Engineering Acoustics. Malcolm J. Crocker

the piston is vibrating with simple harmonic motion at the left‐hand side of the tube (see Figure 3.1) and that it has been oscillating back and forth for some time. We shall only consider the piston motion and the oscillatory motion it induces in the fluid from when we start our clock. Let us choose to start our clock when the piston is moving with its maximum velocity to the right through its normal equilibrium position at x = 0. See the top of Figure 3.1, at t = 0. As time increases from t = 0, the piston straight away starts slowing down with simple harmonic motion, so that it stops moving at t = T/4 at its maximum excursion to the right. The piston then starts moving to the left in its cycle of oscillation, and at t = T/2 it has reached its equilibrium position again and has a maximum velocity (the same as at t = 0) but now in the negative x direction. At t = 3 T/4, the piston comes to rest again at its maximum excursion to the left. Finally at t = T the piston reaches its equilibrium position at x = 0 with the same maximum velocity we imposed on it at t = 0. During the time T, the piston has undergone one complete cycle of oscillation. We assume that the piston continues vibrating and makes f oscillations each second, so that its frequency f = 1/T (Hz).

Figure 3.1 Schematic illustration of the sound pressure distribution created in a tube by a piston undergoing one complete simple harmonic cycle of operation in period T seconds.

      As the piston moves backward and forward, the gas in front of the piston is set into motion. As we all know, the gas has mass and thus inertia and it is also compressible. If the gas is compressed into a smaller volume, its pressure increases. As the piston moves to the right, it compresses the gas in front of it, and as it moves to the left, the gas in front of it becomes rarified. When the gas is compressed, its pressure increases above atmospheric pressure, and, when it is rarified, its pressure decreases below atmospheric pressure. The pressure difference above or below the atmospheric pressure, p₀, is known as the sound pressure, p, in the gas. Thus the sound pressure p = p_tot – p₀, where p_tot is the total pressure in the gas. If these pressure changes occurred at constant temperature, the fluid pressure would be directly proportional to its density, ρ, and so p/ρ = constant. This simple assumption was made by Sir Isaac Newton, who in 1660 was the first to try to predict the speed of sound. But we find that, in practice, regions of high and low pressure are sufficiently separated in space in the gas (see Figure 3.1) so that heat cannot easily flow from one region to the other and that the adiabatic law, p/ρ^γ = constant, is more closely followed in nature.

As the piston moves to the right with maximum velocity at t = 0, the gas ahead receives maximum compression and maximum increase in density, and this simultaneously results in a maximum pressure increase. At the instant the piston is moving to the left with maximum negative velocity at t = T/2, the gas behind the piston, to the right, receives maximum rarefaction, which results in a maximum density and pressure decrease. These piston displacement and velocity perturbations are superimposed on the much greater random motion of the gas molecules (known as the Brownian motion). The mean speed of the molecular random motion in the gas depends on its absolute temperature. The disturbances induced in the gas are known as acoustic (or sound) disturbances. It is found that momentum and energy pulsations are transmitted from the piston throughout the whole region of the gas in the tube through molecular interactions (sometimes simply termed molecular collisions).

      If a disturbance in a thin cross‐sectional element of fluid in a duct is considered, a mathematical description of the motion may be obtained by assuming that (i) the amount of fluid in the element is conserved, (ii) the net longitudinal force is balanced by the inertia of the fluid in the element, (iii) the compressive process in the element is adiabatic (i.e. there is no flow of heat in or out of the element), and (iv) the undisturbed fluid is stationary (there is no fluid flow). Then the following equation of motion may be derived:

(3.1)

      where p is the sound pressure, x is the coordinate, and t is the time.

This equation is known as the one‐dimensional equation of motion, or acoustic wave equation. Similar wave equations may be written if the sound pressure p in Eq. (3.1) is replaced with the particle displacement ξ, the particle velocity u, condensation s, fluctuating density ρ′, or the fluctuating absolute temperature T′. The derivation of these equations is in general more complicated. However, the wave equation in terms of the sound pressure in Eq. (3.1) is perhaps most useful since the sound pressure is the easiest acoustical quantity to measure (using a microphone) and is the acoustical perturbation we sense with our ears. It is normal to write the wave equation in terms of sound pressure p, and to derive the other variables, ξ, u, s, ρ′, and T′ from their relations with the sound pressure p [16]. The sound pressure p is the acoustic pressure perturbation or fluctuation about the time‐averaged, or undisturbed, pressure p₀.

      The speed of sound waves c is given for a perfect gas by

(3.2)

The speed of sound is proportional to the square root of the absolute temperature T. The ratio of specific heats γ and the gas constant R are constants for any particular gas. Thus Eq. (3.2) may be written as

(3.3)

where, for air, c₀ = 331.6 m/s, the speed of sound at 0 °C, and T_c is the temperature in degrees Celsius. Note that Eq. (3.3) is an approximate formula valid for T_c near room temperature. The speed of sound in air is almost completely dependent on the air temperature and is almost independent of the atmospheric pressure. For a complete discussion of the speed of sound in fluids, see chapter 5 in the Handbook of Acoustics [1].

      A solution to Eq. (3.1) is

(3.4)

where f₁ and f₂ are arbitrary functions such as sine, cosine, exponential, log, and so on. It is easy to show that Eq. (3.4) is a solution to the wave equation Eq. (3.1) by differentiation and substitution into Eq. (3.1). Varying x and t in Eq.