for any plane wave traveling in the negative x‐direction
(3.13)
The quantity ρ c is known as the characteristic impedance of the fluid, and for air, ρ c = 428 kg s/m2 at 0 °C and 415 kg s/m2 at 20 °C.
The intensity of sound, I, is the time‐averaged sound energy that passes through unit cross‐sectional area in unit time. For a plane progressive wave, or far from any source of sound (in the absence of reflections):
where ρ = the fluid density (kg/m3) and c = speed of sound (m/s).
In the general case of sound propagation in a three‐dimensional field, the sound intensity is the (net) flow of sound energy in unit time flowing through unit cross‐sectional area. The intensity has magnitude and direction
where p is the total fluctuating sound pressure and ur is the total fluctuating sound particle velocity in the r‐direction at the measurement point. The total sound pressure p and particle velocity ur include the effects of incident and reflected sound waves.
We note, in general, for sound propagation in three dimensions that the instantaneous sound intensity I is a vector quantity equal to the product of the scalar sound pressure and the instantaneous vector particle velocity u. Thus I has magnitude and direction. The vector intensity I may be resolved into components Ix, Iy, and Iz. For a more complete discussion of sound intensity and its measurement see Chapter 8 in this book, chapters 45 and 156 in the Handbook of Acoustics [1] and the book by Fahy [13].
3.3.4 Energy Density
Consider the case again of the oscillating piston in Figure 3.1. We shall consider the sound energy that is produced by the oscillating piston, as it flows along the tube from the piston source. We observe that the wavefront and the sound energy travel along the tube with velocity c metres/second. Thus after one second, a column of fluid of length c m contains all of the sound energy provided by the piston during the previous second. The total amount of energy E in this column equals the time‐averaged sound intensity multiplied by the cross‐sectional area S, which is from Eq. (3.9):
(3.16)
The sound energy per unit volume is known as the energy density ε,
This result in Eq. (3.17) can also be shown to be true for other acoustic fields as well, as long as the total sound pressure is used in Eq. (3.17), and provided the location is not very close to a sound source.
3.3.5 Sound Power
Again in the case of the oscillating piston, we will consider the sound power radiated by the piston into the tube. The sound power radiated by the piston, W, is
(3.18)
But from Eqs. (3.10) and (3.14) the power is
and close to the piston, the rms particle velocity, urms, must be equal to the rms piston velocity. From Eq. (3.19), we can write
(3.20)
where r is the piston and duct radius, and vrms is the rms velocity of the piston.
3.4 Decibels and Levels
The range of sound pressure magnitudes and sound powers of sources experienced in practice is very large. Thus, logarithmic rather than linear measures are often used for sound pressure and sound power. The most common measure of sound is the decibel. Decibels are also used to measure vibration, which can have a similar large range of magnitudes. The decibel represents a relative measurement or ratio. Each quantity in decibels is expressed as a ratio relative to a reference sound pressure, sound power, or sound intensity, or in the case of vibration relative to a reference displacement, velocity, or acceleration. Whenever a quantity is expressed in decibels, the result is known as a level.
The decibel (dB) is the ratio R1 given by
(3.21)
Thus, R1 = 100.1 = 1.26. The decibel is seen to represent the ratio 1.26. A larger ratio, the bel, is sometimes used. The bel is the ratio R2 given by log10 R2 = 1. Thus, R2 = 101 = 10. The bel represents the ratio 10 and is thus much larger than a decibel. For simplicity, in the following sections of this book we denote log( ) = log10( ).
3.4.1 Sound Pressure Level
The sound pressure level Lp is given by
(3.22)
where pref is the reference pressure, pref = 20 μPa = 0.00002 N/m2 (= 0.0002 μbar) for air. This reference pressure was originally chosen to correspond to the quietest sound (at 1000 Hz) that the average young person can hear. The sound pressure level is often abbreviated as SPL. Figure 3.4 shows some sound pressure levels of typical sounds.
