Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 39A (Waves)

The linear homentropic acoustic equations areDifferentiate the first equation in time, take the divergence of the second, and use the third. This eliminates ${\rho }^{\prime }$ and $\mathbf{u}$ and gives the acoustic pressure wave equation

For a progressive plane wave, writeThe momentum equation gives ${\rho }_0\omega U=kP$, while the dispersion relation is $\omega =c_{0}k$. Therefore the pressure and longitudinal velocity amplitudes obeyThe constant $Z$ is the acoustic impedance.

Let $Z_j={\rho }_j{c}_j$, $k_j=\omega /c_j$, and $\varphi =k_{1}L$. Use the harmonic convention $e^{-i\omega t}$ and write the pressure and velocity amplitudes asContinuity of pressure and normal velocity at both interfaces is equivalently the layer transfer relationSubstituting $p(0)=1+R$, $u(0)=(1-R)/Z_0$, $p(L)=T$, and $u(L)=T/Z_0$, then adding the two resulting equations, givesHence the acoustic transmission through a uniform layer isFor either $\lambda \ll 1$ or $\lambda \gg 1$, the strong acoustic impedance mismatch makes transmission small except near $k_{1}L=n\pi$, where $|T|=1$. Away from those resonances,At the antiresonant points $k_{1}L=(n+\frac{1}{2})\pi$, the minimum is $2/(\lambda +{\lambda }^{-1})$.