Ncert Exercises & Solutions

15.5 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 $(x - vt)$ or $(x + vt)$ , i.e. $y = f (x \pm vt)$ . Is the converse true? Examine if the following functions for y can possibly represent a travelling wave :

$(a) {(x - vt)}^{2}$
$(b) \log (\frac{x + vt}{x_{0}})$
$(c) \frac{1}{(x + vt)}$

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,

$\log (\frac{x + vt}{x_{0}}) = \log 0 = \infty$

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

$\frac{1}{x + vt} = \frac{1}{0} = \infty$

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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