_{C}\end{eqnarray}$$

that satisfies $\unicode[STIX]{x1D706}_{C}^{-}(0)=-\unicode[STIX]{x03C0}/2$ and the condition

(7.39) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{C}^{-}\equiv \text{Im}[\unicode[STIX]{x1D719}_{-}(w,\unicode[STIX]{x1D706})]|_{\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{C}^{-}(w)}\leqslant 0,\end{eqnarray}$$

for all relevant values of $(\hat{z},\hat{r}_{P},\hat{z}_{P})$ and of the curve parameter $w$ . This path is designated as ${\mathcal{L}}_{C}$ in figure 16. Note that the requirement $\unicode[STIX]{x1D706}_{C}^{-}(0)=-\unicode[STIX]{x03C0}/2$ on the path issuing from $w=0$ is dictated by the fact that, of the two solutions $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}(w)$ of (7.38) through $\hat{r}=\hat{r}_{C}$ , in $-\unicode[STIX]{x03C0}/2\leqslant \unicode[STIX]{x1D706}\leqslant 0$ and in $0\leqslant \unicode[STIX]{x1D706}\leqslant \unicode[STIX]{x03C0}/2$ , only the one reducing to $-\unicode[STIX]{x03C0}/2$ at $w=0$ for which $\text{d}\unicode[STIX]{x1D706}/\text{d}w$ is negative can satisfy (7.39) (see (7.33)).

In contrast to ${\mathcal{L}}_{S}$ that passes through the point $\hat{r}=\hat{r}_{S}$ itself, the path ${\mathcal{L}}_{C}$ cannot be used as a contour of integration that includes the point $\hat{r}=\hat{r}_{C}$ because the functions $G_{nj}^{\text{out}}|_{\unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{\pm }}$ which appear in the integrand of the integral in (5.8) are both divergent at $\hat{r}=\hat{r}_{C}$ . To be able to apply Cauchy’s integral theorem, we need to confine the domain of integration in the complex plane to one in which the integrand is analytic. This may be done in the present case by determining the nature of the singularities of $G_{nj}^{\text{out}}|_{\unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{\pm }}$ at $\hat{r}=\hat{r}_{C}$ and accordingly indenting the paths of steepest descent through this point to excise the singularities of $G_{nj}^{\text{out}}|_{\unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{\pm }}$ from the domain of integration.

To approximate $G_{nj}^{\text{out}}|_{\unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{\pm }}$ in the neighbourhood of $\hat{r}=\hat{r}_{C}$ (in order to determine the nature of their singularities at this point), we may note that $\unicode[STIX]{x1D719}_{\pm }$ can be expanded into a Taylor series in powers of $(\hat{r}-\hat{r}_{C})^{1/2}$ to obtain

(7.40) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{\pm } & = & \displaystyle \unicode[STIX]{x1D719}_{C}+\frac{\hat{r}_{C}^{2}-1}{\hat{r}_{C}\hat{R}_{C}}\left(\hat{r}-\hat{r}_{C}\right)\pm \frac{[2\hat{r}_{C}(\hat{r}_{P}^{2}-1)]^{3/2}}{3\hat{R}_{C}^{3}}\left(\hat{r}-\hat{r}_{C}\right)^{3/2}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{(\hat{r}_{C}^{2}-1)[\hat{r}_{C}^{2}(\hat{r}_{P}^{2}-1)(\hat{R}_{C}^{2}+4)+\hat{r}_{C}^{2}-1]}{2\hat{r}_{C}^{2}\hat{R}_{C}^{5}}(\hat{r}-\hat{r}_{C})^{2}\nonumber\\ \displaystyle & & \displaystyle \mp \,\frac{\hat{r}_{C}^{1/2}(\hat{r}_{P}^{2}-1)^{3/2}[3\hat{R}_{C}^{4}+4(\hat{r}_{C}^{2}-1)(3\hat{R}_{C}^{2}+5)]}{5\sqrt{2}\hat{R}_{C}^{7}}(\hat{r}-\hat{r}_{C})^{5/2}+\cdots \,,\end{eqnarray}$$

for fixed $(\hat{z},\hat{r}_{P},\hat{z}_{P})$ , where $\hat{R}_{C}=(\hat{r}_{C}^{2}\hat{r}_{P}^{2}-1)^{1/2}$ . Insertion of this in (4.26) shows that the function $c_{1}$ appearing in the expression for $G_{nj}^{\text{out}}|_{\unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{\pm }}$ in (4.55) has the value

(7.41) $$\begin{eqnarray}c_{1}\simeq \frac{2^{1/6}}{\hat{R}_{C}}\left[\hat{r}_{C}(\hat{r}_{P}^{2}-1)\left(\hat{r}-\hat{r}_{C}\right)\right]^{1/2}\left[1-\frac{3\hat{R}_{C}^{4}+4(\hat{r}_{C}^{2}-1)(3\hat{R}_{C}^{2}+5)}{20\hat{r}_{C}\hat{R}_{C}^{4}}\left(\hat{r}-\hat{r}_{C}\right)\right]\end{eqnarray}$$

near $\hat{r}=\hat{r}_{C}$ . Evaluating $c_{1}$ , $p_{j}$ and $q_{j}$ near $\hat{r}=\hat{r}_{C}$ from (7.41), (4.52) and (4.53) and inserting the results in (4.55), we arrive at

(7.42) $$\begin{eqnarray}\displaystyle & & \displaystyle G_{nj}^{\text{out}}|_{\unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{\pm }}\simeq \frac{\hat{R}_{C}^{2-n}}{3\hat{r}_{C}^{2}\hat{r}_{P}(\hat{r}_{P}^{2}-1)(\hat{r}-\hat{r}_{C})}\left[\left(\begin{array}{@{}c@{}}1\\ -\hat{R}_{C}\\ \hat{r}_{C}\hat{r}_{P}\end{array}\right)\right.\nonumber\\ \displaystyle & & \displaystyle \qquad \left.\pm \,\frac{2^{1/2}\hat{r}_{C}^{1/2}(\hat{r}_{P}^{2}-1)^{1/2}(\hat{r}-\hat{r}_{C})^{1/2}}{\hat{R}_{C}^{2}}\left(\begin{array}{@{}c@{}}2\hat{R}_{C}^{2}+3^{n-1}\\ (-1)^{n-1}\hat{R}_{C}\\ 3^{n-1}\hat{r}_{C}\hat{r}_{P}\end{array}\right)\right],\quad 0\leqslant \hat{r}-\hat{r}_{C}\ll 1.\qquad\end{eqnarray}$$

The integrand in (5.8) therefore has both a simple pole and a branch point at $\hat{r}=\hat{r}_{C}$ which should be circumvented by an indentation of the integration contour.

The semi-circular indentation designated as ${\mathcal{L}}_{\unicode[STIX]{x1D716}}$ in figure 16 is described by

(7.43) $$\begin{eqnarray}\hat{r}=\hat{r}_{C}+\unicode[STIX]{x1D716}(\hat{r}_{S}-\hat{r}_{C})\exp (\text{i}\unicode[STIX]{x1D706}),\quad \unicode[STIX]{x1D706}_{0}^{-}\leqslant \unicode[STIX]{x1D706}\leqslant -\unicode[STIX]{x03C0}/2,\end{eqnarray}$$

where $\unicode[STIX]{x1D716}\ll 1$ is a real constant and $\unicode[STIX]{x1D706}_{0}^{-}$ is the value of $\unicode[STIX]{x1D706}$ at which this circular arc intersects the path ${\mathcal{L}}_{C}$ . Since $\unicode[STIX]{x1D716}$ is small, the value of $\unicode[STIX]{x1D706}_{0}^{-}$ can be determined from the approximate solution to $\text{Re}(\unicode[STIX]{x1D719}_{-}-\unicode[STIX]{x1D719}_{C})=0$ for $w\ll 1$ .

To derive approximate solutions to (7.38) in the vicinity of $w=0$ , i.e. to find the paths ${\mathcal{L}}_{C}$ and ${\mathcal{K}}_{C}$ that are shown in figures 16 and 17 when $w\ll 1$ , we can insert (7.40) in (7.38) and set the first two terms of the resulting Taylor expansions of $\text{Re}(\unicode[STIX]{x1D719}_{\pm }-\unicode[STIX]{x1D719}_{C})$ in powers of $w^{1/2}$ equal to zero. This leads to the following two equations for the dependences $\unicode[STIX]{x1D706}_{C}^{\pm }(w)$ of $\unicode[STIX]{x1D706}$ on $w$ along ${\mathcal{K}}_{C}$ and ${\mathcal{L}}_{C}$ respectively:

(7.44) $$\begin{eqnarray}\unicode[STIX]{x1D705}\cos \left({\textstyle \frac{3}{2}}\unicode[STIX]{x1D706}_{C}^{\pm }\right)\pm 3\cos \unicode[STIX]{x1D706}_{C}^{\pm }\simeq 0,\quad w\ll 1,\end{eqnarray}$$

where

(7.45) $$\begin{eqnarray}\unicode[STIX]{x1D705}=\frac{2^{3/2}\hat{r}_{C}^{5/2}(\hat{r}_{P}^{2}-1)^{3/2}w^{1/2}}{(\hat{r}_{C}^{2}-1)(\hat{r}_{C}^{2}\hat{r}_{P}^{2}-1)}.\end{eqnarray}$$

The function $\cos ((3/2)\unicode[STIX]{x1D706}_{\pm })$ in (7.44) can be written in terms of $\cos ((1/2)\unicode[STIX]{x1D706}_{\pm })$ to obtain a cubic equation for $\cos ((1/2)\unicode[STIX]{x1D706}_{\pm })$ whose relevant roots, i.e. the roots satisfying $\lim _{w\rightarrow 0}\unicode[STIX]{x1D706}_{C}^{\pm }=-\unicode[STIX]{x03C0}/2$ and the constraints (7.39) and (7.54) below, are given by

(7.46) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{C}^{\pm }\simeq -2\arccos \left\{\mp \unicode[STIX]{x1D705}^{-1}\left[\left(1+\unicode[STIX]{x1D705}^{2}\right)^{1/2}\cos \left(\unicode[STIX]{x1D707}\pm {\textstyle \frac{2}{3}}\unicode[STIX]{x03C0}\right)+{\textstyle \frac{1}{2}}\right]\right\},\quad w\ll 1,\end{eqnarray}$$

with

(7.47) $$\begin{eqnarray}\unicode[STIX]{x1D707}=\frac{1}{3}\arccos \frac{1-\frac{3}{2}\unicode[STIX]{x1D705}^{2}}{(1+\unicode[STIX]{x1D705}^{2})^{3/2}}.\end{eqnarray}$$

Note that, in contrast to their Taylor expansions in powers of $w^{1/2}$ , the above expressions for $\unicode[STIX]{x1D706}_{C}^{\pm }$ are valid also at $\hat{z}=\hat{z}_{P}$ where $\unicode[STIX]{x1D705}$ diverges.