Consider a plane wave at frequency $\omega$ propagating on an infinite isotropic plate and driving the motion of air on one side of the plate. The plate motion is governed by this equation, from section 3.2.3:

$$\rho h \dfrac{\partial^2 w}{\partial t^2}+EK \left[\frac{\partial^4 w}{\partial x^4}+2\frac{\partial^4 w}{\partial x^2 \partial y^2} +\frac{\partial^4 w}{\partial y^4} \right]=0\tag{1}$$

for a plate of thickness $h$, with a constant $K$ given by

$$K=\frac{h^3}{12(1-\nu^2)} \tag{2}$$

where $E$ is Young’s modulus, $\rho$ is density and $\nu$ is Poisson’s ratio. If the wave takes the form $w=e^{i \omega t – i k x}$ then eq. (1) requires

$$EK k^4=\rho h \omega^2. \tag{3}$$

The speed of the wave will be given by

$$\frac{\omega}{k} = \left( \dfrac{EK}{\rho h} \right)^{1/4} \sqrt{\omega} ,\tag{4}$$

rising proportional to the square root of frequency.

Now pressure in the air above the plate will satisfy a two-dimensional version of the wave equation:

$$\frac{\partial^2 p}{\partial t^2}= c^2 \left[ \frac{\partial^2 p}{\partial x^2} + \frac{\partial^2 p}{\partial z^2} \right] \tag{5}$$

where $z$ is the coordinate normal to the plate. The pressure will vary harmonically in time at the same frequency as the plate wave, and it must have a wavenumber in the $x$ direction equal to $k$, for continuity with the plate motion. What remains is to find out what happens in the $z$ direction. We can look for a solution of the form

$$p=p_n(z) e^{i \omega t -i k x} . \tag{6}$$

Equation (5) then requires

$$\omega^2 = c^2\left( k^2 -\frac{d^2 p_n}{d z^2} \right)$$

so that

$$\frac{d^2 p_n}{d z^2} + (k_a^2-k^2) p_n =0 \tag{7}$$

where $k_a=\omega/c$ is the wavenumber of sound waves at frequency $\omega$.

The solutions of eq. (7) depend on the sign of the bracketed term. If $k_a > k$, the term is positive and solutions are sinusoidal. In that case, waves propagate away from the plate at an angle governed by the ratio of $k$ to $\sqrt{(k_a^2-k^2)}$. If, on the other hand, $k_a < k$ then the bracketed term is negative, and the solution of interest is an exponential decay of sound pressure away from the plate. There is no propagating wave: the sound field is evanescent. Examples of these two cases were illustrated with animations in Figs. 11 and 12 of section 4.3.

The crossover between the two regimes occurs when $k_a = k$, in other words when

$$\frac{\omega_c}{c} = \left( \dfrac{\rho h}{EK} \right)^{1/4} \sqrt{\omega_c} \tag{8}$$

from eq. (3). It follows that this *critical frequency* is given by

$$\omega_c = c^2 \left( \dfrac{\rho h}{EK} \right)^{1/2} . \tag{9}$$