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 - vt or x + vt, i.e. y = f(x ± vt). Is the converse true? Examine if the following functions for y can possibly represent a travelling wave:

Accepted Answer

The converse is not true. While every travelling wave must have the form y = f(x ± vt), not every function of this form represents a valid travelling wave. A travelling wave must be bounded and represent a finite disturbance in the medium. (a) (x - vt)²: This function is unbounded. As x or t increases, the value grows to infinity. Hence, it cannot represent a travelling wave. (b) log[(x + vt)/x₀]: This function is unbounded. It approaches infinity as the argument increases, and negative infinity as the argument approaches zero. Hence, it cannot represent a travelling wave. (c) 1/(x + vt): This function is unbounded near x + vt = 0, where it approaches infinity. Hence, it cannot represent a travelling wave. — The question tests understanding of necessary and sufficient conditions for a travelling wave. The necessary condition is that x and t must appear in the combination (x ± vt). However, this is not sufficient - the function must also be bounded to represent a physical wave. All examples in the textbook (sinusoidal functions like y = a sin(kx - ωt + φ)) are bounded, indicating this is a key requirement for travelling waves.

Explanation

Apar Academy