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Chapter 16: Q.15 (page 451)

Show that the displacement D(x,t)=cx2+dt2, where c and d are constants, is a solution to the wave equation. Then find an expression in terms of c and d for the wave speed.

Short Answer

Expert verified

The given displacement is a solution to the wave equation.

The expression for the wave speed isv=dc.

Step by step solution

01

Given :

The displacement is given byD(x,t)=cx2+dt2where c, d are constants.

02

Calculating the partial derivatives to show that the given displacement satisfy the wave equation:

The given displacement must satisfy the wave equation Take the partial derivatives of D(x, t) with respect to x,

鈭�D鈭�x=鈭�cx2+dt2鈭�x=2cx鈭�2D鈭�x2=2c

Take the partial derivatives of D(x, t) with respect to t,

鈭�D鈭�t=鈭�cx2+dt2鈭�t=2dt鈭�2D鈭�t2=2d

Substituting the above values in the wave equation:

鈭�2D鈭�t2=v2鈭�2D鈭�x22d=v2.2cv2=dc

Therefore, it satisfies the wave equation.

03

Calculating the speed of the wave:

From the above equation, the wave speed is obtained as

v2=dcv=dc

Therefore, the wave speed isv=dc.

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