SPH3U Wave Anatomy and the Wave Equation: Notes and Practice

Key things to remember

1A wave transfers energy without transferring matter.

2Transverse waves vibrate perpendicular to travel (light, string); longitudinal waves vibrate parallel (sound).

3Sound is longitudinal and needs a medium — it cannot cross a vacuum.

4Amplitude is measured from the rest position to a crest, not trough to crest.

f = \frac{# cycles}{time} = \frac{1}{T}

— frequency and period are reciprocals.

v = f λ

applies to every wave.

7The medium fixes the speed and the source fixes the frequency, so entering a new medium changes the wavelength.

8Sound travels fastest in solids, slower in liquids, slowest in gases.

A wave is a disturbance that transfers energy from place to place without transferring matter. Drop a stone in a pond and the ripple travels outward, but a floating leaf bobs up and down and stays where it is. The water goes nowhere; the energy goes everywhere.

Energy travels, matter does not

This is the defining feature of a wave and the source of most confusion about them. Sound crossing a room does not blow air from the speaker to your ear — each air molecule jiggles about its own position and passes the disturbance on. A crowd doing 'the wave' in a stadium is the same idea: the wave circles the stands while every person stays in their seat.

Two kinds of wave

Fig 1The difference is the direction the particles move relative to the direction the wave travels.

ATransverse

Particles vibrate perpendicular to the direction of travel
Has crests and troughs
Light, radio, waves on a string, water surface waves
Cannot travel through the interior of a fluid

BLongitudinal

Particles vibrate parallel to the direction of travel
Has compressions and rarefactions
Sound, ultrasound, a pushed slinky
Travels through solids, liquids and gases

Sound is longitudinal, which has a consequence

Because sound is a compression wave in a material, it needs a medium — it cannot cross a vacuum. Space really is silent. Light is transverse and electromagnetic, so it crosses vacuum perfectly well, which is why you see a distant explosion but never hear it.

Describing a wave

Fig 2Amplitude is measured from the rest position, not from trough to crest.

Quantity	Symbol	Unit	Definition
Amplitude	$A$	m	Maximum displacement from the rest position
Wavelength	$λ$	m	Length of one complete cycle — crest to crest
Period	$T$	s	Time for one complete cycle to pass a point
Frequency	$f$	Hz	Cycles per second. $1 Hz = 1 s^{- 1}$
Speed	$v$	m/s	How fast the disturbance travels through the medium

f = \frac{# cycles}{time} = \frac{1}{T}

Frequency and period are reciprocals. A 4 Hz wave has a period of 0.25 s.

Amplitude is measured from the middle

It is the distance from the rest position to a crest, not the full distance from trough to crest. That full swing is twice the amplitude, and quoting it instead is a routine way to be out by a factor of two. Amplitude determines how much energy the wave carries — and for sound, how loud it is.

The wave equation

v = f λ

Speed equals frequency times wavelength. Metres per second from hertz times metres.

This follows directly from the definitions: one wavelength passes in one period, so the speed is $\frac{λ}{T}$ , and since $\frac{1}{T} = f$ , that is $f λ$ . It applies to every wave there is — sound, light, water, seismic — which makes it the most-used equation in the unit.

The medium sets the speed, the source sets the frequency

This is the idea that makes the equation useful. A wave's speed is fixed by what it is travelling through; its frequency is fixed by whatever is making it. So when a wave passes from one medium into another, the frequency stays the same and the wavelength changes to match the new speed. Nothing is free to vary independently — change the speed and

λ

must follow.

Loading interactive…

Change the frequency and wavelength and watch the speed readout follow, then switch tabs for interference and standing waves.

Wave	Typical speed	Note
Sound in air at $20^{\circ} C$	$344 m/s$	Rises with temperature
Sound in water	$1480 m/s$	Faster — particles are closer together
Sound in steel	$5000 m/s$	Fastest in stiff solids
Light in vacuum	$3.00 \times 1 0^{8} m/s$	The universal speed limit

Sound travels faster in solids than in gases, which reverses most people's intuition. The reason is stiffness: in a solid the particles are tightly bound, so a disturbance is passed on almost immediately, while in a gas the molecules must travel some distance before colliding.

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

A wave has a frequency of

250 Hz

and a wavelength of

1.4 m

. Find its speed and its period.

Example 2

A guitar string vibrates 36 times in

0.30 s

. Find the frequency and the period.

Example 3

A sound of frequency

512 Hz

travels from air (

v = 344 m/s

) into water (

v = 1480 m/s

). Find the wavelength in each.

Where marks get lost

Measuring amplitude from trough to crest. It is measured from the rest position, so that value is twice the amplitude.

Thinking the medium travels with the wave. Only energy moves; the particles oscillate about fixed positions.

Assuming frequency changes when a wave enters a new medium. The frequency is set by the source — the wavelength changes instead.

Believing sound travels faster in air than in solids. Stiffer media carry sound faster, so steel beats water beats air.

Confusing period with frequency. They are reciprocals, so a large frequency means a small period.

Claiming sound can travel through a vacuum. It is a mechanical wave and needs particles to compress.

Key things to remember

Worked examples

Where marks get lost

Flashcards 11

Practice