Chapter 16, Problem 22P

### Physics for Scientists and Enginee...

10th Edition
Raymond A. Serway + 1 other
ISBN: 9781337553278

Chapter
Section

### Physics for Scientists and Enginee...

10th Edition
Raymond A. Serway + 1 other
ISBN: 9781337553278
Textbook Problem

# (a) Show that the function y(x, t) = x2 + v2t2 is a solution to the wave equation. (b) Show that the function in part (a) can be written as f(x + vt) + g(x − vt) and determine the functional forms for f and g. (c) What If? Repeat parts (a) and (b) for the function y(x, t) = sin (x) cos (vt).

(a)

To determine

To shows: That the wave function y(x,t)=x2+v2t2 is a solution to the linear wave equation.

Explanation

Any function is a solution of linear wave equation in general if it satisfies the equation completely.

The linear wave equation in general is,

â€‚Â âˆ‚2yâˆ‚x2=1v2âˆ‚2yâˆ‚t2

The given wave function is,

â€‚Â y(x,t)=x2+v2t2Â Â Â Â Â Â Â Â (I)

Differentiate equation (I) partially with respect to x.

â€‚Â âˆ‚y(x,t)âˆ‚x=âˆ‚âˆ‚x(x2+v2t2)=2x

Again differentiate partially with respect to x.

â€‚Â âˆ‚2y(x,t)âˆ‚x2=âˆ‚âˆ‚x(2x)=2Â Â Â Â Â Â Â Â (II)

Differentiate equation (I) partially with respect to t

(b)

To determine

To shows: That the wave function y(x,t)=x2+v2t2 can be written as f(x+vt)+g(xvt) and determine the functional form of f and g.

(c)

To determine

Repeat part (a) and part (b) for the function y(x,t)=sin(x)cos(vt).

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