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# Characteristics of waves

## Displacement and amplitude in terms of oscillations

In the context of mechanical waves, the displacement $(x)$ refers to that of a single particle undergoing oscillatory motion at a certain position in the wave.

The wave itself does not have a displacement (although it does cause displacement of the particles of the medium).

The amplitude of a wave is the maximum increase of an oscillating quantity from its equilibrium magnitude. It is equivalent to the maximum displacement $x_{0}$ of a single oscillating particle in a mechanical wave.

The amplitude of a longitudinal wave, when expressed graphically as lines on a single axis, is more difficult to determine.

One must compare the positions of the particles before the wave arrives (i.e. at equilibrium) and at a particular instant after the wave arrives and then determine the displacement of each particle. The greatest displacement experienced by the particles is the amplitude.

## Amplitude and wavelength

The amplitude $(A)$ of a wave is the height of the maximum point (crest) of a wave from the equilibrium position (in between the highest and lowest points).

The amplitude of an ocean wave is the height of a crest of the wave from the undisturbed ocean height (the height of the ocean if there were no waves present).

The wavelength $(\lambda)$ of a wave is the distance between two successive maximum or two successive minimum points.

The wavelength of an ocean wave is the distance between two successive crests of the wave.

The SI unit of both amplitude and wavelength is the metre $(\text{m})$. They have the same unit as height, length and distance.

The amplitude $A$ and the wavelength $\lambda$ of a wave. This graph represents the shape of a wave at one particular instant.

## Period of a wave - definition

The period of a wave is the time it takes to complete one full oscillation (up-and-down movement) at a given point on the wave.

The period of a light wave is the time it takes for the electric field at a fixed point to drop from its maximum value to its minimum value and return back again.

The period of a wave on a rope is the time taken for one complete up-and-down movement of the rope.

All particles in a mechanical wave have the same period $(T).$ Every particle takes the same amount of time to go through one full oscillation.

The period is a quantity of time so its SI unit is the second $(\text{s})$.

The period of a wave is the time taken for the wave to complete one cycle. The graph has time $t$ on the horizontal axis.

## Frequency of a wave - definition

The frequency of a wave is the number of oscillations (up-and-down movements) completed per unit time at a particular point on the wave.

The frequency of a wave on a slinky spring is the number of up-and-down movements completed in one second.

The frequency is equivalent to 1 divided by the period (as one wave passes through a point during the period $T$): \begin{align*} \Tred{\text{frequency}}&=\frac{1}{\Tblue{\text{period}}}\\ \Tred{f}&=\frac{1}{\Tblue{T}} \end{align*}

The SI unit of frequency is called the Hertz $(\text{Hz})$. It is equal to $1 / \text{s}.$

The period of a water wave is $\Tblue{10 \us}$. Its frequency is then $\frac{\Tred{1}}{\Tred{10}} \Tred{\text{ Hz}= 0.1 \text{ Hz}}.$

Waves of different frequencies. The frequencies of the waves at the bottom are greater than those at the top.

## Wave speed

The speed of a wave $(v)$ is the speed at which a wave travels between two points.

A wave transfers energy from one point to another and therefore, like a moving object, has a speed.

The SI unit of wave speed is metres per second $(\text{m/s})$.

When lightning strikes, thunder is created. If you stand $1000 \um$ away, you will only hear the noise about $3 \us$ later. This is because sound has a speed of $330 \umps$ $$\text{time taken} = \frac{\text{distance}}{\text{speed}} = \frac{1000\text{ m}}{330\text{ m/s}}\approx 3\text{ s}$$

The speed of a wave depends on the type of wave as well as the medium through which it travels.

Sound waves travel faster in solids than in gases.

Light waves travel faster in a vacuum than in air.

Type of wave Speed $(\text{m/s})$ Speed $(\text{km/h})$
Average speed of an ocean wave $20$ $72$
Sound wave in air $330$ $1188$
Light wave in a vacuum $3\times 10^{8}$ $1.1\times 10^{9}$

The speed of a wave does not depend on its amplitude.

## Equation for wave speed

The speed of a wave $(v)$ is the distance travelled by the wave per second. It is equivalent to the wavelength $\lambda$ (the distance travelled by a wave in one period) divided by the period.

As $\Tblue{\text{frequency}} = \dfrac{1}{\Tviolet{\text{period}}}$ the wave speed can also be written in terms of the frequency.

\begin{align*} \\ \Tred{\text{speed}} &=\Tblue{\text{frequency}}\times\Torange{\text{wavelength}} \\ \Tred{v} &=\Tblue{f}\Torange{\lambda} \\ \Tred{\text{speed}} &=\dfrac{\Torange{\text{wavelength}}}{\Tviolet{\text{period}}} \\ \Tred{v}&=\dfrac{\Torange{\lambda}}{\Tviolet{T}} \end{align*}

The ripples spread out in all directions with a uniform wavespeed.