含液体层的反射透射系数矩阵详细表达式

• Author: Zhu Dengda

• Email: zhudengda@mail.iggcas.ac.cn

import numpy as np
np.set_printoptions(linewidth=200)

import sympy as sp
from sympy.printing.latex import latex
from IPython.display import display, Math

# 定义基本变量
k, b1, a1, mu1, Omg1, kb1, ka1, w = sp.symbols(r'k b_1 a_1 mu_1 \Omega_1 k_{b1} k_{a1} \omega')
b2, a2, mu2, Omg2, kb2, ka2 = sp.symbols(r'b_2 a_2 mu_2 \Omega_2 k_{b2} k_{a2}')
lam1, lam2 = sp.symbols(r'\lambda_1 \lambda_2')
rho1, rho2 = sp.symbols(r'\rho_1 \rho_2')

动态解

液体-液体界面

# left
Pl = sp.Matrix([
    [-a1, -a2],
    [rho1*w**2, -rho2*w**2]
])

# right
Pr = sp.Matrix([
    [-a2, -a1],
    [rho2*w**2, -rho1*w**2]
])

Q = sp.together(sp.expand(Pl.inv() * Pr))

TU = Q[:1, :1]
RU = Q[1:, :1]
RD = Q[:1, 1:]
TD = Q[1:, 1:]

display(Math(latex(RD)))
display(Math(latex(RU)))
display(Math(latex(TD)))
display(Math(latex(TU)))

$\displaystyle \left[\begin{matrix}\frac{- \rho_{1} a_{2} + \rho_{2} a_{1}}{\rho_{1} a_{2} + \rho_{2} a_{1}}\end{matrix}\right]$

$\displaystyle \left[\begin{matrix}\frac{\rho_{1} a_{2} - \rho_{2} a_{1}}{\rho_{1} a_{2} + \rho_{2} a_{1}}\end{matrix}\right]$

$\displaystyle \left[\begin{matrix}\frac{2 \rho_{1} a_{1}}{\rho_{1} a_{2} + \rho_{2} a_{1}}\end{matrix}\right]$

$\displaystyle \left[\begin{matrix}\frac{2 \rho_{2} a_{2}}{\rho_{1} a_{2} + \rho_{2} a_{1}}\end{matrix}\right]$

液体-固体界面

# left
Pl = sp.Matrix([
    [(-1)*a1,           -a2,             k,           ],
    [(-1)*-rho1*w**2 ,  2*mu2*Omg2,     -2*k*mu2*b2,  ],
    [0,                -2*k*mu2*a2,      2*mu2*Omg2,  ]
])

# right
Pr = sp.Matrix([
    [(-1)*a2,            (-1)*k,            -a1        ],
    [(-1)*2*mu2*Omg2,    (-1)*2*k*mu2*b2,   -rho1*w**2 ],
    [(-1)*2*k*mu2*a2,    (-1)*2*mu2*Omg2,   0          ]
])

Q = sp.together(sp.expand(Pl.inv() * Pr))

TU = Q[:1, :2]
RU = Q[1:, :2]
RD = Q[:1, 2:]
TD = Q[1:, 2:]

display(Math(latex(RD)))
display(Math(latex(RU)))
display(Math(latex(TD)))
display(Math(latex(TU)))

$\displaystyle \left[\begin{matrix}\frac{2 \Omega_{2}^{2} a_{1} \mu_{2} + \Omega_{2} \omega^{2} \rho_{1} a_{2} - \omega^{2} \rho_{1} a_{2} k^{2} - 2 a_{1} a_{2} b_{2} k^{2} \mu_{2}}{2 \Omega_{2}^{2} a_{1} \mu_{2} - \Omega_{2} \omega^{2} \rho_{1} a_{2} + \omega^{2} \rho_{1} a_{2} k^{2} - 2 a_{1} a_{2} b_{2} k^{2} \mu_{2}}\end{matrix}\right]$

$\displaystyle \left[\begin{matrix}\frac{- 2 \Omega_{2}^{2} a_{1} \mu_{2} - \Omega_{2} \omega^{2} \rho_{1} a_{2} + \omega^{2} \rho_{1} a_{2} k^{2} - 2 a_{1} a_{2} b_{2} k^{2} \mu_{2}}{2 \Omega_{2}^{2} a_{1} \mu_{2} - \Omega_{2} \omega^{2} \rho_{1} a_{2} + \omega^{2} \rho_{1} a_{2} k^{2} - 2 a_{1} a_{2} b_{2} k^{2} \mu_{2}} & - \frac{4 \Omega_{2} a_{1} b_{2} k \mu_{2}}{2 \Omega_{2}^{2} a_{1} \mu_{2} - \Omega_{2} \omega^{2} \rho_{1} a_{2} + \omega^{2} \rho_{1} a_{2} k^{2} - 2 a_{1} a_{2} b_{2} k^{2} \mu_{2}}\\- \frac{4 \Omega_{2} a_{1} a_{2} k \mu_{2}}{2 \Omega_{2}^{2} a_{1} \mu_{2} - \Omega_{2} \omega^{2} \rho_{1} a_{2} + \omega^{2} \rho_{1} a_{2} k^{2} - 2 a_{1} a_{2} b_{2} k^{2} \mu_{2}} & \frac{- 2 \Omega_{2}^{2} a_{1} \mu_{2} + \Omega_{2} \omega^{2} \rho_{1} a_{2} - \omega^{2} \rho_{1} a_{2} k^{2} - 2 a_{1} a_{2} b_{2} k^{2} \mu_{2}}{2 \Omega_{2}^{2} a_{1} \mu_{2} - \Omega_{2} \omega^{2} \rho_{1} a_{2} + \omega^{2} \rho_{1} a_{2} k^{2} - 2 a_{1} a_{2} b_{2} k^{2} \mu_{2}}\end{matrix}\right]$

$\displaystyle \left[\begin{matrix}- \frac{2 \Omega_{2} \omega^{2} \rho_{1} a_{1}}{2 \Omega_{2}^{2} a_{1} \mu_{2} - \Omega_{2} \omega^{2} \rho_{1} a_{2} + \omega^{2} \rho_{1} a_{2} k^{2} - 2 a_{1} a_{2} b_{2} k^{2} \mu_{2}}\\- \frac{2 \omega^{2} \rho_{1} a_{1} a_{2} k}{2 \Omega_{2}^{2} a_{1} \mu_{2} - \Omega_{2} \omega^{2} \rho_{1} a_{2} + \omega^{2} \rho_{1} a_{2} k^{2} - 2 a_{1} a_{2} b_{2} k^{2} \mu_{2}}\end{matrix}\right]$

$\displaystyle \left[\begin{matrix}\frac{4 \Omega_{2} a_{2} \mu_{2} \left(\Omega_{2} - k^{2}\right)}{2 \Omega_{2}^{2} a_{1} \mu_{2} - \Omega_{2} \omega^{2} \rho_{1} a_{2} + \omega^{2} \rho_{1} a_{2} k^{2} - 2 a_{1} a_{2} b_{2} k^{2} \mu_{2}} & \frac{4 a_{2} b_{2} k \mu_{2} \left(\Omega_{2} - k^{2}\right)}{2 \Omega_{2}^{2} a_{1} \mu_{2} - \Omega_{2} \omega^{2} \rho_{1} a_{2} + \omega^{2} \rho_{1} a_{2} k^{2} - 2 a_{1} a_{2} b_{2} k^{2} \mu_{2}}\end{matrix}\right]$

固体-液体界面

# left
Pl = sp.Matrix([
    [a1,           a2,           k,  ],
    [rho1*w**2 ,   2*mu2*Omg2,     2*k*mu2*b2,  ],
    [0,            2*k*mu2*a2,    2*mu2*Omg2,  ]
])

# right
Pr = sp.Matrix([
    [(-1)*-a2,            (-1)*k,            a1],
    [(-1)*2*mu2*Omg2,     (-1)*-2*k*mu2*b2,  -rho1*w**2 ],
    [(-1)*-2*k*mu2*a2,    (-1)*2*mu2*Omg2,  0]
])

Q = sp.together(sp.expand(Pl.inv() * Pr))

# 注意索引和之前不同
TD = Q[:1, :2]
RD = Q[1:, :2]
RU = Q[:1, 2:]
TU = Q[1:, 2:]

# 最终表达式和液体-固体界面的情况很类似，
# 仅替换 TD <-> TU, RD <-> RU ，以及部分非对角线上的符号
# 这在编程上提供了便利
display(Math(latex(RD)))
display(Math(latex(RU)))
display(Math(latex(TD)))
display(Math(latex(TU)))

$\displaystyle \left[\begin{matrix}\frac{- 2 \Omega_{2}^{2} a_{1} \mu_{2} - \Omega_{2} \omega^{2} \rho_{1} a_{2} + \omega^{2} \rho_{1} a_{2} k^{2} - 2 a_{1} a_{2} b_{2} k^{2} \mu_{2}}{2 \Omega_{2}^{2} a_{1} \mu_{2} - \Omega_{2} \omega^{2} \rho_{1} a_{2} + \omega^{2} \rho_{1} a_{2} k^{2} - 2 a_{1} a_{2} b_{2} k^{2} \mu_{2}} & \frac{4 \Omega_{2} a_{1} b_{2} k \mu_{2}}{2 \Omega_{2}^{2} a_{1} \mu_{2} - \Omega_{2} \omega^{2} \rho_{1} a_{2} + \omega^{2} \rho_{1} a_{2} k^{2} - 2 a_{1} a_{2} b_{2} k^{2} \mu_{2}}\\\frac{4 \Omega_{2} a_{1} a_{2} k \mu_{2}}{2 \Omega_{2}^{2} a_{1} \mu_{2} - \Omega_{2} \omega^{2} \rho_{1} a_{2} + \omega^{2} \rho_{1} a_{2} k^{2} - 2 a_{1} a_{2} b_{2} k^{2} \mu_{2}} & \frac{- 2 \Omega_{2}^{2} a_{1} \mu_{2} + \Omega_{2} \omega^{2} \rho_{1} a_{2} - \omega^{2} \rho_{1} a_{2} k^{2} - 2 a_{1} a_{2} b_{2} k^{2} \mu_{2}}{2 \Omega_{2}^{2} a_{1} \mu_{2} - \Omega_{2} \omega^{2} \rho_{1} a_{2} + \omega^{2} \rho_{1} a_{2} k^{2} - 2 a_{1} a_{2} b_{2} k^{2} \mu_{2}}\end{matrix}\right]$

$\displaystyle \left[\begin{matrix}\frac{2 \Omega_{2}^{2} a_{1} \mu_{2} + \Omega_{2} \omega^{2} \rho_{1} a_{2} - \omega^{2} \rho_{1} a_{2} k^{2} - 2 a_{1} a_{2} b_{2} k^{2} \mu_{2}}{2 \Omega_{2}^{2} a_{1} \mu_{2} - \Omega_{2} \omega^{2} \rho_{1} a_{2} + \omega^{2} \rho_{1} a_{2} k^{2} - 2 a_{1} a_{2} b_{2} k^{2} \mu_{2}}\end{matrix}\right]$

$\displaystyle \left[\begin{matrix}\frac{4 \Omega_{2} a_{2} \mu_{2} \left(\Omega_{2} - k^{2}\right)}{2 \Omega_{2}^{2} a_{1} \mu_{2} - \Omega_{2} \omega^{2} \rho_{1} a_{2} + \omega^{2} \rho_{1} a_{2} k^{2} - 2 a_{1} a_{2} b_{2} k^{2} \mu_{2}} & \frac{4 a_{2} b_{2} k \mu_{2} \left(- \Omega_{2} + k^{2}\right)}{2 \Omega_{2}^{2} a_{1} \mu_{2} - \Omega_{2} \omega^{2} \rho_{1} a_{2} + \omega^{2} \rho_{1} a_{2} k^{2} - 2 a_{1} a_{2} b_{2} k^{2} \mu_{2}}\end{matrix}\right]$

$\displaystyle \left[\begin{matrix}- \frac{2 \Omega_{2} \omega^{2} \rho_{1} a_{1}}{2 \Omega_{2}^{2} a_{1} \mu_{2} - \Omega_{2} \omega^{2} \rho_{1} a_{2} + \omega^{2} \rho_{1} a_{2} k^{2} - 2 a_{1} a_{2} b_{2} k^{2} \mu_{2}}\\\frac{2 \omega^{2} \rho_{1} a_{1} a_{2} k}{2 \Omega_{2}^{2} a_{1} \mu_{2} - \Omega_{2} \omega^{2} \rho_{1} a_{2} + \omega^{2} \rho_{1} a_{2} k^{2} - 2 a_{1} a_{2} b_{2} k^{2} \mu_{2}}\end{matrix}\right]$