There are two distinct types of wave, longitudinal and transverse. Both can be demonstrated using a toy spring lying along a bench.
Push the end of the spring back and forth; the segments of the spring become compressed and then stretched out, along the length of the spring. Wave pulses run along the spring. These are longitudinal waves.
Waggle the end of the spring from side to side. The segments of the spring move from side to side as the wave travels along the spring. These are transverse waves.
So, the distinction between longitudinal and transverse waves is as follows.
- In longitudinal waves, the particles of the medium vibrate parallel to the direction of the wave velocity.
- In transverse waves, the particles of the medium vibrate at right angles to the direction of the wave velocity.
Sound waves are an example of a longitudinal wave. Light and all other electromagnetic waves are transverse waves. Waves in water are quite complex. Particles of the water may move both up and down and from side to side as a water wave travels through the water. You can investigate water waves in a ripple tank. There is more about water waves in Table 12.1 and in Chapter 13.

Representing waves

Figure 12.8 shows how we can represent longitudinal and transverse waves. The longitudinal wave shows how the material through which it is travelling is alternately compressed and expanded. This gives rise to high and low pressure regions, respectively.

Figure 12.8: a Longitudinal waves and b transverse waves. A = amplitude, $\lambda $ = wavelength

However, this can be difficult to draw, so you will often see a longitudinal wave represented as if it were a sine wave. The displacement referred to in the graph is the displacement of the particles in the wave.
We can compare the compressions and rarefactions (or expansions) of the longitudinal wave with the peaks and troughs of the transverse wave.

Phase and phase difference

All points along a wave have the same pattern of vibration. However, different points do not necessarily vibrate in step with one another. As one point on a wave vibrates, the point next to it vibrates slightly outof-step with it. We say that they vibrate out of phase with each other – there is a phase difference between them. This is the amount by which one oscillation leads or lags behind another.
Two particles oscillating in step have a phase difference of ${0^ \circ }$, ${360^ \circ }$ and so on (or 0 rad, $2\pi $ rad and so on).
Two particles oscillating in antiphase have a phase difference of ${180^ \circ }$, ${270^ \circ }$ and so on (or $\pi $ rad, $3\pi $ rad and so on).
Phase difference is measured in degrees or in radians. As you can see from Figure 12.9, two points A and B, with a separation of one whole wavelength λ, vibrate in phase with each other. The phase difference between these two oscillating particles at A and B is ${360^ \circ }$. (You can also say it is ${0^ \circ }$.) The phase difference between any other two points between A and B can have any value between ${0^ \circ }$ and ${360^ \circ }$. A complete cycle of the wave is thought of as ${360^ \circ }$. The separation between points C and D is quarter of a wavelength – the phase difference between these two points is ${90^ \circ }$. In general, when the separation between two oscillating particles on a wave is x, then the phase difference ϕ between these particles in degrees can be calculated using the expression:

$\phi  = \frac{x}{\lambda } \times {360^ \circ }$

where $\lambda $ is the wavelength of the wave.
The idea of phase difference is revisited in Chapter 13.