Wave Motion

1. Introduction

Wave motion is the process by which energy is transferred from one point of a medium to another in the form of a disturbance, without any bulk (net) transport of the medium itself. When a stone is dropped in still water, ripples spread outwards carrying energy, yet the water as a whole is not carried along: each particle only oscillates about its mean position while the disturbance travels on.

Waves are broadly classified as:

• Mechanical waves, which require a material medium (solid, liquid, or gas) for their propagation, for example sound waves, waves on a string, and water waves.
• Electromagnetic waves, which need no medium and can travel through vacuum, for example light and radio waves.

This chapter deals with progressive (travelling) waves and stationary (standing) waves, their mathematical description, and how the two are related.

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2. Transverse and Longitudinal Waves

Mechanical waves occur in two forms depending on the direction of particle vibration relative to the direction of propagation.

a) Transverse wave

A wave in which the particles of the medium vibrate perpendicular to the direction of propagation of the wave. It travels as a series of crests (upward humps) and troughs (downward hollows). Examples: waves on a stretched string, ripples on water, and all electromagnetic waves.

b) Longitudinal wave

A wave in which the particles of the medium vibrate along (parallel to) the direction of propagation. It travels as alternate regions of compression (particles crowded together, high pressure) and rarefaction (particles spread apart, low pressure). Examples: sound waves in air, waves along a long stretched spring (slinky).

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3. Terms Associated with a Wave

• Wavelength ($\lambda$): the distance between two consecutive points that are in the same phase (for example, crest to crest, or compression to compression). It is the distance travelled by the wave in one complete time period.
• Amplitude ($a$): the maximum displacement of a particle of the medium from its mean position.
• Time period ($T$): the time taken by a particle to complete one full oscillation.
• Frequency ($f$): the number of complete oscillations made by a particle in one second; $f = \dfrac{1}{T}$. SI unit: hertz (Hz).
• Angular frequency ($\omega$): $\omega = 2\pi f = \dfrac{2\pi}{T}$.
• Wave number / propagation constant ($k$): $k = \dfrac{2\pi}{\lambda}$, in $\text{rad m}^{-1}$.
• Phase: the state of vibration (position and direction of motion) of a particle at a given instant, expressed as an angle.

The fundamental wave relation connecting speed $v$, frequency $f$, and wavelength $\lambda$ is:

$$ \begin{aligned} v &= f\lambda \ &= \frac{\lambda}{T} \end{aligned} $$

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