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))

You have learnt that a travelling wave in one dimension is represented

1.	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))
Answer» Solution :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.