Progressive wavesAQA A-Level Physics: Revision notes

A progressive wave transfers energy from one place to another through a medium without transferring the medium itself. Each particle (or each point in a field) oscillates about a fixed equilibrium position and passes its motion on to its neighbour, a little later.

The oscillation of every particle has the same frequency as the source, and, if energy losses are negligible, the same amplitude. What travels along the medium is the pattern of displacement, and with it the energy.

• Displacement is the distance of a point from its equilibrium position, in a stated direction.
• Amplitude $A$ is the maximum displacement from the equilibrium position.
• Wavelength $\lambda$ is the shortest distance along the wave between two points oscillating in phase, for example crest to crest.
• Period $T$ is the time for one complete oscillation of a point; frequency $f$ is the number of oscillations per second, measured in hertz (Hz).
• Wave speed $c$ (or $v$) is the speed at which the wave pattern, and its energy, travels.

Period and frequency are linked by $f = \dfrac{1}{T}$.

In one period $T$ the wave travels one wavelength $\lambda$, so the speed is $c = \dfrac{\lambda}{T}$. Using $f = 1/T$ this gives the wave equation

$c = f\lambda$

The speed depends on the medium (and, for a string, on tension and mass per unit length), not on the source. If the frequency is changed while the medium stays the same, the wavelength changes so that $c$ stays constant: doubling $f$ halves $\lambda$.

Worked example. Water waves of frequency 12 Hz have wavelength 2.5 cm. Then $c = f\lambda = 12 \times 0.025 = 0.30\ \text{m s}^{-1}$. Always convert the wavelength to metres first.

The phase of a point is the stage it has reached in its oscillation cycle. Two points on a wave are in phase if they are at the same stage of the cycle, so they have the same displacement and move in the same direction at the same time.

The phase difference between two oscillations measures how much one is ahead of or behind the other. It can be given as:

• a fraction of a cycle, e.g. 1/4 cycle
• an angle in degrees, where one cycle is $360°$
• an angle in radians, where one cycle is $2\pi$ rad.

Points a whole number of wavelengths apart are in phase (phase difference $0$, $2\pi$, $4\pi$, …). Points an odd number of half wavelengths apart are in antiphase (phase difference $\pi$, $3\pi$, …): they always move in opposite directions.

Two points a distance $x$ apart along a progressive wave of wavelength $\lambda$ have a phase difference

$\Delta\phi = \frac{x}{\lambda} \times 2\pi \ \text{(radians)} \qquad\text{or}\qquad \frac{x}{\lambda}\times 360° .$

The point further from the source lags: it repeats the motion of the nearer point after a time $x/c$ has passed.

Worked example. A wave of wavelength 8.0 m travels at 4.0 m s⁻¹. Two points 3.0 m apart are $3.0/8.0 = 3/8$ of a cycle out of phase, so $\Delta\phi = \tfrac{3}{8}\times 2\pi = \tfrac{3\pi}{4} = 2.4$ rad (135°). The time lag is $3.0/4.0 = 0.75$ s, which is 3/8 of the period (2.0 s), as expected.