2.3.1. Non-equilibrium parallel regime

As an inviscid Rayleigh instability mode propagates downstream, its magnitude amplifies exponentially until it approaches the neutral position, $x_{3,n}$ say. Due to the accumulated growth, the mode is likely to enter a nonlinear stage in the vicinity of the neutral position (Goldstein & Leib Reference Goldstein and Leib1989; Wu 2019). Let this region be represented as

(2.19) \begin{equation} x_3\approx x_{3,n}+\tilde \mu \bar {x}_1, \end{equation}

where $\tilde \mu \ll 1$ is to be determined in terms of $\tilde \epsilon$ or $Re$ , and $\bar {x}_1=O(1)$ is negative. The local base-flow velocity and temperature profiles are expanded as

(2.20) \begin{equation} \Big ( {\bar {U}}(x_3, y), {\bar {T}}(x_3, y) \Big ) \approx \Big ( {\bar {U}}(x_{3,n}, y), {\bar {T}}(x_{3,n}, y) \Big ) +\tilde \mu \Big ( {\bar {U}}_1(y), {\bar {T}}_1(y) \Big )\bar {x}_1. \end{equation}

In this region, the growth rate of the mode is $O(\tilde \mu )$ , correspondingly, the amplitude develops over the length scale of $O(\tilde {\mu }^{-1})$ , and so we introduce the slow variable

(2.21) \begin{equation} \tilde {x}=\tilde \mu (x-x_0)=O(1) \quad \mbox{with}\quad x_0=Re(x_{3,n}+\tilde \mu \bar {x}_1). \end{equation}

The nonlinear evolution takes place in a region centred at $x_0$ with a length scale of $O(\tilde {\mu }^{-1})$ , which is much larger than the boundary-layer thickness.

In the presence of the instability mode and an incident sound, the disturbance in the main layer can be expressed, to leading order, as

(2.22) \begin{align} (\tilde \rho , \tilde {u}, \tilde {v}, \tilde {p}, \tilde \theta )&=\tilde {\epsilon }A(\tilde {x}) \Big ( \hat {\rho }_0(y), \hat {u}_0(y), \hat {v}_0(y), \hat {p}_0(y), \hat {\theta }_0(y) \Big )E &+\tilde \epsilon _s(\check \rho _s, \check {u}_s, \check {v}_s, \check {p}_s, \check \theta _s)E_s\nonumber\\ &\quad +\textrm{c.c.}+\cdots , \end{align}

where $E={\textrm{e}}^{\textrm{i}\alpha \zeta }$ , $\zeta =x-ct$ is the coordinate moving at the phase speed with $\alpha$ and $c=\omega /\alpha$ being the streamwise wavenumber and phase speed, respectively, and $\tilde \epsilon \ll 1$ measures the magnitude of the radiating mode with $A(\tilde {x})$ being the amplitude function describing its evolution; the second term in (2.22) represents the acoustic signature with $E_s={\textrm{e}}^{\textrm{i}\alpha _{s}(x-c_s t)}$ being its carrier wave, and $\tilde \epsilon _s\ll 1$ measures the magnitude of the sound. The derivative with respect to $x$ then becomes

(2.23) \begin{equation} \frac {\partial }{\partial {x}} \rightarrow \frac {\partial }{\partial \zeta }+\tilde \mu \frac {\partial }{\partial \tilde {x}}. \end{equation}

To derive the scaling, let us first write down the streamwise momentum equation (of the incompressible N-S equations for convenience) for the perturbation,

(2.24) \begin{equation} \bigg [ ({\bar {U}}-c)\frac {\partial }{\partial \zeta } +\tilde \mu {\bar {U}}\frac {\partial }{\partial \tilde {x}} \bigg ]\tilde {u}+\bar {U}^\prime \tilde {v} -Re^{-1}\frac {\partial ^2\tilde {u}}{\partial \tilde {y}^2}=-\frac {\partial \tilde {p}}{\partial \zeta } -\tilde {u}\frac {\partial \tilde {u}}{\partial \zeta }-\tilde {v}\frac {\partial \tilde {u}}{\partial y} -\tilde \mu \tilde {u}\frac {\partial \tilde {u}}{\partial \tilde {x}} +\cdots . \end{equation}

The scaling is fixed by considering the main and critical layers. In the main layer, the disturbance (the radiating mode and acoustic response included) remains, to the required order of accuracy, linear and inviscid, and the streamwise velocity of the radiating mode exhibits a jump of $O(\tilde {\epsilon }\tilde \mu )$ , which is to be determined by analysis of the critical layer (Goldstein & Leib Reference Goldstein and Leib1989). Suppose that the critical-layer width is $O(\delta _c)$ . It follows that the advection term in (2.24) is of $O(\delta _c)$ , whereas the terms associated with the non-equilibrium and viscous effects are $O(\tilde \mu )$ and $O(Re^{-1}/\delta _c^2)$ , respectively. The requirement that these terms are all balanced leads to

(2.25) \begin{equation} \delta _c=O\left(\tilde \mu \right), \qquad \delta _c=O\Big(Re^{-1/3}\Big). \end{equation}

In the critical layer, the wall-normal velocities of the radiating mode and acoustic response are of $O(\tilde {\epsilon })$ and $O(\tilde \epsilon _s)$ , respectively, but the logarithmic singularity $\ln (y-y_c)$ of the inviscid solutions for the streamwise velocities suggests that the corresponding vorticities, $\tilde \epsilon \hat {u}_{0,y}$ and $\tilde \epsilon _s\check {u}_{s,y}$ , are of $O(\tilde {\epsilon }\delta _c^{-1})$ and $O(\tilde \epsilon _s\delta _c^{-1})$ , respectively (Leib Reference Leib1991). Consideration of the self-nonlinear interactions of the radiating mode leading to the regeneration of the fundamental shows that if $\tilde \mu =O(\tilde {\epsilon }^{2/5})$ , the nonlinear effect enters the amplitude equation (Leib Reference Leib1991; Wu & Cowley 1995). Under this scaling, the temperature fluctuation also contributes a nonlinear effect (Goldstein & Leib Reference Goldstein and Leib1989).

However, in the common critical layer, the vorticity of the acoustic disturbance, $\tilde \epsilon _s\check {u}_{s,y}$ , is $O(\tilde \epsilon _s\delta _c^{-1})$ , as noted earlier. The forcing proportional to the product of the wall-normal velocity of the radiating mode $\tilde \epsilon \hat {v}_{0}$ and $\tilde \epsilon _s\check {u}_{s,y}$ is thus $O(\tilde \epsilon \tilde \epsilon _s\delta _c^{-1})$ . Note that this forcing is of the form $E^*E_s=E$ according to (2.17), and represents the quadratic interaction of subharmonic resonance type between the radiating mode and the acoustic signature. It induces an $O(\tilde \epsilon \tilde \epsilon _s\delta _c^{-2})$ streamwise velocity of the fundamental as is deduced by balancing the forcing and the non-equilibrium terms in (2.24). This streamwise velocity exhibits a jump across the critical layer. If it is comparable with the $O(\tilde {\epsilon }\tilde \mu )$ jump in the main layer, the subharmonic resonance effect enters the amplitude equation, leading to the characteristic threshold intensity of the sound

(2.26) \begin{equation} \tilde \epsilon _s=O\Big(\tilde \mu \delta _c^2\Big)=O\Big(\tilde \mu Re^{-2/3}\Big), \end{equation}

where use has been made of the second relation in (2.25).

With (2.25) and (2.26), we write

(2.27) \begin{equation} \tilde \mu =\tilde {\epsilon }^{2/5}=\delta _c, \qquad Re^{-1}=\lambda \tilde \mu ^3, \qquad \tilde \epsilon _s=\tilde {\epsilon }^{6/5}, \end{equation}

where $\lambda$ is the $O(1)$ Haberman parameter measuring the importance of viscosity. In terms of $Re$ , the threshold sound intensity $\tilde \epsilon _s=O(Re^{-1})$ . The scaling relations (2.27) form the basis of non-equilibrium critical-layer theory describing the mutual interaction between the incident sound and the radiating mode as well as the self-interaction of the latter.

2.3.2. Equilibrium non-parallel regime

Non-parallelism is a salient feature of boundary-layer flows, and its effect on instability may be significant especially as a mode evolves through the vicinity of its neutral positions (Smith Reference Smith1979b ; Bodonyi & Smith Reference Bodonyi and Smith1981; Smith 1989). An asymptotic approach is required to characterise systematically the combined effects of non-parallelism and nonlinearity as shown by Smith (1979a) and Hall & Smith (Reference Hall and Smith1984) for lower-branch viscous T-S instability modes. For inviscid instability modes, such as the present radiating mode, the equilibrium non-parallel regime is pertinent to the region where the length scales over which the growth rate and the amplitude vary are comparable (Wu 2005). This region is represented by

(2.28) \begin{equation} x_3=x_{3,n}+Re^{-1/2}\bar {x} \quad \mbox{with}\quad \bar {x}=O(1). \end{equation}

We take $\delta ^*$ to be the boundary-layer thickness at the neutral position, and it follows that $x_{3,n}=1$ . Inspection of (2.24) shows that in this region, the non-equilibrium effect, $\bar \mu =O(Re^{-1/2})$ , is much smaller than the viscous effect. The critical layer is thus of equilibrium type and viscosity dominated, resulting in a critical layer width $\delta _c=O(Re^{-1/3})$ . A similar scaling argument shows that the self-nonlinear effect of the radiating mode comes into play when its amplitude reaches the threshold

(2.29) \begin{equation} \tilde {\epsilon }=\delta _c^2\bar \mu ^{1/2}=O\Big(Re^{-11/12}\Big). \end{equation}

The quadratic interaction between the sound and the radiating mode regenerates the fundamental of the radiating mode with streamwise velocity of $O(\tilde \epsilon \tilde \epsilon _s\delta _c^{-2})$ . When this is comparable with the $O(\tilde \epsilon Re^{-1/2})$ jump in the outer solution, i.e. $\tilde \epsilon \tilde \epsilon _s\delta _c^{-2}=O(\tilde \epsilon Re^{-1/2})$ , the evolution of the radiating mode is affected by the incident sound wave. It follows that the threshold magnitude of the incident sound is

(2.30) \begin{equation} \tilde {\epsilon }_s=O\Big(Re^{-1/2}\delta _c^2\Big)=O\Big(Re^{-7/6}\Big), \end{equation}

which is asymptotically smaller than the $O(Re^{-1})$ threshold intensity in the non-equilibrium regime.

The local mean velocity and temperature profiles can be approximated by

(2.31) \begin{equation} \Big ( {\bar {U}}(x_3, y), {\bar {T}}(x_3, y) \Big ) \approx \Big ( {\bar {U}}(x_{3,n}, y), {\bar {T}}(x_{3,n}, y) \Big ) +Re^{-1/2}\Big ( {\bar {U}}_1(y), {\bar {T}}_1(y) \Big )\bar {x}, \end{equation}

to the required order. With the key scalings identified, the effects of the sound wave on the evolution of the radiating mode will be analysed in a self-consistent manner.

The non-equilibrium parallel and equilibrium non-parallel regimes, while distinguished, are intrinsically linked. The connection can be understood by noting that (2.30), also follows from (2.26) by setting $\tilde \mu =O(Re^{-1/2})$ . The relation (2.26) holds in both regimes, and may be rewritten as

(2.32) \begin{equation} \tilde \mu =\tilde \epsilon _s Re^{2/3}, \end{equation}

which determines the growth-rate modification by the impinging sound in terms of its intensity. The modification is comparable to the local unmodified growth rate in the streamwise region specified by (2.19). The non-equilibrium parallel and equilibrium non-parallel regimes correspond to the distinguished thresholds $\tilde \epsilon _s=O(Re^{-1})$ and $O(Re^{-7/6})$ , respectively. As $\tilde \epsilon _s$ is reduced from $O(Re^{-1})$ to $O(Re^{-7/6})$ , the non-equilibrium parallel regime acquires the character of the equilibrium non-parallel regime. This observation forms the basis for constructing the composite amplitude equation in § 4.

The scaling relations derived earlier are pertinent to a two-dimensional radiating mode and an associated incident wave. We realise that a planar radiating mode can be continued to a band of oblique ones, each of which, $(\alpha , \beta , \omega )$ say, may be in resonance with an impinging sound wave $(2\alpha , 2\beta , 2\omega )$ . Moreover, when a pair of oblique modes $(\alpha , \pm \beta , \omega )$ are present, subharmonic triadic resonance (cf. Wu Reference Wu and Cowley1995) can take place between them and a planar sound wave $(2\alpha , 0, 2\omega )$ . In these cases, the interactions are associated with the simple-pole singularity of the spanwise and streamwise velocities of the oblique mode(s), and the required threshold amplitude of the incident sound wave(s) would be $\tilde \epsilon _s=O(Re^{-4/3})$ and $O(Re^{-3/2})$ in the non-equilibrium parallel and equilibrium non-parallel regimes, respectively, as can be deduced by a similar scaling argument. These scenarios are interesting but are left for future investigation.

3.1.1. Main layer

In the main layer, the disturbance expands as

(3.2) \begin{align} \big(\tilde \rho , \tilde {u}, \tilde {v}, \tilde {p}, \tilde \theta \big)&= \tilde {\epsilon }\big [ \tilde {A}\big(\tilde {x}\big)\big(\hat {\rho }_0, \hat {u}_0, \hat {v}_0, \hat {p}_0, \hat {\theta }_0\big) +\tilde \mu \big(\hat {\rho }_1, \hat {u}_1, \hat {v}_1, \hat {p}_1, \hat {\theta }_1\big) \big ]E \\ &\quad \nonumber +\tilde \epsilon _s\big(\check \rho _s, \check {u}_s, \check {v}_s, \check {p}_s, \check \theta _s\big)E_s+\textrm{c.c.}+\cdots , \end{align}

where the first term represents a nearly neutral radiating mode, which propagates from upstream with $\tilde {A}(\tilde {x})$ being its amplitude function, and $E={\textrm{e}}^{\textrm{i}\alpha (x-ct)}$ its carrier wave. Here, the second term is the deviation from neutrality, while the third term represents the impinging acoustic perturbation with $E_s={\textrm{e}}^{2\textrm{i}\alpha (x-ct)+2\textrm{i}\tilde {\alpha }_d\tilde {x}}$ being its carrier wave.

Substitution of (3.2) into (2.2) followed by linearisation yields the leading-order equations for the mode,

(3.3a) \begin{align} &\textrm{i}\alpha ({\bar {U}}-c)\hat {\rho }_0+\bar {R}^\prime \hat {v}_0+\bar {R}(\textrm{i}\alpha \hat {u}_0+\hat {v}_0^\prime )=0,\end{align}

(3.3b) \begin{align} &\qquad\qquad\textrm{i}\alpha ({\bar {U}}-c)\hat {u}_0+{\bar {U}}^\prime \hat {v}_0=-\textrm{i}\alpha {\bar {T}}\hat {p}_0,\end{align}

(3.3c) \begin{align} & \qquad\qquad \qquad\textrm{i}\alpha ({\bar {U}}-c)\hat {v}_0=-{\bar {T}}\hat {p}_0^\prime ,\end{align}

(3.3d) \begin{align} &\textrm{i}\alpha ({\bar {U}}-c)\hat {\theta }_0+{\bar {T}}^\prime \hat {v}_0 =\textrm{i}\alpha (\gamma -1)Ma^2({\bar {U}}-c){\bar {T}}\hat {p}_0,\end{align}

(3.3e) \begin{align} &\qquad\qquad\qquad\gamma Ma^2 \hat {p}_0=\bar {R}\hat {\theta }_0+{\bar {T}}\hat {\rho }_0.\end{align}

Elimination of $\hat {\rho }_0$ , $\hat {u}_0$ , $\hat {v}_0$ and $\hat {\theta }_0$ leads to the compressible Rayleigh equation for $\hat {p}_0$ ,

(3.4) \begin{equation} \mathscr{L}\hat {p}_0\equiv \left \{ \frac {\partial ^2}{\partial y^2} +\left ( \frac {{\bar {T}}^\prime }{{\bar {T}}}-\frac {2{\bar {U}}^\prime }{{\bar {U}}-c} \right )\frac {\partial }{\partial y} -\alpha ^2\left [ 1-\frac {Ma^2({\bar {U}}-c)^2}{{\bar {T}}} \right ] \right \}\hat {p}_0=0. \end{equation}

As $y\rightarrow \infty$ ,

(3.5) \begin{equation} \hat {p}_0\sim \mathscr{C}_{\infty }{\textrm{e}}^{-\textrm{i}\alpha q y} \quad \mbox{with}\quad q=\sqrt {Ma^2(1-c)^2-1}, \end{equation}

which represents an outgoing wave that persists away from the main layer. Here, $\mathscr{C}_{\infty }$ is a constant that is determined by normalisation of the eigenfunction.

The solution near the critical level $y_c$ is obtained using the Frobenius method (Leib Reference Leib1991; Boyce & DiPrima Reference Boyce and DiPrima2012) as

(3.6) \begin{equation} \hat {p}_0\sim \frac {\bar {U}_c^\prime }{\bar {T}_c}\bigg \{ \frac {\alpha ^2}{3}a^{\pm }\phi _a+\phi _b +\frac {\alpha ^2}{3}\bigg ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \bigg )\ln |{\hat {\eta }}|\phi _a \bigg \}, \end{equation}

where ${\hat {\eta }}\equiv y-y_c\rightarrow 0$ , and

(3.7) \begin{equation} \phi _{a}={\hat {\eta }}^3+\chi _{a}{\hat {\eta }}^4+\cdots , \qquad \phi _{b}=1-\frac {\alpha ^2}{2}{\hat {\eta }}^2+\chi _{b}{\hat {\eta }}^4+\cdots , \end{equation}

with $\chi _a$ and $\chi _b$ being constants, whose expressions are given by (3.8) and (3.9) of Qin & Wu (Reference Qin and Wu2024) with $\alpha$ replacing $\alpha _{s}$ , and are reproduced in Appendix A for completeness; here, the subscript $c$ represents the value evaluated at the critical level. It follows from (3.3b )–(3.3d ) that as ${\hat {\eta }}\rightarrow 0$ ,

(3.8) \begin{equation} \hat {u}_0\sim -\bigg ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \bigg )\ln |{\hat {\eta }}| +\frac {5}{6}\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime }-\frac {1}{3}\frac {\bar {T}_c^\prime }{\bar {T}_c}-a^{\pm }+\cdots , \end{equation}

(3.9) \begin{equation} \hat {v}_0\sim -\textrm{i}\alpha \bigg \{ 1-\bigg ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \bigg ){\hat {\eta }}\ln |{\hat {\eta }}| +\bigg ( \frac {2}{3}\frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {1}{6}\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime }-a^{\pm } \bigg ){\hat {\eta }} +\cdots \bigg \}, \end{equation}

(3.10) \begin{equation} \hat {\theta }_0\sim \frac {\bar {T}_c^\prime }{\bar {U}_c^\prime {\hat {\eta }}}. \end{equation}

The temperature perturbation exhibits the same simple-pole singularity as that of an inflectional mode (Goldstein & Leib Reference Goldstein and Leib1989), while the logarithmic singularity is present only for a supersonic mode, whose critical level does not correspond to the generalised inflection point (Leib Reference Leib1991). Equation (3.8) indicates a jump of $\hat {u}_0$ ,

(3.11) \begin{equation} \hat {u}_0^+ - \hat {u}_0^- = -(a^+ - a^-)+\cdots . \end{equation}

At the next order, the second terms in the expansion (3.2), $(\hat {\rho }_1, \hat {u}_1, \hat {v}_1, \hat {p}_1, \hat {\theta }_1)$ , are found to satisfy the inhomogeneous version of (3.3a )–(3.3e ), which are given in Appendix A. By eliminating $\hat {\rho }_1$ , $\hat {u}_1$ , $\hat {v}_1$ and $\hat {\theta }_1$ , we can show that $\hat {p}_1$ satisfies an inhomogeneous Rayleigh equation (Wu 2005; Qin & Wu Reference Qin and Wu2024),

(3.12) \begin{equation} \mathscr{L}\hat {p}_1=\tilde {A}^\prime \frac {2\textrm{i} c}{\alpha }\bigg \{ \frac {\bar {U}^\prime \hat {p}_0^\prime }{({\bar {U}}-c)^2} +\frac {\alpha ^2}{c}\bigg [ \frac {Ma^2{\bar {U}}({\bar {U}}-c)}{{\bar {T}}}-1 \bigg ]\hat {p}_0 \bigg \} -\bar {x}_1 \tilde {A}\Delta _1, \end{equation}

where we have put

(3.13) \begin{equation} \varDelta _1\!=\!\bigg \{\! \frac {2\bar {U}^\prime }{{\bar {U}}-c}\bigg ( \frac {{\bar {U}}_1}{{\bar {U}}-c}-\frac {{\bar {U}}_1^\prime }{\bar {U}^\prime } \bigg ) +\frac {\bar {T}^\prime }{{\bar {T}}}\bigg ( \frac {{\bar {T}}_1^\prime }{\bar {T}^\prime }-\frac {{\bar {T}}_1}{{\bar {T}}} \bigg )\! \bigg \}\hat {p}_0^\prime +\alpha ^2 Ma^2\frac {({\bar {U}}-c)^2}{{\bar {T}}}\bigg ( \!\frac {2{\bar {U}}_1}{{\bar {U}}-c}-\frac {{\bar {T}}_1}{{\bar {T}}}\! \bigg )\hat {p}_0. \end{equation}

As $y\rightarrow \infty$ ,

(3.14) \begin{equation} \hat {p}_1\sim p_R(\tilde {x}){\textrm{e}}^{-\textrm{i}\alpha q y} +q^{-1}[1-Ma^2(1-c)]\mathscr{C}_{\infty }\tilde {A}^\prime y{\textrm{e}}^{-\textrm{i}\alpha q y}. \end{equation}

However, by the method of dominant balance, we deduce that as $y\rightarrow y_c$ ,

(3.15) \begin{align} \hat {p}_1 &\sim \frac {\alpha ^2}{\bar {T}_c}\bigg ( \frac {\textrm{i} c}{\alpha }\tilde {A}^\prime -{\bar {U}}_{1c}\bar {x}_1 \tilde {A} \bigg ) \bigg \{ {\hat {\eta }}-\bigg ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \bigg ){\hat {\eta }}^2\ln |{\hat {\eta }}| \\ \nonumber &\quad \, -\bigg [ a^{\pm }+\frac {1}{3}\bigg ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \bigg ) \bigg ]{\hat {\eta }}^2 +\frac {1}{3}j{\hat {\eta }}^3\ln |{\hat {\eta }}| \bigg \}+\frac {\bar {U}_c^\prime }{\bar {T}_c}(\textrm{i}\alpha \tilde {A}^\prime ){\hat {\eta }}^2 \\ \nonumber &\quad \, +\left ( \frac {\alpha ^2\bar {U}_c^\prime }{3\bar {T}_c}\bar {x}_1 \tilde {A} \right )j_1{\hat {\eta }}^3\ln |{\hat {\eta }}| +c^{\pm }\phi _a +d\bigg [ \phi _b+\frac {\alpha ^2}{3}\bigg ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \bigg )\ln |{\hat {\eta }}|\phi _a \bigg ], \end{align}

where $c^{\pm }$ and $d$ are arbitrary functions of $\tilde {x}$ , and

(3.16) \begin{align} j=\frac {\bar {T}_c^{\prime \prime }}{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime \prime }}{\bar {U}_c^\prime } -\bigg ( \frac {\bar {T}_c^\prime }{\bar {T}_c} \bigg )^2 +\bigg ( \frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \bigg )^2 +3\bigg ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \bigg )^2 -2\bigg ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \bigg )\frac {\bar {U}_c^\prime }{\bar {U}_c},\nonumber \\ j_1=\frac {\bar {T}_c^\prime }{\bar {T}_c}\bigg ( \frac {{\bar {T}}_{1c}^\prime }{\bar {T}_c^\prime }-\frac {{\bar {T}}_{1c}}{\bar {T}_c} \bigg ) +\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime }\bigg ( \frac {{\bar {U}}_{1c}^\prime }{\bar {U}_c^\prime }-\frac {{\bar {U}}_{1c}^{\prime \prime }}{\bar {U}_c^{\prime \prime }} \bigg ) -2\bigg ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \bigg )\frac {{\bar {U}}_{1c}}{\bar {U}_c}. \end{align}

It follows from the $x$ - and $y$ -momentum equations, (A3b) and (A3c ), that

(3.17) \begin{equation} \hat {u}_1=-\frac {{\bar {T}}}{{\bar {U}}-c}\hat {p}_1-\frac {{\bar {T}}\bar {U}^\prime }{\alpha ^2({\bar {U}}-c)^2}\hat {p}_{1,y}+\cdots . \end{equation}

Thus, the jump of $\hat {u}_1$ is found to relate to $(c^+ - c^-)$ by the equation

(3.18) \begin{equation} \hat {u}_1^+ - \hat {u}_1^- = -\frac {3\bar {T}_c}{\alpha ^2\bar {U}_c^\prime }(c^+ - c^-)+\cdots . \end{equation}

For the inhomogeneous equation (3.12) to have an acceptable solution, it has to satisfy a solvability condition. This can be derived by multiplying both sides of (3.12) by $\hat {p}_0{\bar {T}}/({\bar {U}}-c)^2$ and integrating from $0$ to $\infty$ , leading to

(3.19) \begin{align} 3(c^{+}-c^{-})-\frac {2\alpha ^2}{\bar {T}_c} \left ( \frac {\textrm{i} c}{\alpha }\tilde {A}^\prime -{\bar {U}}_{1c}\bar {x}_1 \tilde {A} \right ) \left ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \right )(a^{+}-a^{-}) -\alpha ^2 d(a^{+}-a^{-})\nonumber \\ =-\bar {U}_c^\prime \left (\frac {2\textrm{i} c}{\alpha }I_2 \tilde {A}^\prime -I_1\bar {x}_1 \tilde {A}\right ), \end{align}

where use has been made of the impermeability condition $\hat {p}_0^\prime (0)=\hat {p}_{1,y}|_{y=0}=0$ and the far-field condition (3.5) and (3.14), and the expressions for the integrals, $I_1$ and $I_2$ , are given by (4.19) and (4.20) of Qin & Wu (Reference Qin and Wu2024), respectively, and are reproduced in Appendix A for completeness. The jumps $(a^{+}-a^{-})$ and $(c^{+}-c^{-})$ will be determined by analysing the critical-layer dynamics.

The acoustic components, $(\check \rho _s, \check {u}_s, \check {v}_s, \check {p}_s, \check \theta _s)$ , satisfy the same equation as (3.3) with $\alpha _{s}$ and $c_s$ replacing $\alpha$ and $c$ , respectively. Elimination of $\check \rho _s$ , $\check {u}_s$ , $\check {v}_s$ and $\check \theta _s$ leads to the familiar compressible Rayleigh equation for pressure $\check {p}_s$ , $\mathscr{L}{\check {p}_s}=0$ . This equation must be solved subject to the impermeability condition, $\check {p}_{s,y}(0)=0$ . In the far field, the pressure takes the form

(3.20) \begin{equation} \check {p}_s\sim p_I\left[{\textrm{e}}^{\textrm{i}\gamma _{s} y}+\mathscr{R}\,{\textrm{e}}^{-\textrm{i}\gamma _{s} y}\right] \quad \mbox{as}\quad y\rightarrow \infty , \end{equation}

which consists of an incident wave and a reflected wave, where $\gamma _{s}=\alpha _{s}\sqrt {Ma^2(1-c_s)^2-1}$ , and $\mathscr{R}$ is the reflection coefficient. As ${\hat {\eta }}\rightarrow 0$ , the local solution to $\mathscr{L}{\check {p}_s}=0$ is constructed by using the Frobenius method as

(3.21) \begin{equation} \check {p}_s=a_s^\pm \phi _{sa}+b_s^\pm \left [ \phi _{sb} +\frac {\alpha _{s}^2}{3}\left ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \right )\ln |{\hat {\eta }}|\phi _{sa} \right ], \end{equation}

where $\phi _{sa}$ and $\phi _{sb}$ have the same expressions as $\phi _a$ and $\phi _b$ given by (3.7), with $\alpha _{s}$ replacing $\alpha$ . The pressure, velocities and temperature of the acoustic disturbance have the expressions

(3.22a) \begin{equation} \check {p}_s=b_s^\pm \left [ 1-\frac {\alpha _{s}^2}{2}{\hat {\eta }}^2+\frac {\alpha _{s}^2}{3} \left ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \right ){\hat {\eta }}^3\ln |{\hat {\eta }}| +(a_s^\pm /b_s^\pm ){\hat {\eta }}^3 \right ]+O({\hat {\eta }}^4\ln |{\hat {\eta }}|), \end{equation}

(3.22b) \begin{equation} \check {v}_s=-\textrm{i}\alpha _{s} b_s^\pm \frac {\bar {T}_c}{\bar {U}_c^\prime }\left [ \!1- \left ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \right ){\hat {\eta }}\ln |{\hat {\eta }}| +\left ( \frac {2}{3}\frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {1}{6}\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } -\frac {3a_s^\pm }{\alpha _{s}^2 b_s^\pm } \right ){\hat {\eta }}\! \right ]\!+\!O({\hat {\eta }}^2\ln |{\hat {\eta }}|), \end{equation}

(3.22c) \begin{equation} \check {u}_s=b_s^\pm \frac {\bar {T}_c}{\bar {U}_c^\prime }\left [ -\left ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \right )\ln |{\hat {\eta }}| +\frac {5}{6}\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime }-\frac {1}{3}\frac {\bar {T}_c^\prime }{\bar {T}_c} -\frac {3a_s^\pm }{\alpha _{s}^2 b_s^\pm } \right ]+O({\hat {\eta }}\ln |{\hat {\eta }}|), \end{equation}

(3.22d) \begin{equation} \check \theta _s=b_s^\pm \frac {\bar {T}_c\bar {T}_c^\prime }{(\bar {U}_c^\prime )^2}\left [ \frac {1}{{\hat {\eta }}} -\left ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \right )\ln |{\hat {\eta }}| \right ]+O(1). \end{equation}

It is worth noting that in the main layer and free stream, the acoustic wave and radiating mode bear much resemblance, both being governed by linearised Euler equations and having similar forms of solutions. Clearly, the temperature perturbation $\check \theta _s$ and the streamwise velocity $\check {u}_s$ exhibit a simple-pole singularity and a logarithmic singularity, respectively, indicating that the main-layer solution breaks down as ${\hat {\eta }}\rightarrow 0$ . Thus we need to analyse the critical layer.

3.1.2. Critical layer

The singularity of the main-layer solution is to be removed by introducing viscous effects within the critical layer, which determine the critical layer width to be of order $Re^{-1/3}$ , which is $O(\tilde {\epsilon }^{2/5})$ , and so the appropriate local transverse coordinate is

(3.23) \begin{equation} Y=(y-y_c)/\tilde {\epsilon }^{2/5}. \end{equation}

The asymptote of the inviscid solution, (3.6) and (3.8)–(3.10), suggests that the perturbation in the critical layer expands as

(3.24a) \begin{align} \tilde {u}=\tilde {\epsilon }\Big(U_{1}E+\tilde {\epsilon }^{1/5}U_{M}+\tilde {\epsilon }^{2/5}U_{2}E\Big) +\tilde \epsilon _s U_{s1}E_s+\textrm{c.c.}+\cdots , \\[-12pt] \nonumber \end{align}

(3.24b) \begin{align} \tilde {v}=\tilde {\epsilon }\Big(V_{0}E+\tilde {\epsilon }^{2/5}V_{1}E+\tilde {\epsilon }^{3/5}V_{M} +\tilde {\epsilon }^{4/5}V_{2}E\Big)+\tilde \epsilon _s V_{s0}E_s+\textrm{c.c.}+\cdots , \\[-12pt] \nonumber \end{align}

(3.24c) \begin{align} \tilde {p}=\tilde {\epsilon }\Big(P_{0}E+\tilde {\epsilon }^{2/5}P_{1}E+\tilde {\epsilon }^{4/5}P_{2}E\Big) +\tilde \epsilon _s P_{s0}E_s+\textrm{c.c.}+\cdots , \\[-12pt] \nonumber \end{align}

(3.24d) \begin{align} \tilde \theta =\tilde {\epsilon }^{3/5}\Big(\Theta _{1}E+\tilde {\epsilon }^{1/5}\Theta _{M} +\tilde {\epsilon }^{2/5}\Theta _{2}E\Big)+\tilde \epsilon _s\tilde {\epsilon }^{-2/5}\Theta _{s1}E_s+\textrm{c.c.}+\cdots ,\end{align}

where the subscript $M$ denotes the mean-flow distortion. Strictly speaking, the expansions actually contain logarithm terms, but they are not needed in the calculation of the jumps and hence are not written out for brevity. Substituting (3.24) into (2.2) and noting (2.23) and (3.23), we then obtain the equations governing the terms at different orders in the expansion.

At leading order, inspection of the $x$ - and $y$ -momentum equations gives

(3.25) \begin{equation} P_0=\Big(\bar {U}_c^\prime /\bar {T}_c\Big)\tilde {A}, \qquad V_0=-\textrm{i}\alpha \tilde {A}. \end{equation}

Expansion of the energy equation shows that

(3.26) \begin{equation} \mathscr{L}_p\Theta _{1}+{\bar {T}_c^\prime }V_{0}=0, \end{equation}

with the operator $\mathscr{L}_p$ being defined by

(3.27) \begin{equation} \mathscr{L}_p=c\frac {\partial }{\partial \tilde {x}}+ \textrm{i}\alpha \Big(\bar {U}_c^\prime Y+{\bar {U}}_{1c}\bar {x}_1\Big)-\lambda \bar {T}_c\bar {\mu }_c Pr^{-1}\frac {\partial ^2}{\partial Y^2}. \end{equation}

Equation (3.26) is solved by use of Fourier transform to give

(3.28) \begin{equation} \Theta _1=\textrm{i}\alpha \bar {T}_c^\prime \int _{0}^{\infty } \exp \Big(-s_p\xi ^3-\textrm{i}\alpha \bar {U}_c^\prime \bar {Y}\xi \Big)\tilde {A}(\tilde {x}-c\xi )\,\textrm {d}\xi , \end{equation}

where we have put $\bar {Y}\equiv Y+({\bar {U}}_{1c}/\bar {U}_c^\prime )\bar {x}_1$ and $s_p={\textstyle {1}/{3}}\lambda (\alpha \bar {U}_c^\prime )^2\bar {T}_c\bar {\mu }_c Pr^{-1}$ .

At the next order, expansion of the continuity and $x$ -momentum equations yields

(3.29a) \begin{align} & -c\Theta _{1,\tilde {x}}/\bar {T}_c^2+\textrm{i}\alpha \Big(\bar {U}_c^\prime Y+{\bar {U}}_{1c}\bar {x}_1\Big)\Big(-\Theta _{1}/\bar {T}_c^2\Big) +\frac {1}{\bar {T}_c}\left(\textrm{i}\alpha U_1+V_{1,Y}\right)-\frac {\bar {T}_c^\prime }{\bar {T}_c^2}V_{0}=0,\end{align}

(3.29b) \begin{align}& \mathscr{L}_\mu U_{1}+{\bar {U}_c^\prime }V_{1}+\left({\bar {U}_c^{\prime \prime }}Y+{\bar {U}}_{1c}^\prime \bar {x}_1\right)V_{0} =-\textrm{i}\alpha \bar {T}_c P_1-\bar {T}_c P_{0,\tilde {x}}-\textrm{i}\alpha \left(\bar {T}_c^\prime Y+{\bar {T}}_{1c}\bar {x}_1\right)P_0\nonumber \\ &\qquad\qquad\qquad +\lambda \bar {T}_c\bar {\mu }_c^\prime \bar {U}_c^\prime \Theta _{1,Y},\end{align}

where $\bar {\mu }_c^\prime =(\textrm {d}\bar \mu /\textrm {d}{\bar {T}})|_{y=y_c}$ , and $\mathscr{L}_\mu$ is the same as $\mathscr{L}_p$ provided that $Pr$ is set to unity; here, the subscript $Y$ denotes the differentiation with respect to $Y$ . These two equations are combined to obtain an equation for $U_{1,Y}$ , which is solved to give

(3.30) \begin{align} U_{1,Y}\!=\!\textrm{i}\alpha \bar {U}_c^\prime \frac {\bar {T}_c^\prime \left(\bar {T}_c\bar {\mu }_c^\prime\! -\!\bar {\mu }_c Pr^{-1}\right)}{\bar {T}_c\bar {\mu }_c(1\!-\!Pr^{-1})} \!\!\int _{0}^{\infty } \left[ \!1\!-\!{\textrm{e}}^{-(s_p-s)\xi ^3} \right ] \exp \Big(\!-s\xi ^3-\textrm{i}\alpha \bar {U}_c^\prime \bar {Y}\xi \Big)\tilde {A}\!\left(\tilde {x}-c\xi \right)\textrm {d}\xi\nonumber \\ -\textrm{i}\alpha \bar {U}_c^\prime \bigg ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \bigg ) \!\!\int _{0}^{\infty } \exp \Big(-s\xi ^3-\textrm{i}\alpha \bar {U}_c^\prime \bar {Y}\xi \Big)\tilde {A}\left(\tilde {x}-c\xi \right)\textrm {d}\xi , \end{align}

where $s={\textstyle {1}/{3}}\lambda (\alpha \bar {U}_c^\prime )^2\bar {T}_c\bar {\mu }_c$ . Matching $U_1$ with its outer counterpart determines the jump

(3.31) \begin{equation} a^+ - a^- = \bigg ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \bigg )\pi \textrm{i}. \end{equation}

We now consider the acoustic component. At leading order, the $y$ -momentum equation gives $P_{s0,Y}=0$ . Matching with the main-layer solution leads to

(3.32) \begin{equation} b_s^+=P_{s0}=b_s^-\equiv b_s. \end{equation}

The continuity, $x$ -momentum and energy equations for the leading-order terms read $V_{s0,Y}=0$ , and

(3.33a) \begin{align} \bar {U}_c^\prime V_{s0}&=-\textrm{i}\alpha _{s}\bar {T}_c P_{s0},\end{align}

(3.33b) \begin{align} \textrm{i}\alpha _{s}\bar {U}_c^\prime Y \Theta _{s0}+\bar {T}_c^\prime V_{s0}&=\lambda \bar {T}_c\bar {\mu }_c Pr^{-1}\Theta _{s0,YY},\end{align}

respectively. The solution is found to be

(3.34a) \begin{align} & \qquad\qquad\qquad\qquad\qquad V_{s0}=-\textrm{i}\alpha _{s} b_s\bar {T}_c/\bar {U}_c^\prime ,\end{align}

(3.34b) \begin{align} \Theta _{s0}&=\textrm{i}\alpha _{s}\frac {\bar {T}_c\bar {T}_c^\prime }{\bar {U}_c^\prime }b_s \int _{0}^{\infty } \exp \Big(-s_{sp}\xi ^3-\textrm{i}\alpha _{s}\bar {U}_c^\prime Y\xi \Big)\textrm {d}\xi \, \mbox{with}\quad s_{sp}={\textstyle \frac {1}{3}}\lambda \left(\alpha _{s}\bar {U}_c^\prime \right)^2\bar {T}_c\bar {\mu }_c Pr^{-1}.\end{align}

At the next order, the continuity and $x$ -momentum equations yield

(3.35a) \begin{align} \textrm{i}\alpha _{s}\bar {U}_c^\prime Y(-\Theta _{s0}/\bar {T}_c^2)+(\textrm{i}\alpha _{s} U_{s1}+V_{s1,Y})/\bar {T}_c -(\bar {T}_c^\prime /\bar {T}_c^2)V_{s0}=0, \\[-10pt] \nonumber \end{align}

(3.35b) \begin{align} \mathscr{L}_s U_{s1}+\bar {U}_c^\prime V_{s1}+\bar {U}_c^{\prime \prime } Y V_{s0} =-\textrm{i}\alpha _{s}\bar {T}_c^\prime Y P_{s0}+\lambda \bar {T}_c\bar {U}_c^\prime \bar {\mu }_c^\prime \Theta _{s0,Y},\end{align}

where the operator $\mathscr{L}_s$ is defined by

(3.36) \begin{equation} \mathscr{L}_s=\textrm{i}\alpha _{s}\bar {U}_c^\prime Y-\lambda \bar {T}_c\bar {\mu }_c\frac {\partial ^2}{\partial Y^2}. \end{equation}

Eliminating $V_{s1}$ between (3.35a ) and (3.35b ), and solving the resulting equation, we obtain

(3.37) \begin{align} U_{s1,Y}=\textrm{i}\alpha _{s}\frac {\bar {T}_c^\prime (\bar {T}_c\bar {\mu }_c^\prime -\bar {\mu }_c Pr^{-1})}{\bar {\mu }_c(1-Pr^{-1})}b_s \!\!\int _{0}^{\infty }\! \Big [ 1\!-\!\exp (-(s_{sp}-s_s)\xi ^3) \Big ]\exp (-s_s\xi ^3\!-\!\textrm{i}\alpha _{s}\bar {U}_c^\prime Y\xi )\,\textrm {d}\xi\nonumber \\ -\textrm{i}\alpha _{s}\bar {T}_c\bigg ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \bigg )b_s \int _{0}^{\infty } \exp (-s_s\xi ^3-\textrm{i}\alpha _{s}\bar {U}_c^\prime Y\xi )\,\textrm {d}\xi , \end{align}

where $s_s={\textstyle {1}/{3}}\lambda (\alpha _{s}\bar {U}_c^\prime )^2\bar {T}_c\bar {\mu }_c$ . Matching $U_{s1}$ with its outer counterpart (3.22c ) determines the jump

(3.38) \begin{equation} a_s^+ - a_s^- = \frac {\alpha _{s}^2}{3} \bigg ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \bigg )b_s\pi \textrm{i}. \end{equation}

This jump is used together with (3.21) to solve the Rayleigh equation numerically where by we obtain the reflection coefficient $\mathscr{R}$ and the boundary-layer response $b_s$ ; the details were given by Qin & Wu (Reference Qin and Wu2024). There it was showed that both $\mathscr{R}$ and $b_s$ are extraordinarily large for a small subset of the sound incident angle and frequency. Furthermore, resonant over-reflection occurs for a specific pairing of incidence angle and frequency, and the reflected wave coincides with a locally neutral radiating mode.

The solution for the mean-flow distortion is presented in Appendix B.1. We now proceed to consider the fundamental regenerated by the cubic interaction, as well as the interaction between the acoustic component and the leading-order fundamental. The governing equations are found to be

(3.39a) \begin{align} \bigg [ c\frac {\partial }{\partial \tilde {x}}+\textrm{i}\alpha (\bar {U}_c^\prime Y+{\bar {U}}_{1c}\bar {x}_1) \bigg ](-\Theta _{2}/\bar {T}_c^2) +\frac {1}{\bar {T}_c}(\textrm{i}\alpha U_2+V_{2,Y}) =\frac {V_0}{\bar {T}_c^2}\Theta _{M,Y}\nonumber \\ +\frac {1}{\bar {T}_c^2}(V^*_0\Theta _{s1,Y}+V_{s0}\Theta ^*_{1,Y}){\textrm{e}}^{2\textrm{i}\tilde {\alpha }_d\tilde {x}}+\cdots , \end{align}

(3.39b) \begin{align} \mathscr{L}_\mu U_{2}+{\bar {U}_c^\prime }V_{2} =-\textrm{i}\alpha \bar {T}_c P_2-V_0 U_{M,Y}-\textrm{i}\alpha P_0\Theta _M+\lambda \bar {T}_c\bar {\mu }_c^\prime \bar {U}_c^\prime \Theta _{2,Y} \\ \nonumber +(-V^*_0 U_{s1,Y}-V_{s0} U^*_{1,Y}+\textrm{i}\alpha P^*_0\Theta _{s1}-2\textrm{i}\alpha P_{s0}\Theta ^*_{1}) {\textrm{e}}^{2\textrm{i}\tilde {\alpha }_d\tilde {x}}+\cdots , \end{align}

(3.39c) \begin{align} \mathscr{L}_p\Theta _2=-V_0\Theta _{M,Y}+(-V^*_0\Theta _{s1,Y}-V_{s0}\Theta ^*_{1,Y}) {\textrm{e}}^{2\textrm{i}\tilde {\alpha }_d\tilde {x}}+\cdots .\end{align}

Equation (3.39c ) is solved first to give

(3.40) \begin{align} \Theta _2 = -\textrm{i}\alpha ^5(\bar {U}_c^\prime )^2\bar {T}_c^\prime \int _{0}^{\infty }\!\!\!\int _{0}^{\infty }\!\!\!\int _{-\zeta }^{\infty } \zeta ^2\exp [-s_p\xi ^3-\textrm{i}\alpha \bar {U}_c^\prime \bar {Y}\xi -2s_p\zeta ^3-3s_p\zeta ^2\eta ] \nonumber \\ \times \tilde {A}(\tilde {x}-c\xi -c\zeta )\tilde {A}(\tilde {x}-c\xi -c\zeta -c\eta ) \tilde {A}^*(\tilde {x}-c\xi -2c\zeta -c\eta )\,\textrm {d}\xi \,\textrm {d}\eta \,\textrm {d}\zeta \nonumber \\ +\textrm{i}\alpha ^3\bar {T}_c\bar {T}_c^\prime b_s\bigg [ \int _{0}^{\infty }\!\!\!\int _{-\zeta }^{\infty } 2\zeta \exp [-s_p\xi ^3-\textrm{i}\alpha \bar {U}_c^\prime \bar {Y}\xi -2s_p\zeta ^3] \tilde {A}^*(\tilde {x}-c\xi -2c\zeta )\,\textrm {d}\xi \,\textrm {d}\zeta \nonumber \\ +\int _{0}^{\infty }\!\!\!\int _{\zeta }^{\infty }\! \zeta \exp [-s_p\xi ^3-\textrm{i}\alpha \bar {U}_c^\prime \bar {Y}\xi +s_p\zeta ^3/2] \tilde {A}^*(\tilde {x}-c\xi +c\zeta )\,\textrm {d}\xi \,\textrm {d}\zeta \bigg ]. \end{align}

Equations (3.39a ) and (3.39b ) can be reduced to

(3.41) \begin{align} \mathscr{L}_\mu U_{2,Y}=&-V_0 U_{M,YY}-\textrm{i}\alpha P_0\Theta _{M,Y} +\lambda \bar {U}_c^\prime (\bar {T}_c\bar {\mu }_c^\prime -\bar {\mu }_c Pr^{-1})\Theta _{2,YY} \nonumber \\ &+(-V^*_0 U_{s1,YY}-V_{s0} U^*_{1,YY}+\textrm{i}\alpha P^*_0\Theta _{s1,Y}-2\textrm{i}\alpha P_{s0}\Theta ^*_{1,Y}) {\textrm{e}}^{2\textrm{i}\tilde {\alpha }_d\tilde {x}}+\cdots . \end{align}

The jump of $U_2$ is obtained by Fourier transforming (3.41) and setting the transform variable in the solution to zero. Matching $U_2$ with the outer solution (3.18) yields the jump

(3.42) \begin{align} c^{+}-c^{-}=\frac {1}{3}\frac {\alpha ^2}{\bar {T}_c} \bigg ( \frac {\textrm{i} c}{\alpha }\tilde {A}^\prime -{\bar {U}}_{1c}\bar {x}_1 \tilde {A} \bigg )j\pi \textrm{i} +\bigg ( \frac {\alpha ^2\bar {U}_c^\prime }{3\bar {T}_c}\bar {x}_1 \tilde {A} \bigg )j_1\pi \textrm{i} +d\frac {\alpha ^2}{3}\bigg ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \bigg )\pi \textrm{i}\nonumber \\ -\frac {2\pi \textrm{i}\alpha ^6(\bar {U}_c^\prime )^3\bar {T}_c^\prime }{3\bar {T}_c^2} \int _{0}^{\infty }\!\!\!\int _{0}^{\infty }\!\! K(\xi ,\eta )\tilde {A}(\tilde {x}-c\xi ) \tilde {A}(\tilde {x}-c\xi -c\eta )\tilde {A}^*(\tilde {x}-2c\xi -c\eta )\,\textrm {d}\eta \textrm {d}\xi\nonumber \\ +\frac {4\pi \textrm{i}\alpha ^4\bar {U}_c^\prime \bar {T}_c^\prime b_s}{3\bar {T}_c} \int _{0}^{\infty } {\textrm{e}}^{2\textrm{i}\tilde {\alpha }_d(\tilde {x}-c\xi )} K_b(\xi )\tilde {A}^*(\tilde {x}-2c\xi )\,\textrm {d}\xi , \end{align}

where the linear part of the jump corresponds to the $(-\pi )$ phase jump of the logarithmic singularity in (3.15), and the kernel functions

(3.43) \begin{align} & K(\xi ,\eta )\!=\!\xi ^2\exp [-s(2\xi ^3\!+\!3\xi ^2\eta )]\bigg \{ \!\exp [-(s_p\!-\!s)\xi ^3]\! +\!\exp [-(s_p\!-\!s)\xi ^3-3(s_p-s)\xi ^2\eta ]\nonumber \\ & \quad -\frac {\bar {T}_c\bar {\mu }_c^\prime -\bar {\mu }_c Pr^{-1}}{\bar {\mu }_c(1-Pr^{-1})} \Big [ 1-\exp (-(s_p-s)(2\xi ^3+3\xi ^2\eta )) \Big ] +\frac {\bar {T}_c}{\bar {T}_c^\prime }\bigg ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \bigg ) \bigg \}, \\[-12pt]\nonumber \end{align}

(3.44) \begin{align} & K_b(\xi )=\xi {\textrm{e}}^{-2s\xi ^3}\bigg \{ {\textrm{e}}^{-(s_p-s)\xi ^3} -\frac {\bar {T}_c\bar {\mu }_c^\prime -\bar {\mu }_c Pr^{-1}}{\bar {\mu }_c(1-Pr^{-1})}\Big [ 1-{\textrm{e}}^{-2(s_p-s)\xi ^3} \Big ] +\frac {\bar {T}_c}{\bar {T}_c^\prime }\bigg ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \bigg ) \bigg \}.\end{align}

The last term in (3.42) is contributed by the subharmonic parametric resonance between the radiating mode and the boundary-layer response to the incident wave.

Inserting the jumps (3.31) and (3.42) into (3.19), we obtain the evolution equation for the amplitude function $\tilde {A}$ (cf. Qin & Wu Reference Qin and Wu2024),

(3.45) \begin{align} \tilde {A}^\prime (\tilde {x})& =\sigma \bar {x}_1 \tilde {A}+\Upsilon \int _{0}^{\infty }\!\!\!\int _{0}^{\infty } K(\xi ,\eta ) \tilde {A}(\tilde {x}-c\xi )\tilde {A}(\tilde {x}-c\xi -c\eta )\tilde {A}^*(\tilde {x}-2c\xi -c\eta )\,\textrm {d}\eta \textrm {d}\xi\nonumber \\ & \quad +\Upsilon _b\int _{0}^{\infty } {\textrm{e}}^{2\textrm{i}\tilde {\alpha }_d(\tilde {x}-c\xi )} K_b(\xi )\tilde {A}^*(\tilde {x}-2c\xi )\,\textrm {d}\xi , \end{align}

where

(3.46) \begin{equation} \sigma =(-\textrm{i}\alpha /c)\bigg \{ I_1+\frac {\alpha ^2}{\bar {T}_c} \bigg [ \frac {{\bar {U}}_{1c}}{\bar {U}_c^\prime } \bigg ( j-2\bigg ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \bigg )^2 \bigg ) -j_1 \bigg ]\pi \textrm{i} \bigg \}\bigg /G, \end{equation}

(3.47) \begin{equation} \Upsilon =2\pi \alpha ^7(\bar {U}_c^\prime )^2\bar {T}_c^\prime /(c\bar {T}_c^2G), \qquad \Upsilon _b=-4\pi \alpha ^5\bar {T}_c^\prime b_s/(c\bar {T}_c G), \end{equation}

with $b_s=b_s(p_I)$ given by (3.32) and

(3.48) \begin{align} G=2I_2+\frac {\alpha ^2}{\bar {T}_c\bar {U}_c^\prime }\bigg [ j-2\bigg ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \bigg )^2 \bigg ]\pi \textrm{i}. \end{align}

In addition to the cubic self-nonlinear term, the amplitude equation (3.45) consists also of a linear non-local term, which accounts for the impact of the incident sound on the evolution of the radiating mode.

3.1.3. Effects of incident sound on the linear growth rate

The solution to the amplitude equation (3.45) can be written as

(3.49) \begin{equation} \tilde {A}=A_0(\tilde {x}){\textrm{e}}^{\textrm{i}\tilde \alpha _d\tilde {x}}. \end{equation}

By substituting (3.49) into (3.45), we obtain

(3.50) \begin{align} A_0^\prime (\tilde {x})& =(\kappa _{0}-\textrm{i}\tilde \alpha _d)A_0\nonumber \\ & \quad +\Upsilon \int _{0}^{\infty }\!\!\!\int _{0}^{\infty }\! K(\xi ,\eta ) A_0(\tilde {x}-c\xi )A_0(\tilde {x}-c\xi -c\eta )A_0^*(\tilde {x}-2c\xi -c\eta )\,\textrm {d}\eta \textrm {d}\xi \nonumber \\ & \quad +\Upsilon _b\int _{0}^{\infty }\! K_b(\xi )A_0^*(\tilde {x}-2c\xi )\,\textrm {d}\xi , \end{align}

where we have put $\kappa _{0}=\sigma \bar {x}_1$ . In the upstream regime, where the amplitude of the radiating mode is small, the nonlinear term in (3.50) can be neglected, leading to the equation

(3.51) \begin{equation} A_0^\prime (\tilde {x})=(\kappa _{0}-\textrm{i}\tilde \alpha _d)A_0 +\Upsilon _b\int _{0}^{\infty }\! K_b(\xi )A_0^*(\tilde {x}-2c\xi )\,\textrm {d}\xi \quad \mbox{as}\quad \tilde {x}\rightarrow -\infty . \end{equation}

Let us seek a solution of the form

(3.52) \begin{equation} A_0=A_{0r}+\textrm{i} A_{0i}, \qquad (A_{0r},A_{0i})=(a_r,a_i){\textrm{e}}^{\kappa \tilde {x}}+\textrm{c.c.}, \end{equation}

where $a_r$ and $a_i$ are both complex constants, and so is $\kappa$ with its real part representing the linear growth rate modified by the incident sound. Then, $A_0$ can be written as

(3.53) \begin{equation} A_0=(a_r+\textrm{i} a_i){\textrm{e}}^{\kappa \tilde {x}}+(a_r^*+\textrm{i} a_i^*){\textrm{e}}^{\kappa ^*\tilde {x}}. \end{equation}

Substitution of (3.53) into (3.51) and collection of terms proportional to ${\textrm{e}}^{\kappa \tilde {x}}$ and ${\textrm{e}}^{\kappa ^*\tilde {x}}$ lead to the eigenvalue problem,

(3.54) \begin{equation} \begin{pmatrix} \kappa _{0r}+\Upsilon _{br}\chi _b & -\kappa _{0i}+\tilde \alpha _d+\Upsilon _{bi}\chi _b \\ \kappa _{0i}-\tilde \alpha _d+\Upsilon _{bi}\chi _b & \kappa _{0r}-\Upsilon _{br}\chi _b \end{pmatrix} \begin{pmatrix} a_r \\ a_i \end{pmatrix}=\kappa \begin{pmatrix} a_r \\ a_i \end{pmatrix}, \end{equation}

where we have put

(3.55) \begin{equation} \chi _b(\kappa )=\int _{0}^{\infty }\! K_b(\xi ){\textrm{e}}^{-2\kappa c\xi }\,\textrm {d}\xi . \end{equation}

The requirement for non-zero solutions $(a_r, a_i)$ gives the characteristic equation for $\kappa$ ,

(3.56) \begin{equation} \mathcal{F}\equiv (\kappa -\kappa _{0r})^2+(\kappa _{0i}-\tilde \alpha _d)^2 -(|\Upsilon _{b}|\chi _{b})^2=0. \end{equation}

The function $\mathcal{F}$ is real if $\kappa$ is, and thus if the equation $\mathcal{F}=0$ admits a complex root $\kappa$ , so is its complex conjugate. The equation is solved numerically using Newton iteration. The corresponding eigenvector $(a_r, a_i)$ is obtained from (3.54) and written as

(3.57) \begin{equation} a_r=a_0 a_{r0}\equiv |a_0||a_{r0}|{\textrm{e}}^{\textrm{i}(\theta _1+\phi _0)}, \qquad a_i=a_0 a_{i0}\equiv |a_0||a_{i0}|{\textrm{e}}^{\textrm{i}(\theta _2+\phi _0)}, \end{equation}

where $(a_r, a_i)$ is normalised such that $|a_{r0}|^2+|a_{i0}|^2=1$ , and $a_0=|a_0|{\textrm{e}}^{\textrm{i}\phi _0}$ is a complex constant. Thus, the perturbation $A_0$ can be written as

(3.58) \begin{equation} A_0=2|a_0|{\textrm{e}}^{\kappa _r\tilde {x}}\Big [ |a_{r0}|\cos (\kappa _i\tilde {x}+\theta _1+\phi _0) +\textrm{i}|a_{i0}|\cos (\kappa _i\tilde {x}+\theta _2+\phi _0) \Big ]. \end{equation}

The coefficient of the subharmonic resonance term in (3.45) is proportional to $b_s$ , which depends on $p_I$ . We now estimate its values pertinent to applications. The noise level $\tilde {p}_s$ in conventional wind tunnels is of $O(10^{-3})$ (Masutti et al. Reference Masutti, Spinosa, Chazot and Carbonaro2012; Cerminara et al. Reference Cerminara, Durant, André, Sandham and Taylor2019), but in flight conditions, it can be one to two orders of magnitude lower (Schneider Reference Schneider2008, Reference Schneider2015). Thus, we estimate that $\tilde {p}_s=O(10^{-4})$ . Typical Reynolds numbers are in the range of $10^4$ – $10^5$ . Recall that the pressure of the acoustic wave takes the form $\tilde {p}_s=\tilde \epsilon _s p_I{\textrm{e}}^{\textrm{i}(\alpha _{s}{x}+\gamma _{s}{y}-\omega _{s}{t})}+\textrm{c.c.}$ It follows that

(3.59)

Figure 2.

Effects on $\kappa$ of the impinging sound intensity with $\tilde \alpha _d=-1$ and $\lambda =1$ : ( $a$ ) $\kappa _r$ and ( $b$ ) $\kappa _i$ . The dashed line represents the linear growth rate in the absence of the incident sound.

Figure 3.

Effects on $\kappa$ of the detuning with $p_I=5$ and $\lambda =1$ : ( $a$ ) $\kappa _r$ and ( $b$ ) $\kappa _i$ . The dashed line represents the linear growth rate in the absence of the incident sound.

Figure 2 shows the effects of $p_I$ on the modified growth rate $\kappa _r$ . Depending on the position $\bar {x}_1$ and the detuned parameter $\tilde \alpha _d$ , $\kappa _r$ is either enhanced or reduced by the impinging sound. The modified growth rate increases with $p_I$ in a narrow region corresponding to a moderate range of $\bar {x}_1$ , beyond which it decreases with $p_I$ over a broader range, suggesting that the sound plays a stabilising role. As $|\bar {x}_1|\rightarrow \infty$ , $\kappa _r$ approaches the unperturbed growth rate $\kappa _{0r}$ . This can be inferred by examining (3.56). Since $\kappa _{0r}$ and $\kappa _{0i}$ are proportional to $\bar {x}_1$ , the dominant balance for large $|\bar {x}_1|$ yields $(\kappa -\kappa _{0r})^2+(\kappa _{0i}-\tilde {\alpha }_d)^2=0$ so that $\kappa _r=\kappa _{0r}$ and $\kappa _i=\pm (\kappa _{0i}-\tilde {\alpha }_d)$ . Since the function $\mathcal{F}$ is real for real values of $\kappa$ , it may admit real roots for certain parameters. Indeed, the solution changes from a complex conjugate pair to a single real root at certain positions, as is shown by the trend of $\kappa _i$ in figure 2( $b$ ). Destabilising effect of the subharmonic resonance is observed primarily in the regions where $\kappa$ is real. As $p_I$ increases to a high value, the range over which the acoustic wave destabilises the radiating mode broadens, and the peak modified growth rate becomes significantly larger. These trends align with the behaviour illustrated in figure 2( $a$ ). Figure 3 shows the variation of the modified growth rate with $\bar {x}_1$ for three values of $\tilde \alpha _d$ . Altering $\tilde \alpha _d$ shifts the distribution of both $\kappa _r$ and $\kappa _i$ . Again, the complex roots $\kappa$ develop into real roots in a narrow region in each case. All these results indicate that for the parameters examined, the impinging sound in general weakens slightly the linear development of the radiating mode, but in a small region, it enhances significantly the modal growth.

3.1.4. Nonlinear evolution

We proceed to consider effects of the impinging sound on the nonlinear development by solving numerically the nonlinear amplitude equation (3.50), subject to the appropriate initial condition (3.58) as $\tilde {x}\rightarrow -\infty$ . The six-order Adams–Moulton method is used, and the integral terms are approximated by Simpson’s rule. Results are shown in figures 4 and 5. Note that the impinging sound wave modifies the linear growth rate only slightly in the case shown in figure 4, but significantly in the case displayed in figure 5. The amplitude always develops into a singularity at a finite distance. Increasing the intensity of the sound wave advances appreciably its occurrence (figures 4 $a$ and 5 $a$ ). This is so even though the effect on the linear growth rate is rather limited (figure 4 $a$ ). We also examine the effects of viscosity on the nonlinear development, which generally delays the occurrence of the finite-distance singularity (figures 4 $b$ and 5 $b$ ), similar to what has been found previously for free-mode evolution (Goldstein & Leib Reference Goldstein and Leib1989; Leib Reference Leib1991) and for fundamental resonance with impinging sound (Qin & Wu Reference Qin and Wu2024). One may note that the solutions in figure 4 are more oscillatory than those in figure 5. This is due to the fact that linear modes in figure 4 have non-zero $\kappa _i$ , which renders the solutions oscillatory as (3.58) indicates, whereas those in figure 5 are non-oscillatory with $\kappa _i=0$ .

Figure 4.

Effects of incident sound intensity and viscosity on the nonlinear evolution of the radiating mode. ( $a$ ) Nonlinear development for different values of $p_I$ with $\lambda =1$ . ( $b$ ) Nonlinear development for different values of $\lambda$ with $p_I=40$ . The dashed lines represent the corresponding linear solution (3.58). Here, we have taken $\bar {x}_1=-4$ , $\tilde \alpha _d=-1$ and $a_0=1$ .

Figure 5.

Effects of incident sound intensity and viscosity on the nonlinear evolution of the radiating mode. ( $a$ ) Nonlinear development for different values of $p_I$ with $\lambda =1$ . ( $b$ ) Nonlinear development for different values of $\lambda$ with $p_I=10$ . The dashed lines represent the corresponding linear solution (3.58). Here, we have taken $\bar {x}_1=-7.25$ , $\tilde \alpha _d=-1$ and $a_0=1$ .

3.2.1. Main layer

The analysis in the main layer is only slightly modified with expansion (3.2) being replaced by

(3.61) \begin{align} (\tilde \rho , \tilde {u}, \tilde {v}, \tilde {p}, \tilde \theta )&= \bar {\epsilon }\Big [ \bar {A}(\bar {x})(\hat {\rho }_0, \hat {u}_0, \hat {v}_0, \hat {p}_0, \hat {\theta }_0) +Re^{-1/2}(\hat {\rho }_1, \hat {u}_1, \hat {v}_1, \hat {p}_1, \hat {\theta }_1) \Big ]E \\ &\quad \nonumber +\,\bar \epsilon _s(\check \rho _s, \check {u}_s, \check {v}_s, \check {p}_s, \check \theta _s)E_s+\textrm{c.c.}+\cdots . \end{align}

The corresponding solvability condition (3.19) becomes

(3.62) \begin{align} 3(c^{+}-c^{-})&-\frac {2\alpha ^2}{\bar {T}_c} \left ( \frac {\textrm{i} c}{\alpha }\bar {A}^\prime -{\bar {U}}_{1c}\bar {x} \bar {A} \right ) \left ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \right )(a^{+}-a^{-}) -\alpha ^2 d(a^{+}-a^{-})\nonumber \\ &\quad =-\bar {U}_c^\prime \left (\frac {2\textrm{i} c}{\alpha }I_2 \bar {A}^\prime -I_1\bar {x} \bar {A}\right ), \end{align}

which corresponds to (3.19) but with the parameter $\bar {x}_1$ being replaced by variable $\bar {x}$ . The jumps $(a^{+}-a^{-})$ and $(c^{+}-c^{-})$ are determined by analysing the critical-layer dynamics.

3.2.2. Critical layer

The singularity in the main-layer solution is removed by reintroducing viscosity in the critical layer with an $O(Re^{-1/3})$ width, and so the appropriate local transverse coordinate is

(3.63) \begin{equation} Y=(y-y_c)/Re^{-1/3}. \end{equation}

The perturbation expands as

(3.64a) \begin{align} \tilde {u}&=\bar {\epsilon }(U_{1}E+Re^{-1/6}U_{2}E+Re^{-1/4}U_{M}+Re^{-1/2}U_{3}E) +\bar \epsilon _s U_{s1}E_s+\textrm{c.c.}+\cdots , \qquad\qquad \\[-28pt] \nonumber \end{align}

(3.64b) \begin{align} \tilde {v}&=\bar {\epsilon }(V_{0}E + Re^{-1/3}V_{1}E + Re^{-1/2}V_{2}E + Re^{-7/12}V_{M} + Re^{-5/6}V_{3}E) + \bar \epsilon _s V_{s0}E_s + \textrm{c.c.} + \cdots , \quad \\[-28pt] \nonumber \end{align}

(3.64c) \begin{align} \tilde {p}&=\bar {\epsilon }(P_{0}E+Re^{-1/3}P_{1}E+Re^{-1/2}P_{2}E+Re^{-5/6}P_{3}E) +\bar \epsilon _s P_{s0}E_s+\textrm{c.c.}+\cdots , \qquad\qquad \\[-28pt] \nonumber \end{align}

(3.64d) \begin{align} \tilde \theta &=\bar {\epsilon }Re^{1/3}(\Theta _{1}E\!+\!Re^{-1/6}\Theta _{2}E\!+\!Re^{-1/4}\Theta _{M}\!+\!Re^{-1/2}\Theta _{3}E) \!+\!\bar \epsilon _s Re^{1/3}\Theta _{s1}E_s\!+\!\textrm{c.c.}\!+\!\cdots . \\[-1pt] \nonumber \end{align}

Substituting (3.64) into (2.2) and noting (2.28) and (3.63), we then obtain the equations governing the terms at different orders in the expansion.

At leading order, inspection of the $x$ -momentum and $y$ -momentum equations gives

(3.65) \begin{equation} P_0=(\bar {U}_c^\prime /\bar {T}_c)\bar {A}, \qquad V_0=-\textrm{i}\alpha \bar {A}. \end{equation}

Expansion of the energy equation shows that

(3.66) \begin{equation} \bar {\mathscr{L}}_p\Theta _{1}+{\bar {T}_c^\prime }V_{0}=0, \end{equation}

where the operator $\bar {\mathscr{L}}_p$ is defined by

(3.67) \begin{equation} \bar {\mathscr{L}}_p=\textrm{i}\alpha \bar {U}_c^\prime Y-\bar {T}_c\bar {\mu }_c Pr^{-1}\frac {\partial ^2}{\partial Y^2}. \end{equation}

Equation (3.66) is solved by use of Fourier transform to give

(3.68) \begin{equation} \Theta _1=\textrm{i}\alpha {\bar {T}_c^\prime }\bar {A} \int _{0}^{\infty } \exp (-s_p\xi ^3-\textrm{i}\alpha \bar {U}_c^\prime Y\xi )\,\textrm {d}\xi , \end{equation}

where $s_p$ is the same as that given in (3.28) provided that $\lambda$ is set to unity.

At the next order, expansion of the continuity and $x$ -momentum equations yields two coupled equations for $U_1$ and $V_1$ . These equations are combined to obtain an equation for $U_{1,Y}$ , which is solved to give

(3.69) \begin{align} U_{1,Y}&=\textrm{i}\alpha \bar {U}_c^\prime \frac {\bar {T}_c^\prime (\bar {T}_c\bar {\mu }_c^\prime -\bar {\mu }_c Pr^{-1})}{\bar {T}_c\bar {\mu }_c(1-Pr^{-1})}\bar {A}\! \int _{0}^{\infty } [1-\exp (-(s_p-s)\xi ^3)]\exp (-s\xi ^3-\textrm{i}\alpha \bar {U}_c^\prime Y\xi )\,\textrm {d}\xi\nonumber \\ &\quad -\textrm{i}\alpha \bar {U}_c^\prime \bar {A}\bigg ( \frac {\bar {T}_c^\prime }{\bar {T}_c}-\frac {\bar {U}_c^{\prime \prime }}{\bar {U}_c^\prime } \bigg ) \int _{0}^{\infty } \exp (-s\xi ^3-\textrm{i}\alpha \bar {U}_c^\prime Y\xi )\,\textrm {d}\xi , \end{align}

where $s$ is the same as that given in (3.30) provided that $\lambda$ is set to unity. Matching $U_1$ with its outer counterpart determines the jump, which is the same as (3.31) (as expected).

The acoustic signature is the same as that given in § 3.1.2 provided that $\lambda$ is set to unity. The pressure, wall-normal velocity, temperature and streamwise velocity components of the acoustic disturbance are given by (3.32), (3.34a ), (3.34b ) and (3.37), respectively.

Further analysis of the critical layer is related to Appendix B.2, where we present at quadratic order the solution for the mean-flow distortion, and at the cubic level, the jump $(c^{+}-c^{-})$ is determined and given by (B10).

Inserting the jumps (3.31) and (B10) into (3.62), we obtain the amplitude equation for $\bar {A}$ (cf. Qin & Wu Reference Qin and Wu2024),

(3.70) \begin{equation} \bar {A}^\prime (\bar {x})=\sigma \bar {x}\bar {A}+l\bar {A}|\bar {A}|^2 +l_b\bar {A}^*{\textrm{e}}^{2\textrm{i}\bar \alpha _d\bar {x}}, \end{equation}

where $\sigma$ is given in (3.46), and

(3.71) \begin{equation} l=\textrm{i}\alpha \Lambda /(c\bar {U}_c^\prime G), \qquad l_b=\textrm{i}\alpha \Lambda _b/(c\bar {U}_c^\prime G). \end{equation}

As expected, both the self-nonlinearity and the subharmonic resonance terms are now local. The amplitude equation (3.70) describes the nonlinear evolution of the radiating mode under the influence of the incident sound in the regime where non-parallelism is important.