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)

(c)

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Solution

Answer: 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)²becomes 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

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

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

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