with the flute and string can be considered mathematically to be composed of two waves of equal amplitude traveling in opposite directions. Consider the case of a lateral wave on a string under tension. If we create a wave at one end, it will travel forward to the other end. If this end is fixed, it will be reflected. The original (incident) and reflected waves interact (and if the reflection is equal in strength) a perfect standing wave will be created. In Figure 3.30 we show three different frequency waves that have interacted to cause standing waves of different frequencies on the string under tension. A similar situation can be conceived to exist for one‐dimensional sound waves in a tube or duct. If the tube has two hard ends, we can create similar standing one‐dimensional sound waves in the tube at different frequencies. In a tube, the regions of high sound pressure normally occur at the hard ends of the tube, as shown in Figure 3.31. See Refs. [14, 32, 33].
Figure 3.30 Waves on a string: (a) Two opposite and equal traveling waves on a string resulting in standing waves, (b) first mode, n = 1, (c) second mode, n = 2, and (d) third mode, n = 3.
Figure 3.31 Sound waves in a tube. First mode standing wave for sound pressure in a tube. This mode is called the fundamental and occurs at the fundamental frequency.
A similar situation occurs for bending waves on bars, but because the equation of motion is different (dispersive), the higher natural frequencies are not related by simple integers. However, for the case of a beam with simply supported ends, the higher natural frequencies are given by 22, 32, 42, 52, …, or 4, 9, 16, 25 times the fundamental frequency, …, and the mode shapes are sine shapes again.
The standing waves on two‐dimensional systems (such as bending vibrations of plates) may be considered mathematically to be composed of four opposite traveling waves. For simply supported rectangular plates the mode shapes are sine shapes in each direction. For three‐dimensional systems such as the air volumes of rectangular rooms, the standing waves may be considered to be made up of eight traveling waves. For a hard‐walled room, the sound pressure has a cosine mode shape with the maximum pressure at the walls, and the particle velocity has a sine mode shape with zero normal particle velocity at the walls. See chapter 6 in the Handbook of Acoustics [1] for the natural frequencies and mode shapes for a large number of acoustical and structural systems.
For a three‐dimensional room, normally there are standing waves in three directions with sound pressure maxima at the hard walls.
To understand the sound propagation in a room, it is best to use the three‐dimensional wave equation in Cartesian coordinates:
or
This equation can have solutions that are “random” in time or are for the special case of a pure‐tone, “simple harmonic.”
The simple harmonic solution is of considerable interest to us because we find that in rooms there are solutions only at certain frequencies. It may be of some importance now to mention both the sinusoidal solution and the equivalent solution using complex notation that is very frequently used in acoustics and vibration theory.
For a one‐dimensional wave, the simple harmonic solution to the wave equation is
where k = ω/c = 2πf/c (the wavenumber).
The first term in Eq. (3.88) represents a wave of amplitude
The equivalent expression to Eq. (3.88) using complex notation is
where
For the three‐dimensional case (x, y, and z propagation), the sinusoidal (pure tone) solution to Eq. (3.87) is
Note that there are 23 (eight) possible solutions given by Eq. (3.90). Substitution of Eq. (3.90) into Eq. (3.87) (the three‐dimensional wave equation) gives (from any of the eight (23) equations):
from which the wavenumber k is
and
