Waves: types, terms and the wave equation: WASSCE Physics

The radio playing in a taxi, the light from the Sun, the ripples when a stone drops into a pond and the voice of your teacher all arrive as waves. A wave carries energy from one place to another without carrying the material along with it.

Learning objectives

By the end of this lesson you should be able to:

define a wave and distinguish mechanical from electromagnetic waves
distinguish transverse from longitudinal waves with examples
define amplitude, wavelength, frequency, period and wave speed
use $v = f\lambda$ and $f = \dfrac{1}{T}$

What is a wave?

Definition

A wave is a disturbance that travels through a medium (or through space), transferring energy from one point to another without transferring the medium itself.

When a wave passes along a rope, each part of the rope only moves up and down about its rest position; the rope does not travel to the other end. When a ripple crosses a pond, a floating leaf bobs up and down but stays in roughly the same place.

Mechanical and electromagnetic waves

Mechanical waves	Electromagnetic waves
need a material medium (solid, liquid or gas)	can travel through a vacuum
e.g. sound, water waves, waves on a rope or spring, seismic waves	e.g. light, radio waves, X-rays, microwaves
speed depends on the medium	all travel at $3 \times 10^{8}$ m s⁻¹ in a vacuum

Transverse and longitudinal waves

A transverse wave drawn as a sine curve, with a crest, a trough, the amplitude and one wavelength marked

Figure 1. A transverse wave. WASSCE Tutor diagram.

Definition

In a transverse wave, the particles of the medium vibrate perpendicular (at right angles) to the direction in which the wave travels. Examples: waves on a rope, water ripples, light and all electromagnetic waves.
In a longitudinal wave, the particles vibrate parallel to (along) the direction of travel. Examples: sound waves, waves along a slinky spring pushed and pulled at one end.

A longitudinal wave shown as vertical lines bunched together at compressions and spread apart at rarefactions, with one wavelength marked between two compressions

Figure 2. A longitudinal wave: compressions (C) and rarefactions (R). WASSCE Tutor diagram.

A transverse wave has crests and troughs; a longitudinal wave has compressions (particles crowded together, higher pressure) and rarefactions (particles spread out, lower pressure).

Terms used to describe waves

Term	Symbol	Definition	Unit
Amplitude	$a$	the maximum displacement of a particle from its rest (equilibrium) position	m
Wavelength	λ	the distance between two successive particles in phase, e.g. crest to crest or compression to compression	m
Period	$T$	the time for one complete oscillation (time for one wavelength to pass a point)	s
Frequency	$f$	the number of complete oscillations (waves) per second	hertz (Hz)
Wave speed	$v$	the distance travelled by the wave per second	m s⁻¹

Particles are in phase when they are at the same point in their motion at the same time (for example, two neighbouring crests).

The wave equation

In one period $T$ the wave moves forward one wavelength λ, so

v = \dfrac{\lambda}{T} = f\lambda \qquad\text{since}\qquad f = \dfrac{1}{T}

\begin{gathered}v = \dfrac{\lambda}{T} = f\lambda \\[6pt] \text{since} \\[6pt] f = \dfrac{1}{T}\end{gathered}

Worked example

A radio station broadcasts on a frequency of 98.0 MHz. Find the wavelength of its waves. (speed of radio waves = 3.0 × 10⁸ m s⁻¹)

\lambda = \dfrac{v}{f} = \dfrac{3.0 \times 10^{8}}{98.0 \times 10^{6}} = 3.06\ \text{m}

\begin{aligned}\lambda &= \dfrac{v}{f} \\ &= \dfrac{3.0 \times 10^{8}}{98.0 \times 10^{6}} \\ &= 3.06\ \text{m}\end{aligned}

Worked example

Water waves have a wavelength of 0.4 m. A floating bottle bobs up and down 15 times in 10 s. Find (a) the frequency, (b) the period, (c) the speed of the waves.

(a) $f = \dfrac{15}{10} = 1.5$ Hz. (b) $T = \dfrac{1}{1.5} = 0.67$ s. (c) $v = f\lambda = 1.5 \times 0.4 = 0.6$ m s⁻¹.

Common mistake

Changing the medium changes the speed and the wavelength of a wave, but not its frequency: the frequency is set by the source. Light entering glass slows down and its wavelength shortens; its frequency (and colour) stays the same.

Graphs of waves

There are two kinds of graph, and they look alike — read the horizontal axis carefully.

Graph	Horizontal axis	What you read off
Displacement–distance	distance along the wave	amplitude and wavelength
Displacement–time	time, for one particle	amplitude and period (then $f = 1/T$ )

In Sierra Leone

Okada riders and market traders across Sierra Leone keep radios tuned to stations on FM frequencies near 100 MHz, whose waves are about 3 m long. Mobile phones use microwaves of about 900 MHz to 2 GHz, with wavelengths of only 15–33 cm.

Key points

A wave transfers energy, not matter.
Mechanical waves need a medium; electromagnetic waves do not.
Transverse: vibration ⟂ travel (light, ripples). Longitudinal: vibration ∥ travel (sound).
$v = f\lambda$ , $f = 1/T$ ; frequency is set by the source and does not change between media.

WASSCE examination focus

WAEC tests this

Define a wave, distinguish transverse and longitudinal waves with examples, define amplitude, wavelength, frequency and period, and use $v = f\lambda$ . You may be given a displacement–distance or displacement–time graph and asked for the amplitude, wavelength or period. Check the axis before answering.

Quick check

Give one example of a longitudinal wave.
A wave has frequency 50 Hz and wavelength 6 m. Find its speed.
What happens to the frequency of a wave when it passes from air into water?

Answers: 1. Sound waves. 2. $50 \times 6 = 300$ m s⁻¹. 3. It stays the same.

Waves: types, terms and the wave equation