What is an ideal membrane

An ideal membrane is the exact two-dimensional counterpart of an ideal string.

Like the ideal string, it is characterised by only two parameters: the density (in this case the mass per unit area ${\displaystyle \rho }$), and the surface tension T (which has the dimensions of a force per unit length, that is, it is measured in Nm^-1 in the SI system)

The membrane is ideal if

1. T is constant during the motion of the membrane (that is, whatever its deformation)
2. T has the same magnitude over the whole membrane, and, at every point, acts only in the direction tangent to the membrane (that is, without components perpendicular to the surface).
3. all other external forces (such as gravity) are negligible compared with T

How the equation is derived

• The sum of the two gives

${\displaystyle d{F}_z=d{F}_z^x+d{F}_z^y=Tdxdy(\frac{{\partial }^{2}z}{\partial x^2}+\frac{{\partial }^{2}z}{\partial y^2})}$

This force must be equated to the mass times acceleration term, which is

${\displaystyle dm\cdot a=\rho dxdy\frac{{\partial }^{2}z}{\partial t^2}}$

• This gives the two-dimensional version of the classical wave equation

${\displaystyle \frac{{\partial }^{2}z}{\partial x^2}+\frac{{\partial }^{2}z}{\partial y^2}=\frac{1}{c^2}\frac{{\partial }^{2}z}{\partial t^2}}$,

with

${\displaystyle c=\sqrt{\frac{T}{\rho }}}$,

in complete analogy with the case of the string.

Circular membrane

The equation we have written in Cartesian coordinates describes all waves in a membrane, whatever its shape, but it is particularly suited to square or rectangular membranes, because that is the shape of the grid formed by x and y.

In this case the normal modes can be deduced from the normal modes of a string separately in the x and y directions, and therefore the solution of the motion will in general be a superposition of sines and cosines. The details of which combination obviously depend on the boundary conditions and on the initial conditions of the problem.

However, the case in which the membrane is circular is particularly important, because this is the shape in which it appears in many musical instruments (typically some percussion instruments such as drums and timpani).

Although the equation derived above can be used, it is particularly awkward to express the circle in Cartesian coordinates, and it is more direct to switch to polar coordinates through the transformation

${\displaystyle x=r\cos (\varphi )}$,

${\displaystyle y=r\sin (\varphi )}$.

Leaving out the details, we arrive directly at the form taken by the wave equation after this change of coordinates:

${\displaystyle \frac{{\partial }^{2}z}{\partial r^2}+\frac{1}{r}\frac{\partial z}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}z}{\partial {\varphi }^2}=\frac{1}{c^2}\frac{{\partial }^{2}z}{\partial t^2}}$. (1)

We can now separate the variables by looking for a solution of the form

${\displaystyle z(r,\varphi ,t)=R(r)\mathrm{\Phi }(\varphi )\cos (\omega t)}$,

which, substituted into equation (1), splits it into two separate equations, one for ${\displaystyle \mathrm{\Phi }}$, and one for R.

The equation for ${\displaystyle \mathrm{\Phi }}$ is simply that of harmonic motion

${\displaystyle \frac{{\partial }^2\mathrm{\Phi }}{\partial {\varphi }^2}+m^2\mathrm{\Phi }=0}$.

This means that along every concentric circle the profile of the membrane will always be that of a sine (or cosine), ${\displaystyle \mathrm{\Phi }=\cos (m\varphi )}$. Since after a full turn (${\displaystyle \varphi \to \varphi +2\pi }$) we are back at the same point of the membrane, the profile must repeat: m must therefore be an integer (0, 1, 2, …). The number m counts the nodal diameters of the mode, that is the straight lines through the centre along which the membrane stays still.

The equation for R is a new, more complicated type of equation, whose solutions are special functions (called cylindrical Bessel functions), which determine the radial profile of the drum membrane.

${\displaystyle \frac{{\partial }^{2}R}{\partial r^2}+\frac{1}{r}\frac{\partial R}{\partial r}+(\frac{{\omega }^2}{c^2}-\frac{m^2}{r^2})R=0}$.

Of the two solutions of this equation, only the Bessel function of the first kind ${\displaystyle J_m(\omega r/c)}$ stays finite at the centre of the membrane: it is therefore the only acceptable one.

The natural frequencies are obtained by requiring the membrane, of radius a, to be fixed at the edge: ${\displaystyle R(a)=0}$, that is ${\displaystyle J_m(\omega a/c)=0}$. If ${\displaystyle j_{mn}}$ is the n-th zero of the function ${\displaystyle J_m}$, the natural frequencies are

${\displaystyle {\omega }_{mn}=j_{mn}\frac{c}{a}}$.

The zeros of the Bessel functions are not integer multiples of one another (${\displaystyle j_{01}}$ ≈ 2.405, ${\displaystyle j_{11}}$ ≈ 3.832, ${\displaystyle j_{21}}$ ≈ 5.136, ${\displaystyle j_{02}}$ ≈ 5.520, …): the natural frequencies are therefore in the ratios 1 : 1.59 : 2.14 : 2.30 : … and do not form a harmonic series. This is why the sound of a drum does not usually have a well-defined pitch (see the page natural frequencies of the circular membrane).

A visualisation of some of these functions is shown on the page Normal modes of a circular membrane.

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