Define the properties of a progressive wave, apply the wave equation, distinguish transverse from longitudinal waves, and explain intensity, phase and polarisation

What this dot point is asking

SEAB wants you to define the properties of a progressive wave, apply the wave equation $v = f\lambda$, distinguish transverse from longitudinal waves, and explain intensity, phase difference and polarisation. A progressive wave transfers energy through a medium without transferring matter.

The answer

Wave properties

A progressive wave is described by:

• Amplitude $A$: the maximum displacement from equilibrium.
• Wavelength $\lambda$: the distance between adjacent points in phase (for example, crest to crest).
• Frequency $f$: the number of complete oscillations per second (Hz).
• Period $T$: the time for one complete oscillation, $T = \dfrac{1}{f}$.
• Speed $v$: the speed at which the wave profile (and its energy) travels.

The wave equation

These quantities are linked by the wave equation:

In one period the wave advances one wavelength, which is the physical content of this relation. For a fixed medium $v$ is constant, so frequency and wavelength are inversely related.

Transverse and longitudinal waves

• Transverse waves: the oscillations are perpendicular to the direction of energy transfer. Examples are waves on a string and all electromagnetic waves.
• Longitudinal waves: the oscillations are parallel to the direction of energy transfer, forming compressions and rarefactions. The standard example is sound.

Phase difference

The phase difference between two points (or two waves) measures how far through their cycles they are relative to each other, measured in radians. For two points separated by a distance $\Delta x$ on the same wave:

Points one wavelength apart are in phase ($\Delta\phi = 2\pi$); points half a wavelength apart are in antiphase ($\Delta\phi = \pi$).

Intensity

Intensity is the power transferred per unit area:

For a wave, intensity is proportional to the square of the amplitude, $I \propto A^2$. For a point source radiating uniformly, intensity falls off as the inverse square of distance, $I \propto \dfrac{1}{r^2}$, because the same power spreads over a sphere of area $4\pi r^2$.

Polarisation

Polarisation restricts the oscillations of a transverse wave to a single plane. Only transverse waves can be polarised, because only they have oscillation directions perpendicular to travel from which one plane can be selected. Longitudinal waves, oscillating only along the travel direction, cannot be polarised. This is a key test for distinguishing the two wave types.

Examples in context

Example 1. Polaroid sunglasses. Light reflected from a horizontal surface is partially polarised horizontally. Polaroid sunglasses transmit only vertically polarised light, blocking much of the horizontal glare. This works only because light is a transverse wave that can be polarised, a direct demonstration of the transverse nature of light.

Example 2. Why distant sound is quieter. A loudspeaker radiates sound roughly uniformly, so its intensity falls as $1/r^2$. Doubling your distance from the speaker quarters the intensity, which is the inverse-square spreading of energy over an expanding spherical wavefront.

Try this

Q1. State the wave equation and define each quantity. [2 marks]

• Cue. $v = f\lambda$: $v$ wave speed, $f$ frequency, $\lambda$ wavelength.

Q2. A water wave has wavelength $2.5\ \text{m}$ and travels at $5.0\ \text{m s}^{-1}$. Find its frequency and period. [2 marks]

• Cue. $f = v/\lambda = 5.0/2.5 = 2.0\ \text{Hz}$; $T = 1/f = 0.50\ \text{s}$.

Q3. Explain why a longitudinal wave cannot be polarised. [2 marks]

• Cue. Its oscillations are only along the travel direction, so there is no perpendicular plane of oscillation to restrict, which is what polarisation does.