Short-wavelength linear waves propagation in periodic media
We want to study the problem of linear wave propagation in periodic media. We consider initial conditions with short wavelengths that are equal to or less than the medium period length.
The goal is trying to find a closed form solution for the system at time \(n\), given the initial data. We consider a 1D acoustics system:
\(\begin{eqnarray*} q_t + A(x)q_x=0 \end{eqnarray*}\) where, \(\begin{eqnarray*} q(x,t) = \begin{pmatrix} p(x,t) \\ u(x,t) \end{pmatrix} \end{eqnarray*}\) and \(\begin{eqnarray*} A(x) = \begin{pmatrix} 0& K(x) \\ 1/\rho(x) & 0 \end{pmatrix} \end{eqnarray*}\)
Where \(rho(x)\) and \(K(x)\) are constants \(\rho_A\) and \(K_A\) in layer \(A\) and \(\rho_B\) and \(K_B\) in layer \(B\).
Reflection and transmission coefficients
Consider a pressure wave of magnitude \(p_0\) hitting the interface \(AB\) Then the new wave will have the transmitted part is \(T_{AB} p_0\), and the reflected part is \(R_{AB}p_0\), where
\(\begin{eqnarray*} T_{AB} = \frac{2Z_B}{Z_A+Z_B},\\ R_{AB} = \frac{Z_B - Z_A}{Z_A+Z_B}. \end{eqnarray*}\)
Similarly, we have
\(\begin{eqnarray*} T_{BA} = \frac{2Z_A}{Z_A+Z_B},\\ R_{BA} = \frac{Z_A - Z_B}{Z_A+Z_B}. \end{eqnarray*}\)
From the above four formulas, we can see that there are some relations between those coefficients:
\(\begin{eqnarray*} R_{AB} = - R_{BA},\\ T_{AB}+T_{BA}=2,\\ T_{AB}-R_{AB}=1,\\ T_{BA}-R_{BA}=1. \end{eqnarray*}\)
