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# Vibrations of a Rectangular Membrane

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#### Automatisierte Medienanalyse

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Sprachtranskript
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.
Ebene
Randwert
Wellenlehre
Schwingung
Freie Gruppe
Erschütterung
Computeranimation
Trennungsaxiom
Randwert
Variable
Menge
Konditionszahl
Indexberechnung
Computeranimation
Parametersystem
Zahlenbereich
Hill-Differentialgleichung
Geschwindigkeit
Addition
Gewichtete Summe
Rechteck
Anfangswertproblem
Derivation <Algebra>
Bilinearform
Frequenz
Superposition <Mathematik>
Computeranimation
Konstante
Randwert
Multiplikation
Konditionszahl
Gerade
Geschwindigkeit
Lineares Funktional
Wellenlehre
Ähnlichkeitsgeometrie
Anfangswertproblem
Thermodynamisches Gleichgewicht
Bilinearform
Paarvergleich
Superposition <Mathematik>
Frequenz
Trennschärfe <Statistik>
Schwingung
Inverser Limes
Analytische Fortsetzung

### Metadaten

#### Formale Metadaten

 Titel Vibrations of a Rectangular Membrane Alternativer Titel Schwingungen einer rechteckigen Membran Autor Halter, Eberhard Mitwirkende Gotthard Glatzer (Redaktion) Eberhard Halter (Kamera) L. Rüppel (Schnitt) Lizenz CC-Namensnennung - keine kommerzielle Nutzung - keine Bearbeitung 3.0 Deutschland:Sie dürfen das Werk bzw. den Inhalt in unveränderter Form zu jedem legalen und nicht-kommerziellen Zweck nutzen, vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen. DOI 10.3203/IWF/C-1324eng IWF-Signatur C 1324 Herausgeber IWF (Göttingen) Erscheinungsjahr 1979 Sprache Englisch Produzent Universität Karlsruhe, Mathematisches Institut II Produktionsjahr 1978

#### Technische Metadaten

 IWF-Filmdaten Film, 16 mm, LT, 135 m ; SW, 12 1/2 min

#### Inhaltliche Metadaten

 Fachgebiet Mathematik 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. Schlagwörter vibration / technical examples computer simulation vibration behaviour / membrane self oscillation membrane / vibration behaviour