PhysicsCore20 min read

Waves & SHM

Oscillation, superposition, resonance

This topic appears in:

10th Class Physics · ch.121st Year Physics · ch.7O Level Physics (5054) · ch.3A Level Physics (9702) · ch.7
01

What a wave carries, and what it does not

Definition

Wave — A disturbance that transfers energy from place to place without transferring matter.

The defining feature of a wave is what it leaves behind: nothing. Energy moves from one end to the other, but the material itself only vibrates about a fixed position and stays where it was.

A duck sitting on a lake makes this obvious. Waves travel across the surface and reach the far bank, but the duck bobs up and down in one place. If the water were travelling with the wave, the duck would be carried along with it.

This is the difference between a wave and a current, and it is worth being able to state clearly, because "waves transfer energy without transferring matter" is the mark-scheme answer to a very common opening question.

02

Transverse and longitudinal

Waves come in two kinds, distinguished by the direction of the vibration relative to the direction the wave travels.

In a transverse wave the particles vibrate at right angles to the direction of travel. Light, all electromagnetic waves, water ripples and waves on a rope are transverse. They have identifiable crests and troughs.

In a longitudinal wave the particles vibrate along the same line as the direction of travel. Sound is the important example. Instead of crests and troughs there are compressions, where the particles are bunched together, and rarefactions, where they are spread apart.

A slinky spring shows both. Flick it sideways and a transverse pulse runs along it. Push and pull it end-on and a longitudinal pulse travels instead, visible as a moving squeeze in the coils.

TransverseLongitudinal
Vibration directionperpendicular to travelparallel to travel
Featurescrests and troughscompressions and rarefactions
Exampleslight, all EM waves, water ripplessound, ultrasound
Can travel through a vacuum?EM waves canno — needs a medium
03

Describing a wave

Four quantities describe any wave, and every wave calculation you will meet uses at least two of them.

The wavelength is the distance between two neighbouring points in phase — crest to crest is the easiest to picture. The amplitude is the maximum displacement from the undisturbed position, measured from the centre line to a crest, not from trough to crest. Amplitude is what determines how much energy the wave carries.

The frequency is the number of complete waves passing a point each second, measured in hertz. The period is the time for one complete wave to pass, so frequency and period are reciprocals of each other.

These combine into the wave equation, which follows from simple reasoning: if f waves pass every second and each is λ long, the wave front advances fλ metres every second — and that is its speed.

v = f λf = 1 / Tspeed in m/s, frequency in Hz, wavelength in m — convert cm and mm before substituting
v
wave speedm s⁻¹
f
frequencyHz
λ
wavelengthm
T
periods
Worked example 14 marks

Water waves in a ripple tank have a frequency of 12 Hz. Twenty complete waves span 40 cm. Calculate the wavelength and the wave speed.

  1. Wavelength = 40 / 20 = 2.0 cm.Total length divided by the number of waves.
  2. Convert: 2.0 cm = 0.020 m.The equation needs metres, and this conversion is usually worth a mark.
  3. v = f λ = 12 × 0.020.
  4. v = 0.24 m s⁻¹.Sensible for a ripple tank.

λ = 0.020 m, v = 0.24 m s⁻¹

Two waves are drawn separately and then added. Where crests meet crests the result is larger; where a crest meets a trough they cancel. Change the phase difference and watch the resultant grow and shrink.

04

Reflection, refraction and diffraction

All waves do three things when they meet an obstacle or a boundary, and the ripple tank shows all three.

Reflection bounces the wave back. The angle of incidence equals the angle of reflection, and the wavelength, frequency and speed are all unchanged.

Refraction happens when the wave crosses into a region where it travels at a different speed — in a ripple tank, shallower water. The frequency stays the same, because it is set by the source and cannot change at a boundary. So if the speed drops, v = fλ forces the wavelength to drop with it. If the wave meets the boundary at an angle, one end of each wavefront slows first, the wavefront pivots, and the direction of travel changes.

Diffraction is the spreading out that happens when a wave passes through a gap or around an edge. The narrower the gap, the more the wave spreads — and the spreading is greatest when the gap is about the same size as the wavelength. This is why you can hear round a corner but not see round it: sound wavelengths are metres, comparable to a doorway, while light wavelengths are less than a thousandth of a millimetre.

Frequency never changes at a boundary

The frequency of a wave is fixed by whatever produced it. When a wave slows down entering a new medium, the wavelength shortens to match — frequency stays put. Half the marks in refraction questions rest on this one sentence.

Key points

  1. Waves transfer energy without transferring matter.
  2. Transverse: vibration perpendicular to travel. Longitudinal: vibration along it.
  3. Amplitude sets the energy; frequency and wavelength set the speed via v = fλ.
  4. At a boundary, frequency is unchanged; speed and wavelength change together.
  5. Diffraction is greatest when the gap is about one wavelength wide.
05

The Doppler effect

When a source of waves moves towards you, each successive wavefront is emitted from a little closer, so the fronts arrive bunched together — a shorter wavelength and a higher frequency. Moving away, they are stretched apart. This is the Doppler effect, and it is why a siren drops in pitch as the vehicle passes.

Nothing about the wave changes in the source frame; the source emits at the same frequency throughout. What changes is the spacing of the fronts as they reach the observer, which is why the pitch shifts the instant the vehicle passes rather than gradually.

f_observed = f_source × v / (v ± v_s)minus sign: source approaching → higher frequencyplus sign: source receding→ lower frequencyv = speed of the wave, v_s = speed of the sourcethe sign is chosen by whether the denominator should shrink or grow
v
the wave speeda property of the medium, unchanged by the motion
v_s
the source speedmust be less than v for this form
±
the signminus for approaching, plus for receding

Check the direction of the shift first

Before substituting, decide whether the observed frequency should be higher or lower than the source frequency. Approaching gives higher, receding gives lower. If your answer comes out on the wrong side of the source frequency, the sign in the denominator is the wrong way round — a check that takes two seconds and catches the error every time.

Practice questions

6 questions · 25 marks · full working on every one

Try each one on paper first, then open the working. The marks are shown where they are actually awarded, because that is where they are actually lost.

Short questions

3 · 6 marks

Two marks each, in the style of the short-question section of the paper. Answer in two or three lines.

SQ1[2 marks]
Define the wavelength and the frequency of a wave.
Model answer

Wavelength is the distance between two neighbouring points that are in phase, for example crest to crest. Frequency is the number of complete waves passing a fixed point each second.

Examiner tip. "Crest to crest" is enough for wavelength. For frequency the words "per second" must appear.

SQ2[2 marks]
State two differences between transverse and longitudinal waves, and give one example of each.
Model answer

In a transverse wave the vibration is perpendicular to the direction of travel, for example light. In a longitudinal wave it is parallel to the direction of travel, for example sound.

Examiner tip. The direction of vibration relative to the direction of travel is the whole distinction. Say "relative to".

SQ3[2 marks]
State what happens to the frequency, wavelength and speed of a water wave when it passes into shallower water.
Model answer

The frequency stays the same. The speed decreases and, since v = fλ, the wavelength decreases in proportion.

Examiner tip. Frequency is fixed by the source and never changes at a boundary. That single fact answers half the refraction questions ever set.

Solved numericals

1 · 4 marks

Full working, one step per line, with the marks shown where they are awarded.

N1[4 marks]
A wave on a rope has a frequency of 12 Hz. Fifteen complete waves occupy a length of 3.0 m. Calculate the wavelength and the speed of the wave.
Full working
  1. Uses λ = length ÷ number of waves[1]
  2. λ = 3.0 / 15 = 0.20 m[1]
  3. Uses v = fλ[1]
  4. v = 12 × 0.20 = 2.4 m s⁻¹[1]

λ = 0.20 m, v = 2.4 m s⁻¹

Examiner tip. Read carefully: fifteen waves in 3.0 m, not a wavelength of 3.0 m. Dividing is the first mark.

Long questions

2 · 15 marks

Theory and numerical together, as they appear in the long-question section.

LQ1[8 marks]
A ripple tank is used to study water waves. A straight barrier with a narrow gap is placed in the tank.
  1. Describe and explain what is observed as the waves pass through the gap. [3]
  2. State and explain what happens to the effect when the gap is made narrower. [2]
  3. The waves have a frequency of 8.0 Hz and a wavelength of 25 mm. Calculate their speed. [3]
Mark scheme
  1. The waves spread out after passing through the gap[1]
  2. This is diffraction[1]
  3. The wavelength and frequency are unchanged; only the shape of the wavefront changes[1]
  4. The waves spread out more[1]
  5. Because the gap width is closer to the wavelengthmaximum spreading when gap ≈ λ[1]
  6. Converts 25 mm = 0.025 m[1]
  7. Uses v = fλ[1]
  8. v = 8.0 × 0.025 = 0.20 m s⁻¹[1]

(c) 0.20 m s⁻¹

Examiner tip. The millimetre conversion in (c) carries a mark of its own. Leaving it in mm gives 200, which is absurd for a ripple tank and should catch your eye.

LQ2[7 marks]
A student uses a ripple tank to investigate refraction by placing a flat glass plate on the bottom to make part of the tank shallower.
  1. Describe what happens to the direction of the waves as they cross into the shallow region at an angle. [2]
  2. Explain this change in terms of the speed of the waves. [3]
  3. State what is observed if the waves meet the boundary head-on, and explain why. [2]
Mark scheme
  1. The waves change direction at the boundary[1]
  2. They bend towards the normal[1]
  3. The waves travel more slowly in the shallow water[1]
  4. The end of each wavefront entering the shallow region slows first[1]
  5. So the wavefront pivots, changing the direction of travel[1]
  6. The direction does not change[1]
  7. Because the whole wavefront slows at the same instant, so there is nothing to pivot aboutthe wavelength still shortens[1]

Examiner tip. Part (c) catches people out. Refraction still happens — the wavelength shortens — but with no angle of incidence there is no bending.