Question

Given below are some functions of ‘x’ and ‘t’ to represent the displacement (transverse or longitudinal) of waves. State which of the following represent
a. A travelling wave
b. A stationary wave
c. none at all:
$\text A .\ y=2cos(3x)sin(10t)$
$\text B .\ 2\sqrt{x-vt}$
$\text C . \ 3sin(5x-0.5t)+4cos(5x-0.5t)$
$\text D . \ cos(x)sin(t) + cos(2x)sin(2t)$

Answer 1

Hint: A transverse wave is the one that travels through a medium, making crests and troughs. Wave on a string is an example of a transverse wave. A longitudinal wave is the one that travels through the medium making compressions and rarefactions. The sound wave is an example of a longitudinal wave. When two waves approach each other in the same place, on the same line, standing waves are formed.

Complete step-by-step solution:
The general equation of a traveling wave ( for example sound and light waves) is : $A\ sin \left ( \omega t - kx \right )$ where symbols have their usual meaning.
The general equation of stationary wave is the one (generally) in which the terms ‘kx’ and ‘$\omega t$’ occur separately and not in addition or subtraction. It is of the form: $Asin(\omega t)cos(kx)$.
Comparing these with given equations of waves:
$\text A.\ y=2cos(3x)sin(10t)$ - It is a stationary wave because the term ‘kx’ and ‘$\omega t$’ occur separately.
$\text B.\ 2\sqrt{x-vt}$ - Waves have a unique property to repeat after a certain interval of time, called time period. This is suggested by trigonometric functions present in the wave equation. Hence this equation represents nothing at all.
$\text C. \ 3sin(5x-0.5t)+4cos(5x-0.5t)$ - It is a form of travelling wave. Moreover, it represents the superposition of waves.
$\text D. \ cos(x)sin(t) + cos(2x)sin(2t)$- Clearly it represents the superposition of two standing waves as the term ‘kx’ and ‘$\omega t$’ occur separately.

Note: The superposition of waves generally means the interference of waves. Mathematically it means the addition of waves when two waves travel in the same plane and along the same line. There are many more such conditions needed to be fulfilled before obtaining a sustained interference pattern.