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Chapter 16: Q.16 (page 451)

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

Short Answer

Expert verified

The given displacement is a solution to the wave equation.

The wave speed is b/a .

Step by step solution

01

Given information:

The displacement is given by D(x,t)=ln(ax+bt)where a, b are constants

02

Calculate the partial derivatives to show that the given displacement satisfy 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�ln(ax+bt)鈭�x=aax+bt鈭�2D鈭�x2=鈭�鈭�xaax+bt=-a2(ax+bt)2

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

鈭�D鈭�t=鈭�ln(ax+bt)鈭�t=bax+bt鈭�2D鈭�t2=鈭�鈭�tbax+bt=-b2(ax+bt)2

Substitute the values in the above wave equation:

鈭�2D鈭�t2=v2鈭�2D鈭�x2-b2(ax+bt)2=v2-a2(ax+bt)2v2=b2a2v=ba

Therefore, the given displacement satisfies the wave equation.

03

Calculating the wave speed:

As calculated in the above equation, the wave speed is obtained asv=ba.

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