Standing waves at hard boundaries (modes)
1.5.7 Standing waves at hard boundaries (modes)
The linear superposition of sound can also be used to explain a wave phenomenon known as standing waves, which is applicable to any form of sound wave. Standing waves occur when sound waves bounce between reflecting surfaces. The simplest system in which this can occur consists of two reflecting boundaries as shown in Figure 1.26.
Figure 1.26 Reflection of a sound wave between two parallel surfaces.
In this system the sound wave shuttles backwards and forwards between the two reflecting surfaces1. At most frequencies the distance between the two boundaries will not be related to the wavelength and so the compression and rarefaction peaks and troughs will occupy all positions between the two boundaries, with equal probability, as shown in Figure 1.27.
Figure 1.27 A non-stationary sound wave between two parallel surfaces.
However, when the wavelength is related to the distance between the two boundaries the wave keeps tracing the same path as it travels between the two boundaries. This means that the compressions and rarefactions always end up in the same position between the boundaries. Thus the sound wave will appear to be stationary between the reflecting boundaries, and so is called a standing wave, or, more precisely, a resonant mode. It is important to realise that the wave is still moving at its normal speed, it is merely that, like a toy train, the wave endlessly retraces the same positions between the boundaries with respect to the wavelength, as shown in Figure 1.28.
Figure 1.28 The pressure components of a standing wave between two hard boundaries, this is known as a resonant mode.
Figure 1.29 The velocity components of a standing wave between two hard boundaries.
Figures 1.28 and 1.29 show the pressure and velocity components respectively of a standing wave between two hard reflecting boundaries. In this situation the pressure component is a maximum and the velocity component is a minimum at the two boundaries. The largest wave that can fit these constraints is a half wavelength and this sets the lowest frequency at which a standing wave can exist for a given distance between reflectors, and can be calculated using the following equation:
flowest → L = λ/2 → λ 2L → flowest = v/2L
where flowest = the standing wave frequency (in Hz)
L = the distance between the boundaries (in m)
λ = the wavelength (in m)
and v = the velocity of sound (in ms-1)
Any multiple of half wavelengths will also fit between the two reflectors as well, and so there is in theory an infinite number of frequencies at which standing waves occur which are all multiples of flowest. These can be calculated directly using:
fn = nv/2L (1.20)
where fn = the nth standing wave frequency (in Hz)
and n = 1, 2, ..., ∞
An examination of Figures 1.28 and 1.29 shows that there are points of maximum and minimum amplitude of the pressure and velocity components. For example, in Figure 1.28 the pressure component's amplitude is a maximum at the two boundaries and at the midpoint, while in Figure 1.29 the velocity component is zero at the two boundaries and the midpoint. The point at which the pressure amplitude is zero is called a pressure node and the maximum points are called pressure antinodes.
Note that as the number of half wavelengths in the standing waves increases then the number of nodes and antinodes increases, and for hard reflecting boundaries the number of pressure nodes is equal to, and the number of pressure antinodes is one more than, the number of half wavelengths. Velocity nodes and antinodes also exist, and they are always complementary to the pressure nodes, that is, a velocity antinode occurs at a pressure node and vice versa, as shown in Figure 1.30. This happens because the energy in the travelling wave must always exist at a pressure node carried in the velocity component and at a velocity node the energy is carried in the pressure component.
Figure 1.30 The pressure and velocity components of a standing wave between two hard boundaries.