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) (x – vt )² 

(b) log \([\frac{x + vt}{x_0} ] \)

(c) \(\frac{1}{(x + vt)}\)

Answer 1

No; 

Does not represent a wave Represents a wave 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:

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. 

For x = 0 and t = 0, the function

log \((\frac{x+vt}{x_0})=logo=∞\)

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

For x = 0 and t = 0, the function

 log \(\frac{1}{x+vt}=\frac{1}{0}=∞\)

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