# Characteristics of waves

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.

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 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 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 $$\frac{\Tred{1}}{\Tred{10}} \Tred{\text{ Hz}= 0.1 \text{ Hz}}.$$

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.

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.

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