Progressive waves

A progressive wave is a disturbance that travels from one place to another, transferring energy as it moves.

When a progressive wave travels through a material, the particles oscillate about their equilibrium positions: their positions when they are not disturbed. Each particle moves back and forth as the disturbance passes, but it does not travel along with the wave. Energy is transferred through the material even though the particles do not make the journey with it.

Amplitude, frequency and period

The amplitude is the maximum displacement of an oscillating particle from its equilibrium position.

The frequency $$f$$ is the number of complete oscillations made per unit time. It is measured in hertz ($$\mathrm{Hz}$$), where one hertz means one oscillation per second. The period $$T$$ is the time taken for one complete oscillation, measured in seconds. They are related by:

A higher frequency means more oscillations each second, so the time taken for each oscillation is shorter.

Wavelength and wave speed

The wavelength $$\lambda$$ is the distance between successive points that are at the same stage of oscillation, such as two successive crests. It is measured in metres.

The wave speed $$c$$ is the distance travelled by the disturbance per unit time. It is not the speed at which an individual particle oscillates. In one period, the disturbance travels one wavelength, so its speed is wavelength divided by period. Since $$f = 1/T$$:

When $$f$$ is in hertz and $$\lambda$$ is in metres, $$c$$ is in metres per second.

Wave phase and phase difference

Phase describes the stage reached in an oscillation cycle. For example, an oscillating point passes through its equilibrium position twice in each cycle: once moving in each direction. Its displacement alone therefore does not identify its stage in the cycle.

Phase difference compares the stages of two oscillations. It can also compare the phase of a wave at two positions or at two different times. Oscillations are in phase when they are at corresponding stages of their cycles. They may be different numbers of complete cycles from their starting points and still be in phase.

Expressing phase difference

Phase difference can be given as a fraction of a cycle or as an angle. One complete cycle corresponds to $$360^\circ$$ or $$2\pi$$ radians. For example, a quarter-cycle difference is $$90^\circ$$ or $$\pi/2$$ radians; a half-cycle difference is $$180^\circ$$ or $$\pi$$ radians.

A phase difference of one complete cycle brings the oscillations back to corresponding stages, so it is equivalent to zero phase difference. More generally, phase differences that differ by a whole number of cycles describe the same relative stages.

Phase difference between positions on a wave

For a periodic wave at one instant, one wavelength is the distance covered by one complete cycle. The separation between two positions, as a fraction of the wavelength, therefore gives their phase separation as a fraction of a cycle. For separation $$\Delta x$$ and wavelength $$\lambda$$:

For example, points separated by $$\lambda/4$$ have a phase separation of a quarter cycle, or $$90^\circ$$. Points separated by $$\lambda/2$$ have a phase separation of half a cycle, or $$180^\circ$$. Points separated by one wavelength are in phase because their separation is one complete cycle.