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 + or v + vt, i.e. y = f ( x pm vt).` Is the converse ture ? Examine if the following functions for y can possibly represent a travelling wave:
(a) `(x-vt)^(2)`
(b) `log [ ((x + v _(t)))/(x _(0))]`
(c ) `(1)/((x + vt))`

Explanation

We function for one dimensional traveling wave should be:
`y = a sin (omega t pm kx)`
`=a sin {k ((omega )/(k) t pm x )}`
`y =a sin (k (vt pm x)} (because v = (omega)/(k)=` wavevelocity)
`implies y =f (vt pm x) OR y =f (x pm vt)`
Above equation for traveling wave satisfied following condition:
(i) It is continuous for all values of x and t.
(ii) Its range is finite. (Here, vlaues of y are finite in the range `-a le y le a)`
(iii) It gives definite value foe given vaue of x and t.
Now in present example, the functions given in option (a), (b) , (c) do not satisfy above conditions because of following reasons.
(a) Function `(x-vt)^(2)` has infinite range from 0 to `+ oo`
(b) Function log `((x + vt)/(x _(0)))` does not give definite value for given value of x and t because it has series expansion.
(c) Function `((1)/( x + vt))` is not continuous because here we can not take `x =- vt.`
Thus, all the three given functions do not represent traveling wave.
Hence, it is not necessary that every function `f (x pm vt) OR f (vt pm x)` should represenet travelling wave.