Vibration of a Rectangular Membrane
This Demonstration shows the vibration of a 2D membrane for a selected combination of modal vibration shapes. The membrane is fixed along all four edges. You can select any combination of the first five spatial modes . The fundamental mode is given by , . The system obeys the two-dimensional wave equation, given by , where is the amplitude of the membrane's vibration. You can vary the width and length of the membrane using the sliders, the tension, and the surface density, and see the new motion played in time. You can choose a 3D or a contour plot. The contour plot can be used to observe the vibrational modes as it is commonly found in textbook diagrams on this subject.
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.
 R. D. Belvins, Formula for Natural Frequency and Mode Shape, New York: Van Nostrand, 1979.
 R. Engelstad. "ME 740, Advanced Vibration." Class lecture, University of Wisconsin-Madison, Madison, WI, April 2, 2013.
 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.