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 ${\displaystyle L=L_{x}=L_{y}}$:

${\displaystyle f_{n_{x},n_{y}}={\frac {v}{2L}}{\sqrt {n_{x}^{2}+n_{y}^{2}}}}$

Further details

The general wave vector is two-dimensional

${\displaystyle k={\sqrt {k_{x}^{2}+k_{y}^{2}}}}$,

The boundary conditions require

${\displaystyle k_{x}={\frac {n_{x}\pi }{L_{x}}}\quad n_{x}=1,2,\ldots }$

${\displaystyle k_{y}={\frac {n_{y}\pi }{L_{y}}}\quad n_{y}=1,2,\ldots }$