Question

You have learnt that a travelling wave in one dimension is represented by a function y= f (x, t)where x and t must appear in the combination x – v t or x + v t, i.e. y = f (x ± v t). Is the converse true? Examine if the following functions for y can possibly represent a travelling wave:

(a) ${\displaystyle {(x–vt)}^2}$

(b) ${\displaystyle \log \left[\frac{x+vt}{x_0}\right]}$

(c) ${\displaystyle \frac{1}{x+vt}}$

Answer 1

No, the converse is not true. The basic requirement for a wave function to represent a travelling wave is that for all values of x and t, wave function must have a finite value. Out of the given functions for y none satisfies this condition. Therefore, none can represent a travelling wave

Answer 2

No;

(a) Does not represent a wave

(b) Represents a wave

(c) Does not represent a wave

The converse of the given statement is not true. The essential requirement for a function to represent a travelling wave is that it should remain finite for all values of x and t.

Explanation:

a) For x = 0 and t = 0, the function (x – vt)² becomes 0.

Hence, for x = 0 and t = 0, the function represents a point and not a wave.

b) For x = 0 and t = 0, the function

${\displaystyle \log \left(\frac{x+vt}{x_0}\right)=\log 0=\propto }$

Since the function does not converge to a finite value for x = 0 and t = 0, it represents a travelling wave

(c) For x = 0 and t = 0, the function

${\displaystyle \frac{1}{x+vt}=\frac{1}{0}=\propto }$

Since the function does not converge to a finite value for x = 0 and t = 0, it does not represent a travelling wave.