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) / x0]

(c) 1 / (x+vt)

Answer 1

The is not true, means f any function representing in the form of y = f(x ±vt) it doesn't necessarily express a travelling wave .

The essential condition is that
if any function y = f(x ± vt ) satisfied
d²y/dt² = v²d²y/dx²

(a) it's wave equation ,
y = ( x - vt)²
dy/dt = -2v(x - vt)
dy/dt = 2v²
Again ,
dy/dx =2 (x -vt)
d²y/dx² = 2
d²y/dt² = v².d²y/dx²
Hence , it's wave equation.

(b) log{(x + vt)/xo}
It's not wave equations becoz it doesn't follow d²y/dt² = v².d²y/dx²
Also at x = 0, and t = 0
Log(0) → ∞ infinte
Hence , it's not wave equation.

(c) 1/(x + vt)
It's not wave equation .
d²y/dt² = v².d²y/dx²