Mathematical Methods in the Physical Sciences · Problem 9.16

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · miscellaneous-square-membrane-nodal

Problem

Consider the normal modes of vibration for a square membrane of side π\pi (see Problem 6.3). Sketch the 2,1 and 1,2 modes. Show that the combination sin⁡xsin⁡2y−sin⁡2xsin⁡y\sin x \sin 2y - \sin 2x \sin y of these two modes has a nodal line along y=xy = x. Thus find a vibration frequency of a membrane in the shape of a 45∘45^\circ right triangle.

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