We have already seen a derivation of the modal density for bending modes of a plate, in section 3.2.4. But there is a neat argument due to Weinreich [1] which shows how this result can be generalised to a range of systems of interest to us. Figure 1 shows a repeat of Fig. 1 from section 3.2.4, indicating the modes of a rectangular plate in 2D wavenumber space: they mark out a regular grid. But something very similar can be said for many other systems. The modes of a stretched string or a pinned-pinned bending beam correspond to points that are regularly spaced in wavenumber, along a line (i.e. in 1D wavenumber space). An acoustic volume, on the other hand, has points that form a regular grid filling 3D wavenumber space, according to eq. (6) of section 4.2.
If we now want to ask how many modes have natural frequencies below some chosen value $\omega$, we first convert this into a limit of wavenumber $k$ using the equation of motion of the system in question. For any system satisfying the wave equation, in 1D, 2D or 3D, the answer is that wavenumber is simply proportional to frequency. But for bending problems, either for beams or for plates, the relation is that frequency is proportional to the square of wavenumber. So for all these systems, we can say that
$$k \propto \omega^\alpha \tag{1}$$
where $\alpha=1$ for the wave equation and $\alpha = 1/2$ for bending problems.
Now to find out the number of modes below $\omega$ we have to count the number of points on the wavenumber grid lying within a distance $k$ of the origin. The points are uniformly distributed over the line/plane/space, so for high frequencies this mode count is approximately proportional to the enclosed length/area/volume, depending on whether the dimension $d$ is 1, 2 or 3. The result for the mode count $N(\omega)$ is that
$$N(\omega) \propto k^d \propto \omega^{\alpha d} . \tag{2}$$
Now the modal density $n(\omega)$, which is the inverse of the mean spacing between adjacent modes, is given by the derivative of $N(\omega)$, so that
$$n(\omega) \propto \omega^{\alpha d -1} . \tag{3}$$
The results can be collected in a table:
| $\alpha = 1$ | $\alpha = 1/2$ | |
| $d=1$ | String, pipe | Bending beam |
| constant | $\omega^{-1/2}$ | |
| $d=2$ | Membrane | Bending plate |
| $\omega$ | constant | |
| $d=3$ | Acoustic volume | |
| $\omega^2$ |
A stretched string and an acoustic pipe both have constant modal density on average, and they share this behaviour with a bending plate. A bending beam has modal density that decreases as frequency increases. A membrane (like a drum or the head of a banjo) has modal density increasing proportional to frequency, and an acoustic volume has modal density increasing proportional to the square of frequency, as seen in Fig. 17 of section 4.2.
The strictly regular grid in wavenumber space corresponds to particular special cases for each type of system. However, as was argued for the plate in section 3.2.4, the fact that the points cover the relevant wavenumber space in a statistically uniform manner remains true for the general case of all these systems.
[1] G. Weinreich, section 3.1.7 of “Mechanics of musical instruments”, ed. A. Hirschberg, J. Kergomard and G. Weinreich; CISM Courses and Lectures no. 355, Springer-Verlag (1995)