Definitions and approximations

An ideal rectangular membrane is in every respect the two-dimensional counterpart of an ideal string.

The parameters that describe the membrane are summarised in the following table:

Membrane fixed at the edge

${\displaystyle f_{n_x,n_y}=\frac{v}{2\pi }\sqrt{{\left(\frac{n_x\pi }{L_x}\right)}^2+{\left(\frac{n_y\pi }{L_y}\right)}^2}}$, with ${\displaystyle v=\sqrt{T/\mu }}$

In the particular case of a square membrane with side ${\displaystyle L=L_x=L_y}$:

${\displaystyle f_{n_x,n_y}=\frac{v}{2L}\sqrt{n_x^2+n_y^2}}$

In a square membrane different modes can have the same frequency: for example the modes (1,2) and (2,1), which differ only by a 90° rotation. These modes are said to be degenerate. Any superposition of them still oscillates at the same frequency, but with nodal lines of different shapes, for example along a diagonal of the square.

Further details

The general wave vector is two-dimensional, with components ${\displaystyle k_x}$ and ${\displaystyle k_y}$ and magnitude

${\displaystyle k=\sqrt{k_x^2+k_y^2}}$,

and the corresponding frequency is ${\displaystyle f=vk/(2\pi )}$.

The boundary conditions require

${\displaystyle k_x=\frac{n_x\pi }{L_x}{n}_x=1,2,\dots }$

${\displaystyle k_y=\frac{n_y\pi }{L_y}{n}_y=1,2,\dots }$

Substituting these values into the expression for the frequency gives the formula of the previous section.

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