Higher Dimensions

and Other

Coordinates

CHAPTER

5

5.1 Two-Dimensional Wave Equation: Derivation

For an example of a two-dimensional wave equation, we consider a membrane
that is stretched taut over a flat frame in thexy-plane (Fig. 1). The displace-
ment of the membrane above the point(x,y)at timetisu(x,y,t).Weassume
that the surface tensionσ(dimensionsF/L) is constant and independent of
position. We also suppose that the membrane is perfectly flexible; that is, it
does not resist bending. (A soap film satisfies these assumptions quite accu-
rately.) Let us imagine that a small rectangle (of dimensions xby yaligned
with the coordinate axes) is cut out of the membrane, and then apply Newton’s
law of motion to it. On each edge of the rectangle, the rest of the membrane ex-
erts a distributed force of magnitudeσ(symbolized by the arrows in Fig. 2a);
these distributed forces can be resolved into concentrated forces of magnitude
σ xorσ y, according to the length of the segment involved (see Fig. 2b and
Fig. 3).

Figure 1 Frame in thexy-plane.

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