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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 [(x + vt)/x_0]`
(c) `1/(x + vt)`
Solution 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
Solution 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
`log ((x+vt)/x_0) = log 0 = prop`
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
`1/(x + vt) = 1/0 = prop`
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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