Non-stationary elastic wave scattering and energy transport in a one-dimensional harmonic chain with an isotopic defect

Abstract

The fundamental solution describing non-stationary elastic wave scattering on an isotopic defect in a one-dimensional harmonic chain is obtained in an asymptotic form. The chain is subjected to unit impulse point loading applied to a particle far enough from the defect. The solution is a large-time asymptotics at a moving point of observation, and it is in excellent agreement with the corresponding numerical calculations. At the next step, we assume that the applied point impulse excitation has random amplitude. This allows one to model the heat transport in the chain and across the defect as the transport of the mathematical expectation for the kinetic energy and to use the conception of the kinetic temperature. To provide a simplified continuum description for this process, we separate the slow in time component of the kinetic temperature. This quantity can be calculated using the asymptotics of the fundamental solution for the deterministic problem. We demonstrate that there is a thermal shadow behind the defect: the order of vanishing for the slow temperature is larger for the particles behind the defect than for the particles between the loading and the defect. The presence of the thermal shadow is related to a non-stationary wave phenomenon, which we call the anti-localization of non-stationary waves. Due to the presence of the shadow, the continuum slow kinetic temperature has a jump discontinuity at the defect. Thus, the system under consideration can be a simple model for the non-stationary phenomenon, analogous to one characterized by the Kapitza thermal resistance. Finally, we analytically calculate the non-stationary transmission function, which describes the distortion (caused by the defect) of the slow kinetic temperature profile at a far zone behind the defect.

Appendices

Appendix A: Non-dimensionalization

The equations of motion for the system under consideration are

$$\begin{aligned} \hat{m}_n\frac{\mathrm d^2 \hat{u}_n}{\mathrm d\hat{t}^2} -\hat{C}(\hat{u}_{n+1}-2\hat{u}_n+\hat{u}_{n-1})= \delta _{n-N} \hat{p}\big (\hat{t}\big ). \end{aligned}$$

(A.1)

Here $n \in \mathbb {Z}$, $\hat{t}$ is the time, ${\hat{u}_n}(\hat{t})$ is the displacement of the particle with a number n, $\hat{m}_n$ is the mass of a particle with a number n:

$$\begin{aligned} \hat{m}_n=\hat{M}+\delta _n(\hat{m}-\hat{M}), \end{aligned}$$

(A.2)

${\hat{M}}$ is the mass of a regular particle, ${\hat{m}}$ is the mass of the defect, $\hat{C}$ is the bond stiffness, ${\hat{p}}$ is the external force. The dimensionless equations of motion (3.1) can be obtained by introducing the following dimensionless quantities:

$$\begin{aligned} u_n=\frac{\hat{u}_n}{\hat{A}};\quad t=\omega \hat{t};\quad p=\frac{\hat{p}}{\hat{C}\hat{A}},\quad m=m_0=\frac{\hat{m}}{\hat{M}}. \end{aligned}$$

(A.3)

Here $\hat{A}$ is the lattice constant (the distance between neighbouring particles); $\omega {\mathop {=}\limits ^{\text{ def }}}\sqrt{{\hat{C}}/{\hat{M}}}$.

Appendix B: Asymptotics at alternative moving points of observation

The choice of the moving observation fronts is generally ambiguous. One can try to construct asymptotics on moving observation points different from Eq. (5.14). Consider the case $n\le 0$ (the scattered wave for $n>0$ as in Sect 5.1.2 can be calculated using the evenness of $(\breve{{\mathcal V}}_n^N)^\textrm{pass}$ with respect to n). Let us estimate the integral $(\mathcal I_n^N)^\textrm{pass}$ on a moving point of observation different from the one given by Eq. (5.14). We can try to use the following moving point of observation instead of Eq. (5.14):

$$\begin{aligned} |n|=wt. \end{aligned}$$

(B.1)

The integral $(\mathcal I_n^N)^\textrm{pass}$ (5.13) can be represented in the form of Eq. (5.15), where

$$\begin{aligned} A^{\textrm{pass}}(\Omega )=\frac{(m-1)\Omega \mathrm e^{\mathrm iN \arccos \frac{2-\Omega ^2}{2}} }{-(4-\Omega ^2)+\mathrm i(m-1)\Omega \sqrt{4-\Omega ^2}}, \end{aligned}$$

(B.2)

$\phi (\Omega )$ is given by Eq. (5.17). Applying the procedure of the method of stationary phase in the same way as it is done in Sect. 5.1.1, instead of Eq. (5.25) one gets:

$$\begin{aligned} (\breve{\mathcal V}_n^N)^{\textrm{pass}}= & {} -\frac{H(1-w)(m-1)\root 4 \of {1-w^2}}{\sqrt{\pi t}\big (w^2+(m-1)^2(1-w^2)\big )} \Big ( \Big ((m-1)\sqrt{1-w^2}\cos F+w\sin F\Big ) \cos \Big (\omega t+\frac{\pi }{4}\Big )\nonumber \\{} & {} + \Big (w\cos F-(m-1)\sqrt{1-w^2}\sin F\Big ) \sin \Big (\omega t+\frac{\pi }{4} \Big ) \Big ) +O(t^{-1}). \end{aligned}$$

(B.3)

Here

$$\begin{aligned} F= & {} F_0\arccos (2w^2-1), \end{aligned}$$

(B.4)

$$\begin{aligned} F_0= & {} N. \end{aligned}$$

(B.5)

Formula (B.3) can be transformed into the form of Eq. (5.26), wherein we should substitute

$$\begin{aligned} \psi _*=\arctan \frac{w\sin F+(m-1)\sqrt{1-w^2}\cos F}{w\cos F-(m-1)\sqrt{1-w^2}\sin F}= F+\psi \end{aligned}$$

(B.6)

instead of $\psi$ defined by (5.27).

Consider now the following moving point of observation:

$$\begin{aligned} |n|=w(t-N). \end{aligned}$$

(B.7)

In the latter case

$$\begin{aligned} A^{\textrm{pass}}(\Omega )=\frac{(m-1)\Omega \mathrm e^{\mathrm iN(1-w) \arccos \frac{2-\Omega ^2}{2}} }{-(4-\Omega ^2)+\mathrm i(m-1)\Omega \sqrt{4-\Omega ^2}}, \end{aligned}$$

(B.8)

and asymptotics for $(\breve{\mathcal V}_n^N)^{\textrm{pass}}$ has the form (B.3), wherein

$$\begin{aligned} F_0=N(1-w). \end{aligned}$$

(B.9)

Remark B.1

Asymptotics (5.25) can be formally obtained by substituting

$$\begin{aligned} F_0=0 \end{aligned}$$

(B.10)

into Eqs. (B.3), (B.4).

All asymptotics in the form of Eq. (B.3) with various $F_0$ defined by (B.5), (B.9), (B.10) are formally correct. According to Eqs. (B.4), (B.6), for various $F_0$ the right-hand side of Eq. (B.3) has the same amplitude but different phases. Moreover, formulae (B.1), (B.7), or (5.14) that we use to return to the variables n and t from w and t are also different. Therefore, the applicability of the corresponding asymptotics as an approximate solution in terms of n and t can also be different. To check which approach is better, we calculate the absolute error

$$\begin{aligned} e_n^N{\mathop {=}\limits ^{\text{ def }}} ({\mathcal V}_n^N)_{\textrm{num}}-V_{n-N}-(\breve{\mathcal V}_n^N)^{\textrm{pass}}_{\textrm{approx}} \end{aligned}$$

(B.11)

using various asymptotics (B.3), (B.4) with $F_0$ defined by (B.5), (B.9) or (B.10). Here $({\mathcal V}_n^N)_{\textrm{num}}$ are values for the particle velocities found numerically, $V_{n-N}$ is given by exact formula (3.10), $({\mathcal V}_n^N)^{\textrm{pass}}_{\textrm{approx}}$ are found by Eq. (B.3) wherein w is found in accordance with the corresponding formula from set (B.1), (B.7), (5.14). The plot for the error $e_n^N$ is presented in Fig. 7. One can see that the choice of $F_0$ in the form of Eq. (B.10) gives the best result, whereas asymptotics with $F_0$ in the form of (B.5), (B.9) are practically inapplicable as an approximate solution in terms of variables n and t.

Fig. 7

Error $e_n^N$ versus n. The source position is indicated by the vertical magenta solid line. The right leading reflected wave-front is indicated by the vertical magenta dashed line

Appendix C: Fixed position asymptotics for $n\le 0$: the amplitude expansion near the cut-off frequency in the pass-band

Take $n\le 0$. For $C_n^N$ defined by Eq. (7.5), one has

$$\begin{aligned}{} & {} C_n^N(\Omega )= \big (A(\Omega )+ B^{}(\Omega )\big )E_n^N(\Omega ), \end{aligned}$$

(C.1)

$$\begin{aligned}{} & {} A^{}(\Omega )E_n^N(\Omega )=\Omega \breve{\mathcal {G}}_n^N, \qquad B^{}(\Omega )E_n^N(\Omega )=\Omega G_{n-N}. \end{aligned}$$

(C.2)

In the pass-band, $A^{}(\Omega )$ is defined by Eq. (5.16),

$$\begin{aligned} B(\Omega )= & {} -\frac{1}{\mathrm i\sqrt{4-\Omega ^2}}, \end{aligned}$$

(C.3)

$$\begin{aligned} E_n^N(\Omega )= & {} {\mathrm e}^{\mathrm i({N-n}) \arccos \frac{2-\Omega ^2}{2}}, \end{aligned}$$

(C.4)

see Eqs. (4.5), (4.6), (4.8), (5.1)–(5.5), (5.13). For $\Omega \rightarrow 2-0$, one can obtain the following asymptotic expansions:

$$\begin{aligned} A(\Omega )= & {} \frac{-\mathrm i}{2 \sqrt{2-\Omega }} - \frac{1}{2 (m-1)} + \frac{\mathrm i(-m^2+2 m+7) \sqrt{2-\Omega }}{16 (m-1)^2} +O(2-\Omega ), \end{aligned}$$

(C.5)

$$\begin{aligned} B(\Omega )= & {} \frac{\mathrm i}{2 \sqrt{2-\Omega } }+\frac{\mathrm i\sqrt{2-\Omega }}{16} +O\big ((2-\Omega )^{3/2}\big ),\end{aligned}$$

(C.6)

$$\begin{aligned} E_n^N(\Omega )= & {} (-1)^{N-n}-2 \mathrm i({N-n}) (-1)^{N-n}\sqrt{2-\Omega }+O(2-\Omega ). \end{aligned}$$

(C.7)

Now, one gets

$$\begin{aligned} A(\Omega )+ B(\Omega )= & {} -\frac{1}{2 (m-1)} +\frac{\mathrm i\sqrt{2-\Omega }}{2 (m-1)^2} +O(2-\Omega ), \end{aligned}$$

(C.8)

$$\begin{aligned} C_n^N(\Omega )= & {} -\frac{(-1)^{N-n}}{2 (m-1)} +\frac{\mathrm i(-1)^{N-n}\big (1+2({N-n})(m-1)\big )\sqrt{2-\Omega }}{2 (m-1)^2}+O(2-\Omega ). \end{aligned}$$

(C.9)

Appendix D: Fixed position asymptotics for $n\le 0$: the amplitude expansion near the cut-off frequency in the stop-band

In the stop-band, we again have Eqs. (C.1), (C.2), wherein

$$\begin{aligned} A(\Omega )= & {} \frac{\Omega ^3(m-1) }{\Big (-\Omega ^2+2\mathrm e^{-{\text {arccosh}}\frac{\Omega ^2-2}{2}}+2\Big ) \Big (-m\Omega ^2+ 2\mathrm e^{- {\text {arccosh}}\frac{\Omega ^2-2}{2}}+2\Big )}, \end{aligned}$$

(D.1)

$$\begin{aligned} B(\Omega )= & {} \frac{\Omega }{-\Omega ^2+2\mathrm e^{-{\text {arccosh}}\frac{\Omega ^2-2}{2}}+2}, \end{aligned}$$

(D.2)

$$\begin{aligned} E_n^N(\Omega )= & {} (-1)^{{N-n}}\mathrm e^{-({N-n}) {\text {arccosh}}\frac{\Omega ^2-2}{2}}, \end{aligned}$$

(D.3)

see Eqs. (4.5), (4.7), (4.9), (5.1)–(5.5), (5.36). For $\Omega \rightarrow 2+0$, one can obtain the following asymptotic expansions:

$$\begin{aligned} A(\Omega )= & {} \frac{1}{2 \sqrt{\Omega -2}} -\frac{1}{2 (m-1)} -\frac{\left( m^2-2 m-7\right) \sqrt{\Omega -2}}{16 (m-1)^2} +O(\Omega -2), \end{aligned}$$

(D.4)

$$\begin{aligned} B(\Omega )= & {} -\frac{1}{2 \sqrt{\Omega -2}} +\frac{\sqrt{\Omega -2}}{16} +O\big ((\Omega -2)^{3/2}\big ), \end{aligned}$$

(D.5)

$$\begin{aligned} E_n^N(\Omega )= & {} (-1)^{N-n}\big ( 1 -2 ({N-n})\sqrt{\Omega -2}+O(\Omega -2)\big ). \end{aligned}$$

(D.6)

Now, one gets

$$\begin{aligned} A(\Omega )+ B(\Omega )= & {} -\frac{1}{2 (m-1)}+\frac{\sqrt{\Omega -2}}{2 (m-1)^2} +O(\Omega -2), \end{aligned}$$

(D.7)

$$\begin{aligned} C_n^N(\Omega )= & {} -\frac{(-1)^{N-n}}{2 (m-1)}+\frac{(-1)^{N-n}\big (1+2({N-n})(m-1)\big )\sqrt{\Omega -2}}{2 (m-1)^2} +O(\Omega -2). \end{aligned}$$

(D.8)