# Vibrations of a Rectangular Membrane

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 Title Vibrations of a Rectangular Membrane Alternative Title Schwingungen einer rechteckigen Membran Author Halter, Eberhard Contributors Gotthard Glatzer (Redaktion) Eberhard Halter (Kamera) L. Rüppel (Schnitt) License CC Attribution - NonCommercial - NoDerivatives 3.0 Germany:You are free to use, copy, distribute and transmit the work or content in unchanged form for any legal and non-commercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor. DOI 10.3203/IWF/C-1324eng IWF Signature C 1324 Publisher IWF (Göttingen) Release Date 1979 Language English Producer Universität Karlsruhe, Mathematisches Institut II Production Year 1978

 IWF Technical Data Film, 16 mm, LT, 135 m ; SW, 12 1/2 min

 Subject Area Mathematics Abstract The single steps of the complex vibrational forms of a rectangular membrane were calculated on a computer and filmed from the display. They permit a detailed analysis of different vibrational modes under special initial and boundary values. The superposition is demonstrated separately. Keywords vibration / technical examples computer simulation vibration behaviour / membrane self oscillation membrane / vibration behaviour

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Vibrations of a rectangular membrane. The Boundary Value Problem The free vibrations of a membrane are described by the homogenous wave equation. The membrane is attached along its boundary to a fixed plane curve, that is we have the boundary
condition u=0.
In the case of a rectangular membrane one obtains a set of solutions of this boundary value problem by the method of separation of variables. A Set of Solutions Each solution is characterized by two indexing
numbers m and n. The solution with the indices m=1 and n=1, namely u ll, has the lowest frequency. Further examples: u 21, u 12,
u 22, u 31 and finally u 13. Graphically the parameters m and n signify the number of hills and
valleys, which are encountered while crossing the membrane along a parallel to one of the boundary lines.
Here you see u 32, u 33. Solutions with larger m and n obviously vibrate at higher frequencies. Here is the case m=4, n=5. For comparison, u ll again. The Superposition Principle Constant multiples and sums of solutions are additional solutions of the boundary value problem. The present example envolves u 11, u 12 and u 22. In the initial--boundary value problem in addition to the conditions already stated, the solution u and its time derivative u t must assume prescribed values in the rectangle at t=0. With these initial conditions the solution is uniquely determined. The Initial-Boundary Value Problem First example: The initial form is that of a pyramid. The initial velocity is zero throughout.
The solution is periodic. Despite the presence of sharp edges the solution can be obtained as the limit of a linear superposition of the functions u mn. Second example: The initial form is that of a hiproof. Again the initial velocity is zero everywhere. Here the solution is very similar to the solution of the first example. This demonstrates the continuous dependence of solutions on initial values. Third example: In comparison to the second example we lengthen the ridge of the roof. Now the solution is easily distinguished from the solution of the first example, where the initial form was a pyramid. Fourth example: We lengthen the ridge once again. There is no longer any similarity to the pyramid solution.
Fifth example: Initially the membrane is at equilibrium but it has negative velocity near its center. There is no observable periodicity. Here one sees clearly how the solution is built up through the superposition of many small waves. These examples are only a small selection from the numerous interesting phenomena envolving free vibrations of a rectangular membrane.
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