   Chapter 14.5, Problem 49E

Chapter
Section
Textbook Problem

Assume that all the given functions have continuous second-order partial derivatives.49. Show that any function of the formz = f(x + at) + g(x − at)is a solution of the wave equation ∂ 2 z ∂ t 2 = a 2 ∂ 2 z ∂ x 2 [Hint: Let u = x + at, v = x – at.]

To determine

To show: Any function is of the form z=f(x+at)+g(xat) is satisfies the wave equation 2zt2=a22zx2 .

Explanation

Proof:

The given function is, z=f(x+at)+g(xat) .

Let u=x+atandv=xat .

Thus, the function can be expressed as, z=f(u)+g(v) .

The equation, 2zt2=a22zx2 is to be proved.

The partial derivative, zx is computed as follows,

zx=fuux+gvvx=f'(u)x(x+at)+g'(v)v(xat)=f'(u)(1)+g'(v)(1)=f'(u)+g'(v)

Thus, the partial derivative, zx=f'(u)+g'(v) (1)

Take the partial derivative with respect to x in the equation (1),

2zx2=x[f'(u)+g'(v)]=f''(u).ux+g''(v).vx=f''(u).(1)+g''(v).(1)=f''(u)+g''(v)

Thus, 2zx2=f''(u)+g''(v)

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