Example in Acoustics

There are examples of acoustics systems, made of two or more tightly coupled media, whose wave propagation can be simulated by a multivariable waveguide section. One such system is an elastic, porous solid [28, pp. 609-611], where the coupling between gas and solid is given by the frictional force arising when the velocities in the two media are not equal. The wave equation for this acoustic system is (26), where the matrix ${\mbox{\boldmath$\Phi$}}$ takes form

$${\mbox{\boldmath$\Phi$}}= \left[ \begin{array}{rr} \Phi & -\Phi \\ -\Phi & \Phi \end{array}\right] \; ,$$ (35)

and $\Phi$ is a flow resistance. The stiffness and the mass matrices are diagonal and can be written as

$${\mbox{\boldmath$K$}}= \left[ \begin{array}{ll} k_a & 0 \\ 0 & k_b \end{array}\right] \; ;$$ (36)

$${\mbox{\boldmath$M$}}= \left[ \begin{array}{ll} \mu_a & 0 \\ 0 & \mu_b \end{array}\right] \; .$$ (37)

Let us try to enforce a traveling wave solution with spatial and temporal frequencies $V$ and $\omega_s$, respectively:

$${\mbox{\boldmath$p$}}= {\mbox{\boldmath$p$}}_0 e^{j(V x - \o... ... t)} = \left[ \begin{array}{l} p_a \\ p_s \end{array}\right] ,$$ (38)

where $p_a$ and $p_s$ are the pressure wave components in the gas and in the solid, respectively. We easily obtain from (26) the two relations

$$j \omega_s \Phi {k_s}^{-1} p_s = ({\omega_s}^2 {\alpha_a}^2 - V^2) p_a$$ (39)

$$j \omega_s \Phi {k_a}^{-1} p_a = ({\omega_s}^2 {\alpha_s}^2 - V^2) p_s \; ,$$ (40)

where

$${\alpha_a}^2 = {c_a}^{-2} + j \frac{k_s \Phi}{\omega}$$ (41)

$${\alpha_s}^2 = {c_s}^{-2} + j \frac{k_a \Phi}{\omega}$$ (42)

and $c_a$ and $c_s$ are the sound speeds in the gas and in the solid, respectively. By multiplying together both members of (40) and (41) we get

$$\begin{array}{lcl} V^2 & = & \frac{1}{2} {\omega_s}^2 ({\alp... ...\alpha_s}^2)^2 - {4 {\omega_s}^2 \Phi^2 k_a k_b}} . \end{array}$$ (43)

Equation (44) gives us a couple of complex numbers for $v$, i.e., two attenuating traveling waves forming a vector ${\mbox{\boldmath$p$}}$ as in (32). It can be shown [28, p. 611] that, in the case of small flow resistance, the faster wave propagates at a speed slightly slower than $c_s$, and the slower wave propagates at a speed slightly faster than $c_a$. It is also possible to show that the admitance matrix (35) is non-diagonal and frequency dependent.

This example is illustrative of cases in which the matrices ${\mbox{\boldmath$K$}}$ and ${\mbox{\boldmath$M$}}$ are diagonal, and the coupling among different media is exerted via the resistance matrix ${\mbox{\boldmath$\Phi$}}$. If ${\mbox{\boldmath$\Phi$}}$ approaches zero, we are back to the case of decoupled waveguides. In any case, two pairs of delay lines are adequate to model this kind of system.