Wave properties and wave graphsEdexcel A-Level Physics: Revision notes

A wave transfers energy without transferring matter. Key quantities:

• Amplitude A: the maximum displacement of a point from its equilibrium position
• Wavelength λ: the distance between two adjacent points that are in phase, e.g. crest to crest
• Period T: the time for one complete oscillation, in seconds
• Frequency f: the number of oscillations per second, in hertz, with f = 1/T
• Wave speed v: the distance moved by the wave pattern per second

The wave equation is v = fλ. In one period the wave travels one wavelength, so v = λ/T = fλ. The speed of a wave depends on the medium, while the frequency is fixed by the source.

In a transverse wave the oscillations are perpendicular to the direction of energy transfer. Examples are waves on a string, water ripples and all electromagnetic waves. They show crests and troughs.

In a longitudinal wave the oscillations are parallel to the direction of energy transfer. Sound is the main example. The medium forms compressions (high pressure, particles close together) and rarefactions (low pressure, particles spread out).

In both cases the particles of the medium oscillate about fixed equilibrium positions. There is no net movement of the medium along the wave.

A sound wave in air can be described in two ways: by the pressure variation about atmospheric pressure, or by the displacement of the molecules from their equilibrium positions.

In a progressive wave the two are a quarter of a cycle apart:

• At the centre of a compression the pressure is a maximum but the molecular displacement is zero, because molecules behind are displaced forwards and those ahead are displaced backwards
• At the centre of a rarefaction the pressure is a minimum and the displacement is again zero
• The molecular displacement is largest where the pressure is at its normal value

When plotting displacement for a longitudinal wave, displacement in the direction of travel is taken as positive. The graph then looks like that of a transverse wave.

Two graphs describe a progressive wave, and they give different information.

• Displacement–distance graph: a snapshot of the whole wave at one instant. Read the wavelength (crest to crest) and the amplitude.
• Displacement–time graph: the motion of one point as time passes. Read the period T (and so f = 1/T) and the amplitude.

The two are linked by v = fλ = λ/T.

For a stationary wave the graphs differ from a progressive wave. All points between adjacent nodes oscillate in phase, points in adjacent loops are in antiphase, and the amplitude changes with position from zero at a node to a maximum at an antinode. Every point oscillates at the same frequency.

One method uses a signal generator, a loudspeaker, two microphones and a two-channel oscilloscope.

1. Fix microphone 1 near the loudspeaker. Move microphone 2 away along a metre rule.
2. The traces move in and out of phase. Record the positions where they are exactly in phase.
3. The distance between two in-phase positions n wavelengths apart is nλ. Measuring across many wavelengths reduces the percentage uncertainty.
4. Calculate v = fλ.
5. Repeat for several frequencies and plot λ against 1/f. The gradient is the speed of sound, about 340 m s⁻¹ at room temperature.

A sound of frequency 2.0 kHz has speed 340 m s⁻¹ in air. Find λ and T.

λ = v/f = 340 ÷ 2000 = 0.17 m

T = 1/f = 1 ÷ 2000 = 5.0 × 10⁻⁴ s

When the sound passes into water where v = 1500 m s⁻¹, the frequency stays at 2.0 kHz, so λ = 1500 ÷ 2000 = 0.75 m.