1. Introduction
A dusty plasma consists of charged micrometer-sized dust particles, free electrons, free ions and neutrals (Rao, Shukla & Yu Reference Rao, Shukla and Yu1990; Mendis & Rosenberg Reference Mendis and Rosenberg1994; Barkan et al. Reference Barkan, Merlino and D’Angelo1995, Reference Barkan, D’Angelo and Merlino1996; I et al. Reference I, Juan, Chiang and Chu1996; Murillo Reference Murillo2000; Kalman et al. Reference Kalman, Hartmann, Donkó and Rosenberg2004; Morfill & Ivlev Reference Morfill and Ivlev2009; Ghosh et al. Reference Ghosh, Gupta, Chakrabarti and Chaudhuri2011; Shukla & Eliasson Reference Shukla and Eliasson2012; Melzer et al. Reference Melzer, Schella, Schablinski, Block and Piel2013; Thomas et al. Reference Thomas, Konopka, Merlino, R. and Rosenberg2016). The interaction potential between two dust particles is typically described by the Yukawa potential
$\phi (r) = ({q}/{4\pi \varepsilon _{0}a})\exp ({a}/{\lambda _{D}})$
, where
$a$
is the distance between the particles,
$q$
is the dust particle charge,
$\varepsilon _0$
is the vacuum permittivity and
$\lambda _{D}$
is the Debye length of the dusty plasma.
For simplicity and generality, two parameters are commonly introduced in a dusty plasma (Feng et al. Reference Feng, Lin, Li and Wang2016; Li & Duan Reference Li and Duan2021). One is the coupling parameter,
$\Gamma =({Q^{2}}/{4\pi \varepsilon _{0}a}) ({1}/{k_{B}T_{d}})$
, and the other is the screening parameter,
$\kappa = ({a}/{\lambda _{D}})$
, where
$k_B$
is the Boltzmann constant and
$T_{d}$
is the temperature of the dust particle fluid. A dusty plasma can be in the gas, the liquid or the solid state, which depends on the values of both parameters
$\Gamma$
and
$\kappa$
. For example, if
$\Gamma \geqslant 10^{2}$
, a dusty plasma may form a dusty plasma crystal.
There is a substantial body of research on dusty plasma (Thomas et al. Reference Thomas, Morfill, Demmel, Goree, Feuerbacher and Möhlmann1994; Homann et al. Reference Homann, Melzer, Peters and Piel1997; Melzer et al. Reference Melzer, Nunomura, Samsonov, Ma and Goree2000; Nunomura et al. Reference Nunomura, Goree, Hu, Wang and Bhattacharjee2002; Liu, Avinash & Goree Reference Liu, Avinash and Goree2003; Nosenko, Goree & Piel Reference Nosenko, Goree and Piel2006; Koukouloyannis & Kourakis Reference Koukouloyannis and Kourakis2009; Heinrich, Kim & Merlino Reference Heinrich, Kim and Merlino2009; Feng, Goree & Liu Reference Feng, Goree and Liu2010; Oxtoby et al. Reference Oxtoby, Griffith, Durniak, Ralph and Samsonov2013; Marciante & Murillo Reference Marciante and Murillo2017; Lin, Murillo & Feng Reference Lin, Murillo and Feng2019; Houwe et al. Reference Houwe, Abbagari, Inc, Betchewe, Doka and Crépin2022). However, most of the previous work has focused on dusty plasmas composed of identical dust particles. Nevertheless, experimental results show that dusty plasmas in both space and laboratory environments contain a variety of dust particles (Horanyi & Goertz Reference Horanyi and Goertz1990; Chow, Mendis & Rosenberg Reference Chow, Mendis and Rosenberg1993; Meuris, Verheest & Lakhina Reference Meuris, Verheest and Lakhina1997; Duan Reference Duan2001). It has been found that the dust size distribution plays a crucial role in the characteristics of a dusty plasma (Duan & Parkes Reference Duan and Parkes2003; Duan & Shi Reference Duan and Shi2003; Duan et al. Reference Duan, Yang, Shi and Lü2007).
Recently, dusty plasmas containing two or more different microparticles with different sizes have been studied (Smith et al. Reference Smith, Hyde, Matthews, Reay, Cook and Schmoke2008; Hartmann et al. Reference Hartmann, Donk, Kalman, Kyrkos, Golden and Rosenberg2009; Ivlev et al. Reference Ivlev, Zhdanov, Thomas and Morfill2009; Jiang et al. Reference Jiang, Hou, Ivlev, Li, Du, Thomas, Morfill and Sütterlin2011; Wieben, Schablinski & Block Reference Wieben, Schablinski and Block2017). A series of experiments on binary complex plasmas over a broad range of parameters has also been conducted (Du et al. Reference Du2012; Killer et al. Reference Killer, Bockwoldt, Schütt, Himpel, Melzer and Piel2016).
The wave phenomenon in plasmas has attracted considerable attention (Sergeev & Grach Reference Sergeev and Grach2014; He et al. Reference He, Zhan, Zhang and Ge2016; Xu & Song Reference Xu and Song2019). The waves in a binary dusty plasma have also been studied (Yang et al. Reference Yang, Schwabe, Zhdanov, Thomas, Lipaev, Molotkov, Fortov, Zhang and Du2017; Sun et al. Reference Sun2018). The slow dynamics in dusty plasmas with two different dust sizes were experimentally studied by Du et al. (Reference Du, Nosenko, Thomas, Lin, Morfill and Ivlev2019). Moreover, the reflection and transmission of a solitary wave across an interface in a binary complex plasma have been examined numerically, experimentally and analytically (Hong et al. Reference Hong, Sun, Schwabe, Du and Duan2021).
A dusty plasma usually contains different species of dust particles both in space plasma and in laboratory experiments. However, most previous studies on the dusty plasma focussed on the dust plasma in which all the dust particles are assumed to be the same. How a wave propagates in an inhomogeneous dusty plasma composed of different species of the dust particles remains an interesting question with many unresolved problems. Recently, experimental investigations on a dust acoustic wave propagating to an interface were reported (Du et al. Reference Du, Nosenko, Thomas, Lin, Morfill and Ivlev2019; Kumar et al. Reference Kumar, Bandyopadhyay, Singh, Arora and Sen2021). Subsequently, studies examined how an incident pulse wave is transmitted and reflected by several impurity dust particles (Wei et al. Reference Wei, Peng, Yang, Wang, Yang and Duan2023). Following the previous works, the present paper investigates how both the linear wave and the nonlinear wave are reflected and transmitted by an interface, as reported in experiments (Du et al. Reference Du, Nosenko, Thomas, Lin, Morfill and Ivlev2019). It has been found that both the transmitted and reflected waves not only depend on the incident wave, but also are strongly influenced by the mass ratio between the two regions. It appears that the dust acoustic wave cannot propagate through the dust lattice if the wavelength is smaller than the dust lattice constant. Additionally, it has been observed that at least one transmitted solitary wave is generated when an incident solitary wave reaches a discontinuous interface. However, there is at most one reflected solitary wave. Based on these results, an experiment can be designed to estimate the mass of the dust particles and the distribution of different dust species by sending an incident wave through the inhomogeneous region and measuring the transmitted and reflected wave information.
This paper is organised as follows. Section 2 establishes the model used in the present study. Section 3 gives linear wave propagation and its dispersion relation. Section 4 discusses the continuity condition when a wave propagates across a discontinuous interface. Section 5 examines the transmission and reflection of an incident solitary wave by the discontinuous interface. Section 6 provides the conclusion for the present paper.
2. Model
It has been recently reported that a binary complex plasma was formed by injecting two types of particles under microgravity conditions in the PK-3 Plus laboratory onboard the International Space Station (ISS). One type consists of melamine formaldehyde particles with a diameter of 2.55
$\mu$
m and a mass of
$m = 1.34\times 10^{-14} \text { kg}$
, while the other consists of SiO
$_2$
particles with a diameter of 1.55
$\mu$
m and a mass of
$m = 3.6\times 10^{-15} \text { kg}$
. Due to differences in particle properties, phase separation occurred, leading to the formation of an interface between the two types of particles (Du et al. Reference Du, Nosenko, Thomas, Lin, Morfill and Ivlev2019; Hong et al. Reference Hong, Sun, Schwabe, Du and Duan2021). In this study, we consider a strongly coupled regime where the coupling strength satisfies
$\Gamma \gt 100$
, ensuring that the dusty plasma is in a solid-state condition. Based on this, we now consider an inhomogeneous dust particle chain consisting of an homogeneous chain of
$N$
identical dust particles from
$i=-N+1$
to
$i=0$
, and another homogeneous chain of
$N$
identical dust particles from
$i=1$
to
$i=N$
. However, the dust particles differ in the regions
$i=-N+1$
to
$i=0$
(
$x\lt 0$
) and
$i=1$
to
$i=N$
(
$x\gt 0$
), as shown in figure 1.
Schematic of a one-dimensional dust particle chain, in which the blue spheres are one kind of dust particles, while the magenta spheres are another kind of dust particles.
We now aim to understand the following phenomena. Suppose that a wave propagates in the region
$i\lt 0$
(
$x\lt 0$
) and moves towards the region
$i\gt 0$
(
$x\gt 0$
). When it reaches the discontinuity at
$x=0$
, it will evolve into a transmitted wave in the region
$x\gt 0$
and a reflected wave in the region
$x\lt 0$
. In this case, the region
$x\gt 0$
will contain only the transmitted wave, while the region
$x\lt 0$
will contain both the incident and reflected wave.
When there are perturbations in a dust particle chain, we assume that the position of dust particle
$i$
is given by
$x_i = ia_0 + \xi _i$
, where
$\xi _i$
is the displacement of dust particle
$i$
from its equilibrium position
$ia_0$
and
$a_0$
is the lattice constant representing the distance between neighbouring dust particles in the equilibrium state. For simplicity and convenience, we express
$\xi _i$
as
$\xi _i^Z$
, where
$Z$
can be
$I$
,
$R$
or
$T$
for the incident, reflected or transmitted wave, respectively.
For an arbitrary dust particle
$i$
, the equation of motion is given by
$m_{d}\ddot {\boldsymbol {\xi }}_i= -q \nabla \sum \limits _{j\neq i} \phi _{ij}$
, where
$m_{d}$
and
$q$
are the mass and charge of the dust particle,
$\phi _{ij}$
is the Yukawa interaction potential between dust particles of
$i$
and
$j$
,
$\phi _{ij} = {q} /{4\pi \varepsilon _{0} x_{ij}}e^{-\frac{x_{ij}} {\lambda_D}}$
, where
$x_{ij}=|x_{i}-x_{j}|$
is the distance between dust particles
$i$
and
$j$
,
$\lambda _D$
is the Debye length, and
$\varepsilon _{0}$
is the permittivity of vacuum.
Due to the shielding effect, we consider only the interaction forces between nearest-neighbouring dust particles. We also use two approximations. First, the small amplitude approximation
$a_{0}\gg \xi _{i}-\xi _{i-1}$
. The small amplitude approximation means that the displacement difference between the nearest neighbours of the dust particles (
$ \xi _{i}-\xi _{i-1}$
) is much less than the lattice constant (
$a_{0}$
). Second, the long wavelength approximation
$a_0\ll \lambda$
, where
$\lambda$
is the wavelength of the perturbations. In other words, the long wave approximation means that the wavelength is much larger than the lattice constant. By using the small amplitude approximation and Taylor expansions, we have
$m_{d}({\partial ^{2}{\xi _{i}}}/{\partial {t^{2}}}) =k_{1}(\xi _{i+1}+\xi _{i-1}-2\xi _{i}) +k_{2}[(\xi _{i+1}-\xi _{i})^{2}-(\xi _{i}-\xi _{i-1})^{2}] +O[(\xi _{i+1}-\xi _{i})^3-(\xi _{i}-\xi _{i-1})^3]$
, where
$k_{1} = ({q_{}^{2}}/{4\pi \varepsilon _{0}a_{0}^{3}})e^{-\kappa }[2+2\kappa +\kappa ^{2}]$
,
$k_{2} = - ({q_{}^{2}}/{8\pi \varepsilon _{0}a_{0}^{4}})e^{-\kappa }[6+6\kappa +3\kappa ^{2}+\kappa ^{3}]$
. By using the long wavelength approximation and the continuum approximation
$\xi _i(t)=\xi (x, t)$
, we have
$\xi _{i\pm 1, t}=\xi (x, t)\pm ({\partial {\xi }}/{\partial {x}})a_{0}+ ({a_{0}^{2}}/{2}) ({\partial ^{2}{\xi }}/{\partial {x}^{2}})\pm ({a_{0}^{3}}/{6})({\partial ^{3}{\xi }}/{\partial {x}^{3}})+ ({a_{0}^{4}}/{24}) ({\partial ^{4}{\xi }}/{\partial {x}^{4}})+\cdots$
. Since all the physical quantities are functions of time
$t$
and the spatial coordinate
$i$
, where
$i$
is a discrete integer variable, for simplicity and convenience, we can treat
$i$
as
$x$
, where
$x$
is a continuous variable. This is known as the continuum approximation. Furthermore, due to the small amplitude approximation, Taylor expansion can be applied. Consequently, we obtain the equation of motion for the dust particle as follows (Wei et al. Reference Wei, Peng, Yang, Wang, Yang and Duan2023):
\begin{equation} \frac {\partial ^{2}{\xi }}{\partial {t}^{2}}= B_{1}(\frac {\partial ^{2}{\xi }}{\partial {x}^{2}} +\frac {a_{0}^{2}}{12}\frac {\partial ^{4}{\xi }}{\partial {x}^{4}})+ B_{2}\frac {\partial {\xi }}{\partial {x}} \frac {\partial ^{2}{\xi }}{\partial {x}^{2}}, \end{equation}
where
$B_{1}= {k_{1}a_{0}^{2}}/{m_{d}}$
and
$B_{2}={2k_{2}a_{0}^{3}}/{m_{d}}$
.
3. Linear wave and dispersion relation
For a small amplitude wave (i.e. a linear wave), (2.1) reduces to a linear equation as
\begin{equation} \frac {\partial ^{2}{\xi }}{\partial {t}^{2}}= B_{1}(\frac {\partial ^{2}{\xi }}{\partial {x}^{2}} +\frac {a_{0}^{2}}{12}\frac {\partial ^{4}{\xi }}{\partial {x}^{4}}). \end{equation}
For a linear wave, we assume that
$\xi =\xi _0e^{i(kx-\omega t)}$
. The dispersion relation from (3.1) is given by
$\omega ^2=B_1k^2(1-({a_0^2 /12})k^2)$
, or
$\omega =\pm \sqrt {B_1}k\sqrt {(1-({a_0^2 / 12})k^2)}$
. The phase speed is
$v_p={\omega / k}=\pm \sqrt {B_1} \sqrt {(1-({a_0^2 / 12})k^2)}$
. It appears that there are no real values for
$\omega$
if
$a_0^2k^2\gt 12$
, which indicates that the waves cannot propagate in the dust chain if the wavelength is small enough. The critical wavenumber is
$k_c={2\sqrt {3} / a_0}$
. Waves exist if
$k\lt k_c$
, but do not exist if
$k\gt k_c$
.
The natural frequency in the dust lattice chain is
$\omega _0=\sqrt {{k_1 / m_d}}$
. For the long wavelength approximation, i.e.
$a_0k\lt \lt 1$
, we have
$v_p=\sqrt {B_1}=\omega _0a_0$
.
4. Continuity equation
Suppose there is an incident wave propagating in the positive
$x$
direction in the region
$x\lt 0$
. When it reaches the discontinuity point
$x=0$
, it will be both reflected and transmitted at this point. We now aim to construct both the reflected wave and transmitted waves from the incident wave. To do this, we use the continuity condition at
$x=0$
.
The momentum of the dust particles and the force acting on them at the interface
$x=0$
should be continuous. Neglecting higher-order terms and applying the continuity conditions at the interface
$x=0$
, we have
\begin{align} \frac{m_d^- }{ a_0^{-3}}\frac{\partial \xi ^I }{ \partial t}\mid _{x=0} +\frac{m_d^- }{a_0^{-3}}\frac{\partial \xi ^R }{ \partial t}\mid _{x=0} =\frac{m_d^+ }{ a_0^{+3}}\frac{\partial \xi ^T }{ \partial t}\mid _{x=0}, \\[-10pt]\nonumber\end{align}
\begin{align} k_1^-{ \xi ^I }\mid _{x=0} +k_1^-{ \xi ^R }\mid _{x=0} =k_1^+{ \xi ^T }\mid _{x=0},\\[6pt]\nonumber \end{align}
where
$m_d^-$
and
$m_d^+$
are the mass of the dust particles in the regions
$x\lt 0$
and
$x\gt 0$
, respectively. Here,
$k_1^-$
and
$k_1^+$
correspond to the parameters of
$k_1$
in the regions
$x\lt 0$
and
$x\gt 0$
, respectively. For the three-dimensional case,
$n_d^-={1 /a_0^{-3}}$
and
$n_d^+={1/ a_0^{+3}}$
.
4.1. Reflection and transmission by linear incident wave
We first study the transmission and reflection of an incident linear wave. For convenience and generality, we assume that
$\xi ^I=\xi _0^Ie^{k_Ix-\omega t}$
,
$\xi ^R=\xi _0^Re^{k_Rx-\omega t}$
and
$\xi ^T=\xi _0^Te^{k_Tx-\omega t}$
, where
$\xi _0^I, \xi _0^R \ \mathrm{and}\ \xi _0^T$
are the wave amplitude of the incident wave, reflected and transmitted waves, respectively. Here,
$\omega$
is the wave frequency, and
$k_I$
,
$k_R$
and
$k_T$
are the wavenumbers for the incident, reflected and transmitted wave, respectively. We have
$k_I=k_R=({\omega / \sqrt {B_1^-}})$
,
$k_T= ({\omega / \sqrt {B_1^+}})$
,
$B_1^-=({k_1^-(a_0^-)^2) /r m_d^-}$
and
$B_1^+=({k_1^+(a_0^+)^2) / m_d^+}$
. We derive these results from (4.1) and (4.2),
\begin{align} \xi _0^T=\frac{2 }{ \chi _m+\chi _k}\xi _0^I,\\[-10pt]\nonumber \end{align}
\begin{align} \xi _0^R=\frac{\chi _k-\chi _m }{ \chi _m+\chi _k}\xi _0^I, \\[6pt]\nonumber\end{align}
where
$\chi _m={m_d^+ /m_d^-}({a_)^- / a_0^+})^3$
,
$\chi _k={k_1^+ / k_1^-}$
. The amplitude ratios of both the transmitted wave to the incident wave and the reflected wave to the incident wave are
\begin{align} \frac{\xi _0^T }{ \xi _0^I} =\frac{2 }{ \frac{m_d^+ }{ m_d^-}(\frac{a_)^- }{ a_0^+})^3+\frac{q^+ }{ q^-} (\frac{a_0^- }{ a_0^+})^3\frac{2+2\kappa ^++(\kappa ^+)^2 }{ 2+2\kappa ^-+(\kappa ^-)^2 }e^{(\kappa ^- -\kappa ^+)}},\\[-10pt]\nonumber \end{align}
\begin{align} \frac{\xi _0^R }{ \xi _0^I}=\frac{1-\frac{m_d^+ }{ m_d^-}\frac{q^- }{ q^+} \frac{2+2\kappa ^-+(\kappa ^-)^2 }{ 2+2\kappa ^++(\kappa ^+)^2 }e^{(\kappa ^+-\kappa ^-)} }{ 1+\frac{m_d^+}{ m_d^-}\frac{q^- }{ q^+} \frac{2+2\kappa ^-+(\kappa ^-)^2 }{2+2\kappa ^++(\kappa ^+)^2 }e^{(\kappa ^+-\kappa ^-)}},\\[6pt]\nonumber\end{align}
where
$\kappa ^+={a_0^+ / \lambda _D^+}$
,
$\kappa ^-={a_0^- / \lambda _D^-}$
are the screening parameters in the region of
$x\gt 0$
and
$x\lt 0$
, respectively. Here,
$\lambda _D^+=[({n_e^+e^2 / \epsilon _0k_BT_e^+})+({Z_i^2n_i^+e^2 / \epsilon _0k_BT_i^+})]^{-1/2}$
,
$\lambda _D^-=[({n_e^-e^2 / \epsilon _0k_BT_e^-})+({Z_i^2n_i^-e^2 / \epsilon _0k_BT_i^-})]^{-1/2}$
are the Debye length in the regions of
$x\gt 0$
and
$x\lt 0$
, respectively,
$n_e^+, T_e^+, n_i^+\ \mathrm{and}\ T_i^+$
are the number density, the temperature of the electrons, the number density and the temperature of the ions in the region
$x\gt 0$
, respectively,
$n_e^-, T_e^-, n_i^- \ \mathrm{and}\ T_i^-$
are the number density, the temperature of the electrons, the number density and the temperature of the ions in the region
$x\lt 0$
, respectively, and
$q^+$
and
$q^-$
are the electric charge of a dust particle in the region
$x\gt 0$
and region
$x\lt 0$
, respectively. The electric charge neutrality in two regions yields
$n_e^+e-Z_i^+n_i^+e+({q^+ / (a_0^+)^3})=0$
,
$n_e^-e-Z_i^-n_i^-e+({q^- / (a_0^-)^3})=0$
, where we used the equation of
$a_0=n_d^{-1/3}$
for a three-dimensional dusty plasma (Hong et al. Reference Hong, Sun, Schwabe, Du and Duan2021).
At equilibrium, the variables of
$a_0^+$
and
$a_0^-$
should be interdependent because the force in the dust particle chain must be a constant. Consequently, the following relationship holds:
$k_1^+a_0^++k_2^+(a_0^+)^2 =k_1^-a_0^-+k_2^-(a_0^-)^2$
. Thus, we have
\begin{equation} \frac{a_0^+ }{ a_0^-}=\sqrt {\frac{q_d^- }{ q_d^+}\frac{2+2\kappa ^-+(\kappa ^-)^2 }{ 2+2\kappa ^++(\kappa ^+)^2 }e^{(\kappa ^+-\kappa ^-)}}. \end{equation}
Now consider the limiting case where
$a_0\lt \lt \lambda _D$
. In this case, (4.7) simplifies to
\begin{equation} \frac{a_0^+ }{ a_0^-}=\sqrt {\frac{q_d^- }{ q_d^+} } .\end{equation}
Equations (4.5) and (4.6) become
\begin{align} \frac{\xi _0^T }{ \xi _0^I} =\frac{2 }{ \frac{m_d^+ }{ m_d^-}(\frac{a_)^- }{ a_0^+})^3+ (\frac{a_0^- }{ a_0^+})^5}, \\[-10pt]\nonumber\end{align}
\begin{align} \frac{\xi _0^R }{ \xi _0^I}=\frac{1-\frac{m_d^+ }{ m_d^-} (\frac{a_0^+ }{ a_0^-})^2 }{ 1+\frac{m_d^+ }{ m_d^-} (\frac{a_0^+ }{ a_0^-})^2 }. \\[6pt]\nonumber\end{align}
Furthermore, if all the dust particles are made of the same material and each dust particle is a spherical grain, then
$m_d={4 / 3}\pi r_d^3$
, where
$r_d$
is the dust particle radius. Additionally,
$q\sim r_d^\gamma$
, with
$1\lt \gamma \lt 2$
(Hong et al. Reference Hong, Sun, Schwabe, Du and Duan2021). Therefore, we have
${a_0^+/ a_0^-}=({m_d^+ / m_d^-})^{-\frac{\gamma }{ 6}}$
,
${q^+ / q^-}=({m_d^+ / m_d^-})^{\frac{\gamma }{ 3}}$
.
Equations (4.9) and (4.10) become
\begin{align} \frac{\xi _0^T }{ \xi _0^I} =\frac{2 }{ (\frac{m_d^+ }{ m_d^-})^\frac{4\gamma }{ 3}+(\frac{m_d^+ }{ m_d^-})^{\frac{5\gamma }{6}}},\\[-10pt]\nonumber \end{align}
\begin{align} \frac{\xi _0^R }{ \xi _0^I}=\frac{1-(\frac{m_d^+ }{m_d^-})^{1-\frac{\gamma }{ 3}}}{1+(\frac{m_d^+}{ m_d^-})^{1-\frac{\gamma }{3}}}. \\[6pt]\nonumber\end{align}
Dependence of the amplitude ratio
$\xi _0^T /r \xi _0^I$
of transmitted wave to the incident wave on the mass ratio
$ {m_d^+ / m_d^-}$
, where the parameter
$\gamma =1,1.5,2$
.
Dependence of the amplitude ratio
$\xi _0^R / \xi _0^I$
of reflected wave to the incident wave on the mass ratio
$ {m_d^+ / m_d^-}$
, where the parameter
$\gamma =1,1.5,2$
.
Figure 2 shows the dependence of the amplitude ratio (
$\xi _0^T / \xi _0^I$
) of the transmitted wave to the incident wave on the mass ratio (
$ {m_d^+ / m_d^-}$
), where the parameter
$\gamma =1,1.5,2$
(Hong et al. Reference Hong, Sun, Schwabe, Du and Duan2021). It shows that the amplitude ratio of the transmitted wave to the incident wave decreases as the dust particle mass ratio of that in the region
$x\gt 0$
to that in the region
$x\lt 0$
increases. In other words, the transmitted wave amplitude decreases for a given incident wave if the incident wave propagates from the smaller dust particle mass region to that of the larger dust particle mass region, while the transmitted wave amplitude increases if the incident wave propagates from the larger dust particle mass region to that of the smaller dust particle mass region. It is also shown that the amplitude ratio of the transmitted wave to the incident wave increases as the parameter
$\gamma$
increases in the region
${m_d^+/ m_d^-}\lt 1$
, while it decreases as the parameter
$\gamma$
increases in the region
${m_d^+ / m_d^-}\gt 1$
.
It is found from figure 3 that there are no reflected waves if the mass of both regions is same. However, there are reflected waves if the dust particle mass in both regions is different. The reflected wave amplitude increases as the mass ratio between the two regions increases. Furthermore, the amplitude ratio of the reflected wave to the incident wave decreases as the parameter
$\gamma$
increases in the region
${m_d^+ / m_d^-}\lt 1$
, while it increases as the parameter
$\gamma$
increases in the region
${m_d^+ / m_d^-}\gt 1$
. There is a phase shift of
$\pi$
for the reflected wave in the region
${m_d^+ / m_d^-}\gt 1$
, while there are no phase shifts in the region
${m_d^+ / m_d^-}\lt 1$
.
Notice that there is no reflection in an homogeneous dust lattice chain, while there is reflection in an inhomogeneous one. This result can be used to determine whether the dust lattice is homogeneous or not.
4.2. Approximate nonlinear wave equations for incident, transmitted and reflected waves
In this section, we study the transmission and reflection of a nonlinear wave by using the traditional reductive perturbation method under the condition that the waves satisfy both the small amplitude approximation and the long wavelength approximation (Su & Mirie Reference Su and Mirie1980; Jeffrey & Kawahara Reference Jeffrey and Kawahara1982; Ghosh et al. Reference Ghosh, Chaudhuri, Sarkar, Khan and Gupta2002; Zhang et al. Reference Zhang, Yang, Xu, Yang, Qi and Duan2014; Gao et al. Reference Gao, Zhang, Zhang, Li and Duan2017). We introduce the stretched coordinates
$\eta ^+ = \varepsilon (x-c^+t)$
,
$\tau = \varepsilon ^{3}x$
and
$\xi ^+ = \varepsilon \xi ^+_{1} + \varepsilon ^{3} \xi ^+_{2} + \cdots$
in the region
$x\gt 0$
. However, we introduce
$\eta ^- = \varepsilon (x-c^-t)$
,
$\zeta ^- = \varepsilon (x+c^-t)$
,
$\tau = \varepsilon ^{3}x$
and
$\xi ^- = \varepsilon \xi ^-_{1} + \varepsilon ^{3} \xi ^-_{2} + \cdots$
in the region
$x\lt 0$
, where
$\xi ^+$
and
$\xi ^-$
represent physical quantities in the regions
$x\gt 0$
and
$x\lt 0$
, respectively,
$\varepsilon$
is a small parameter,
$c^+$
is the wave speed in the region
$x\gt 0$
, and
$c^-$
is the wave speed in the region
$x\lt 0$
. By substituting these expansions into (2.1), we obtain
$c^+ = \sqrt {B_{1}^+}$
,
$c^- = \sqrt {B_{1}^-}$
,
$\xi ^+_1(\eta ^+, \tau )=\xi ^T(\eta ^+, \tau )$
,
$\xi ^-_1(\eta ^-, \zeta ^-, \tau )=\xi ^I(\eta ^-, \tau )+\xi ^R(\zeta ^-, \tau )$
, resulting in three Korteweg–de Vries (KdV) equations:
$ ({\partial {w}^I}/{\partial {\tau }})+\alpha ^- w^I ({\partial {w}^I}/{\partial {\eta ^-}}) + \beta ^- ({\partial ^{3}{w}^I}/{\partial {\eta }^{-3}})=0$
,
$ ({\partial {w}^T}/{\partial {\tau }})+\alpha ^+ w^T ({\partial {w}^T}/{\partial {\eta ^+}}) + \beta ^+ ({\partial ^{3}{w}^T}/{\partial {\eta }^{+3}})=0$
,
$ ({\partial {w}^R}/{\partial {\tau }})+\alpha ^- w^R ({\partial {w}^R}/{\partial {\zeta ^-}}) + \beta ^- ({\partial ^{3}{w}^R}/{\partial {\zeta }^{-3}})=0$
, where
$w^I= {\partial {\xi ^I}}/{\partial {\eta ^-}}$
,
$w^T={\partial {\xi ^T}}/{\partial {\eta ^+}}$
,
$w^R= {\partial {\xi ^R}}/{\partial {\zeta ^-}}$
,
$\alpha ^+= {B_{2}^+}/{2B_{1}^+}$
,
$\alpha ^-={B_{2}^-}/{2B_{1}^-}$
,
$\beta ^+= {a_{0}^{+2}}/{24}$
and
$\beta ^-= {a_{0}^{-2}}/{24}$
. One of the solitary wave solutions of the KdV equation can be written as
$w^Z=w^Z_{m} {sech}^{2}({\eta ^{\pm }-u_{0}\tau }/{D^{Z}})$
, where
$\eta ^+$
and
$\eta ^-$
correspond to the regions
$x\gt 0$
and
$x\lt 0$
, respectively. Here,
$Z$
represents
$I$
,
$R$
and
$T$
for the incident, reflected and transmitted solitary waves, respectively. The parameters are defined as follows:
$w^Z_{m} = {3u_{0}}/{\alpha ^{\pm }}$
(the wave amplitude),
$D^{Z} = 2\sqrt { {\beta ^{\pm }}/{u_{0}}}$
(the wave width) and
$u_{0}$
(propagation velocity in the moving coordinate system).
One solitary wave solution in the experimental coordinate system can be rewritten as
$\xi ^Z= \xi ^Z_{m} \mathrm{tanh}({x-v^Z_{g}t +x_0^Z}/{D^Z})$
,
$u^Z= u^Z_{m} \mathrm{sech}^{2} ({x-v^Z_{g}t+x_0^Z}/{D^Z})$
, where
$\xi ^Z_m={ a_0^{\pm }B_1^{\pm }\sqrt {6(\varepsilon ^2u_0)^{Z}} / B_2^{\pm }}$
is the displacement amplitude of the dust particle,
$u^Z_m=-{6(\varepsilon ^2 u_0)^{Z}{B_1^{\pm }}^{\frac {3}{2}} /r B_2^{\pm }}$
is the dust particle velocity amplitude,
$v^Z_g={\sqrt {B_1^{\pm }} / 1-(\varepsilon ^2u_0)^{Z}}$
is the wave speed,
$D^Z={a_0^{\pm } / \sqrt {6(\varepsilon ^2u_0)^{Z}}[1-(\varepsilon ^2u_0)^{Z}]}$
is the wave width, and
$x_0^Z$
is the initial phase of the wave for the incident, reflected and transmitted solitary waves.
4.3. Scattering method for a KdV solitary wave
To understand the generation of both transmitted and reflected solitary nonlinear waves at
$x=0$
, we use the inverse scattering method (Hong et al. Reference Hong, Sun, Schwabe, Du and Duan2021). For the KdV equation:
$ {\partial \phi / \partial T}-6\phi ({\partial \phi / \partial X })+({\partial ^3 \phi / \partial X^3 })=0$
, with its initial condition
$\phi (X,0)=-({2A /r B^2})sech^2({X/ B})$
. The number
$N$
of generated solitary waves is given by the maximum integer satisfies
$\sqrt {A+({1/ 4})}+({1 / 2})-N\gt 0$
. The amplitudes
$A_j$
of the
$j$
th solitary wave is given by
$A_j=2{([\sqrt ({A+{1 / 4})}+({1 / 2})-j]^2) / B^2}$
, where
$j=1,2,\cdots, N$
.
Suppose there is an incident solitary wave propagating in the positive
$x$
direction in the region
$x\lt 0$
, which satisfies the KdV equation of
$ ({\partial {w}^I}/{\partial {\tau }})+\alpha ^- w^I ({\partial {w}^I}/{\partial {\eta ^-}}) + \beta ^- ({\partial ^{3}{w}^I}/{\partial {\eta }^{-3}})=0$
, where one solitary wave solution is
$w^I=w^I_{m}{sech}^{2}({\eta ^- -u_{0}\tau }/{D^{I}})$
, with
$w^I_{m} = {3u_{0}}/{\alpha ^-}$
and
$D^{I} = 2\sqrt {{\beta ^-}/{u_{0}}}$
. By substituting
$W^I=-6({\beta ^- / \alpha ^-}w^I)$
and
$T=\beta ^-\tau$
, we obtain
$({\partial {W}^I}/{\partial {T}})-6 W^I ({\partial {W}^I}/{\partial {\eta ^-}}) + ({\partial ^{3}{W}^I}/{\partial {\eta }^{-3}})=0$
. The solitary wave solution can be rewritten as
$W^I=W^I_{m} {sech}^{2} ({\eta ^- -(\frac{u_{0} }{ \beta ^-})T})/{d^{I}}$
. The initial condition for the incident solitary wave is
\begin{align} \frac {\partial {W}^I}{\partial {T}}-6 W^I \frac {\partial {W}^I}{\partial {\eta ^-}} + \frac {\partial ^{3}{W}^I}{\partial {\eta }^{-3}}=0, \nonumber \\ W^I\mid _{x=0}=W^I_{m} \mathrm{sech}^{2}\frac {\eta ^-}{d^{I}}, \end{align}
where
$W^I_{m} = -{U_0 / 2}$
,
$d^{I} = {2 /r \sqrt {U_{0}}}$
and
$U_0$
is an arbitrary parameter controlling the amplitude of the solitary wave.
5. Nonlinear transmitted wave and the nonlinear reflected wave due to an incident solitary wave
5.1. Construction of the ‘initial conditions’ for both transmitted nonlinear wave and the reflected nonlinear wave from an incident KdV solitary wave
We now attempt to find the transmitted and reflected waves resulting from an incident solitary wave at an interface. We can derive the following equations from (4.1) and (4.2):
\begin{align} \frac{m_d^-c^- }{ a_0^{-3}}\frac{\partial \xi ^I }{ \partial \eta ^-}\mid _{x=0} -\frac{m_d^-c^- }{ a_0^{-3}}\frac{\partial \xi ^R }{ \partial \zeta ^-}\mid _{x=0} =\frac{m_d^+c^+ }{ a_0^{+3}}\frac{\partial \xi ^T }{\partial \eta ^+}\mid _{x=0}, \\[-10pt]\nonumber\end{align}
\begin{align} k_1^-a_0^-\frac{\partial \xi ^I }{\partial \eta ^-}\mid _{x=0} +k_1^-a_0^-\frac{\partial \xi ^R }{ \partial \zeta ^-}\mid _{x=0} =k_1^+a_0^+\frac{\partial \xi ^T }{\partial \eta ^+}\mid _{x=0}. \\[6pt]\nonumber\end{align}
\begin{align} w^T\mid _{x=0}=\frac{2 }{ \chi _2+\chi _1}w^I\mid _{x=0},\\[-10pt]\nonumber \end{align}
\begin{align} w^R\mid _{x=0}=\frac{\chi _2-\chi _1 }{ \chi _2+\chi _1}w^I\mid _{x=0}, \\[6pt]\nonumber\end{align}
where
$\chi _1= \sqrt {\frac{m_d^+(a_0^-)^7 }{ m_d^-(a_0^+)^7}}\frac{q^+ }{ q^-}$
and
$\chi _2= ({q^+a_0^- / q^-a_0^+})^2$
.
The ‘initial conditions’ for the transmitted nonlinear wave can be described by the following equations:
\begin{align} \frac {\partial {W}^T}{\partial {T}}-6 W^T \frac {\partial {W}^T}{\partial {\eta ^+}} + \frac {\partial ^{3}{W}^T}{\partial {\eta }^{+3}}=0, \nonumber \\ W^T\mid _{x=0}=\frac{2 }{ \chi _2+\chi _1}W^I_{m}\mathrm{sech}^{2}\frac {-\epsilon c^-t}{d^{I}}, \end{align}
where
$W^T=-6({\beta ^+ / \alpha ^+})w^T$
and
$T=\beta ^+\tau$
.
Letting
$B^T={{c^+d^{I} /r c^-}}$
, we have
\begin{align} \frac {\partial {W}^T}{\partial {T}}-6 W^T \frac {\partial {W}^T}{\partial {\eta ^+}} + \frac {\partial ^{3}{W}^T}{\partial {\eta }^{+3}}=0, \nonumber \\ W^T\mid _{x=0}=-\frac{2A^T }{ (B^{T})^2}\mathrm {sech}^{2}[\frac {\eta ^+}{B^T}], \end{align}
where
$A^T= ({2 / \chi _1+\chi _2})({m_d^- /m_d^+})(({a_0^+ / a_0^-})^2({k_1^+ / k_1^-})$
. In the limit case where
$a_0\lt \lt \lambda _D$
, we have
$A^T=({2 / \chi _1+\chi _2})({m_d^- / m_d^+}) ({a_0^-/ a_0^+}) ({q^+ / q^-}^2)$
.
Similarly, the ‘initial conditions’ for the reflected wave can be described by the following equations:
\begin{align} \frac {\partial {W}^R}{\partial {R}}-6 W^R \frac {\partial {W}^R}{\partial {\zeta ^+}} + \frac {\partial ^{3}{W}^R}{\partial {\zeta }^{+3}}=0, \nonumber \\ W^R\mid _{x=0}=-\frac{2A^R }{ (B^{R})^2} \mathrm{sech}^{2}[\frac {\eta ^+}{B^R}], \end{align}
where
$A^R= 2({\chi _2-\chi _1 / \chi _1+\chi _2})$
and
$B^R={{d^{I} }}$
.
5.2. Transmitted wave and the reflected wave by an interface
The transmitted nonlinear waves from an indent solitary wave due to the interface can be obtained using (5.6) and the scattering method. The number
$N^T$
of generated transmitted solitary waves is given by the maximum integer satisfying the inequality
$\sqrt {A^T+({1 / 4}})+({1 / 2})-N^T\gt 0$
, where
$A^T=({2 / \chi _1+\chi _2})({m_d^- / m_d^+}) ({a_0^-/ a_0^+}) (({q_d^+ / q_d^-})^2)$
. In the limit case where
$a_0\lt \lt \lambda _D$
, we have
${a_0^+ / a_0^-}=(({m_d^+/ m_d^-}))^{-\frac{\gamma }{ 6}}$
,
${q^+ / q^-}=(({m_d^+/ m_d^-}))^{\frac{\gamma }{ 3}}$
,
$\chi _1=\sqrt {\frac{m_d^+(a_0^-)^7 }{ m_d^-(a_0^+)^7}}({q^+ / q^-})=({m_d^+/ m_d^-})^{\frac{1 }{ 2}+\frac{11 }{ 12}\gamma }$
,
$\chi _2=(({q^+a_0^- / q^-a_0^+})^2)=(({m_d^+/ m_d^-}))^{\gamma }$
. Then,
$A^T=({2 / \chi _1+\chi _2})({m_d^- / m_d^+})^{\frac{5\gamma }{ 6}-1}$
. The corresponding amplitudes of jth transmitted solitary wave,
$A_j^T$
, is given by
$A_j^T=2\frac{\sqrt {A^T+({1 / 4})}+({1 / 2})-j }{ (B^T)^2}$
.
The dependence of the number of transmitted solitary waves on the mass ratio
$ ({m_d^+/ m_d^-})$
is shown in figure 4, with
$\gamma =1,1.5,2$
. It shows that there are two transmitted solitary waves when
$0.2\lt ({m_d^+/ m_d^-})\lt 0.6$
. For
$\gamma =1.0$
, there is only one transmitted solitary wave when
$ ({m_d^+/ m_d^-})\gt 0.6$
. However, if
$\gamma =1.5$
, the region where two transmitted solitary waves exist is
$0.4\lt ({m_d^+/ m_d^-})\lt 0.6$
and the region for only one transmitted solitary wave is
$ ({m_d^+/ m_d^-})\gt 0.6$
. For a larger value of
$\gamma =2$
, the region for two transmitted solitary waves is
$0.6\lt ({m_d^+/ m_d^-})\lt 0.8$
, while the region for only one transmitted solitary wave is
$ ({m_d^+/ m_d^-})\gt 0.8$
. Moreover, the dependence of the amplitude ratio of the transmitted solitary wave to the incident solitary wave on the mass ratio
$ ({m_d^+/ m_d^-})$
is shown in figure 5. It seems that the amplitude ratio of the transmitted solitary wave to the incident solitary wave decreases as the mass ration
$ ({m_d^+/ m_d^-})$
increases.
Dependence of the number of transmitted solitary waves on the mass ratio
$ {m_d^+ / m_d^-}$
, with the parameter
$\gamma$
set to (a) 1, (b) 1.5 and (c) 2.
Dependence of the amplitude ratio,
$A_1^T / A^I$
of the transmitted solitary wave to the incident solitary wave on the mass ratio
$ {m_d^+ / m_d^-}$
, with the parameter
$\gamma$
set to 1, 1.5 and 2.
Dependence of the numbers of the reflected solitary waves on the mass ratio,
$ {m_d^+ / m_d^-}$
, where the parameter
$\gamma$
is set to (a) 1, (b) 1.5 and (c) 2.
Dependence of the amplitude ratio,
$A_1^R / A^I$
, of the reflected solitary wave to the incident solitary wave on the mass ratio
$ ({m_d^+/ m_d^-})$
, with the parameter
$\gamma$
set to 1, 1.5 and 2.
Similarly, the reflected nonlinear waves from an indent solitary wave due to the interface are obtained from (5.7) and the scattering method. The number
$N^R$
of generated reflected solitary waves is given by the maximum integer of the inequality
$\sqrt {A^R+({1 / 4}})+({1 / 2})-N^R\gt 0$
, where
$A^R=2({\chi _2-\chi _1 / \chi _1+\chi _2})$
. The corresponding amplitudes of the jth reflected solitary wave
$A_j^R$
are given by
$A_j^R=2({\sqrt {A^R+({1 / 4}})+({1 / 2})-j (B^R)^2})$
.
The dependence of the number of reflected solitary waves on the mass ratio,
$ ({m_d^+/ m_d^-})$
, is shown in figure 6, where
$\gamma =1,1.5,2$
. It shows that there is either one reflected solitary wave or none. When
$0.2\lt ({m_d^+/ m_d^-})\lt 1.1$
, there is only one reflected solitary wave, while there are no reflected solitary waves when
$1.1\lt ({m_d^+/ m_d^-})$
. It seems that whether there is one or no reflected solitary waves is independent of the parameter
$\gamma$
.
Moreover, the dependence of the amplitude ratio of the reflected solitary wave to the incident solitary wave on the mass ratio of
$ ({m_d^+/ m_d^-})$
is shown in figure 7. It seems that there is no reflection if
$ {m_d^+ = m_d^-}$
. The amplitude ratio of the reflected solitary wave to the incident solitary wave decreases as the mass ratio
$ ({m_d^+/ m_d^-})$
increases in the region
$ 0\lt ({m_d^+/ m_d^-})\lt 1$
, while the amplitude ratio of the reflected solitary wave to the incident solitary wave increases as the mass ratio
$ ({m_d^+/ m_d^-})$
increases in the region
$ ({m_d^+/ m_d^-})\gt 1$
.
6. Conclusions
As is well known, a dusty plasma typically consists of different species of dust particles, both in space plasma and experimental settings on Earth. However, most investigations on dust acoustic wave in dusty plasma assume that all dust particles are the same. The present paper studies dust acoustic wave propagation in an inhomogeneous dusty plasma and seeks to determine how the inhomogeneity of the dusty plasma affects the dust acoustic wave.
It is found in the present paper that the transmitted wave amplitude decreases for a given incident wave if the incident wave propagates from the smaller dust particle mass region to the larger dust particle mass region, while the transmitted wave amplitude increases if the incident wave propagates from the larger dust particle mass region to the smaller dust particle mass region. It is also found that there is no reflected wave if the dust particle mass of both regions is the same. However, there is a reflected wave if the dust particle mass in both regions is different. The reflected wave amplitude increases as the mass ratio between the two regions increases.
It appears that dust acoustic waves cannot propagate in the dust lattice if the wavelength is sufficiently small. The critical wavelength is
$\lambda _c=({\pi / \sqrt {3} }) a_0$
. If
$\lambda \gt \lambda _c$
, the wave exists, but it cannot exist if
$\lambda \lt \lambda _c$
. This indicates that dust acoustic waves cannot propagate in the dust lattice when the wavelength is smaller than the dust lattice constant.
Using the scattering method and the reductive perturbation method, we find that at least one transmitted solitary wave is generated when an incident solitary wave reaches a discontinuous interface. The number of solitary waves and their corresponding amplitudes depend on the mass ratio between the two regions. However, there can be at most one reflected solitary wave. No reflected solitary wave occurs if an incident solitary wave propagates from a region with smaller dust particle mass into a region with larger dust particle mass.
The present results demonstrate that the mass ratio between two distinct regions in an inhomogeneous dusty plasma plays a crucial role in dust acoustic wave propagation. Based on these findings, we can design an experiment to estimate the mass of the dust particles and identify the locations of different dust species by introducing an incident wave, and analysing both the transmitted and reflected waves. For more complex inhomogeneous dusty plasmas, understanding how transmission and reflection depend on both the incident wave and the plasma inhomogeneity remains a topic for future research.
Funding
This work was supported by the Gansu Natural Science Foundation (No. 24JRRP004), the novation Fund for University Teachers in Gansu Province (No. 2025A-212) and the Presidents Research Fund High-Level Talent Recruitment Project (Nos. 2023ZCC-02, 2023PY-18).
Declaration of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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