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 `ax +- bt` or `x - vt` or `x + vt`,i.e. `y = f (x +- vt)`. Is the converse true? Examine if the folliwing function for `y` can possibly represent a travelling wave
(a) `(x - vt)^(2)`
(b) `log[(x + vt)//x_(0)]`
(c) `1//(x + vt)`

Explanation

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