Methods | Part IB, 2002

Write down the wave equation for the displacement y(x,t)y(x, t) of a stretched string with constant mass density and tension. Obtain the general solution in the form

y(x,t)=f(x+ct)+g(xct)y(x, t)=f(x+c t)+g(x-c t)

where cc is the wave velocity. For a solution in the region 0x<0 \leqslant x<\infty, with y(0,t)=0y(0, t)=0 and y0y \rightarrow 0 as xx \rightarrow \infty, show that

E=0[12(yt)2+12c2(yx)2]dxE=\int_{0}^{\infty}\left[\frac{1}{2}\left(\frac{\partial y}{\partial t}\right)^{2}+\frac{1}{2} c^{2}\left(\frac{\partial y}{\partial x}\right)^{2}\right] d x

is constant in time. Express EE in terms of the general solution in this case.

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