The characteristic frequencies for 2D membrane motion are given by

, where

is the length in the

dimension and

is the length in the

direction. These characteristic frequencies or eigenvalues are the frequencies of the membrane's vibrations. The spatial frequencies are given by

and

along the

and

directions, respectively.

The full solution of the PDE is a linear combination of the spatial and time components of the solution obtained by separation of variables and is given by

. The coefficients

and

are found from initial conditions. It simplifies this Demonstration to assume that the initial conditions (position and velocity of the membrane) are such that

and

. Hence the solution becomes

and this is the solution that is animated.

This Demonstration supports modes up to

and

. You select the parameter values of

,

,

, and

from the sliders and see the resulting vibrations. The wave speed is

, where

is the tension the membrane bears per unit length of its boundary. Hence

has units of

and

is the membrane mass per unit of surface area; therefore

has units of

per

. The parameter

represents the wave speed (in the transverse direction) in units of

per

.

Tension is assumed constant, gravity is ignored, and no damping is assumed.

The tension and density parameters are expressed in

and internally converted to the SI unit of meters.

A table of the characteristic frequencies

is on the left in units of Hz. You select the modes to excite by using the dialog shown on the left. A mode is selected and unselected by pressing on the button specific for that mode. Mouseover the 3D plot to see the full analytical solution using the selected modes.

The membrane is fixed on all four edges.

[1] R. D. Belvins,

*Formula for Natural Frequency and Mode Shape*, New York: Van Nostrand, 1979.

[2] R. Engelstad. "ME 740, Advanced Vibration." Class lecture, University of Wisconsin-Madison, Madison, WI, April 2, 2013.

[3] H. Esoy, "Free Vibration Analysis of Rectangular Membranes with Variable Density Using the Discrete Singular Convolution Approach,"

*Asian Journal of Civil Engineering (Building and Housing)*,

**11**(1), 2010 pp. 83–94.