Engineering Acoustics. Malcolm J. Crocker

f₁(ct − x) represents a wave traveling in the positive x‐direction with wave speed c, while f₂(ct + x) represents a wave traveling in the negative x‐direction with wave speed c (see Figure 3.2).

Figure 3.2 Plane waves of arbitrary waveform.

The solution given in Eq. (3.4) is usually known as the general solution since, in principle, any type of sound waveform is possible. In practice, sound waves are usually classified as impulsive or steady in time. One particular case of a steady wave is of considerable importance. Waves created by sources vibrating sinusoidally in time (e.g.a loudspeaker, a piston, or a more complicated structure vibrating with a discrete angular frequency ω) both in time t and space x in a sinusoidal manner (see Figure 3.3):

(3.5)

Figure 3.3 Simple harmonic plane waves.

At any point in space, x, the sound pressure p is simple harmonic in time. The first expression on the right of Eq. (3.5) represents a wave of amplitude A₁ traveling in the positive x‐direction with speed c, while the second expression represents a wave of amplitude A₂ traveling in the negative x‐direction. The symbols ϕ₁ and ϕ₂ are phase angles, and k is the acoustic wavenumber. It is observed that the wavenumber k = ω/c by studying the ratio of x and t in Eqs. (3.4) and (3.5). At some instant t the sound pressure pattern is sinusoidal in space, and it repeats itself each time kx is increased by 2π. Such a repetition is called a wavelength λ. Hence, kλ = 2π or k = 2π/λ. This gives ω/c = 2π/c = 2π/λ, or

(3.6)

      The wavelength of sound becomes smaller as the frequency is increased. In air, at 100 Hz, λ ≈ 3.5 m ≈ 10 ft. At 1000 Hz, λ ≈ 0.35 m ≈ 1 ft. At 10000 Hz, λ ≈ 0.035 m ≈ 0.1 ft. ≈ 1 in.

      At some point x in space, the sound pressure is sinusoidal in time and goes through one complete cycle when ω increases by 2π. The time for a cycle is called the period T. Thus, ωT = 2π, T = 2π/ω, and

      (3.7)

      Example 3.1

      The human audible frequency range is from 20 Hz to 20 kHz. Calculate the extremes of wavelength for audible sounds at 20 °C.

      Solution

From Eq. (3.3) the speed of sound at 20 °C is c = 331.6 + 0.6(20) = 343.6 m/s. Therefore, using Eq. (3.6) the wavelength of a sound of 20 Hz is λ = 343.6/20 = 17.18 m. The wavelength of a sound of 20 kHz is λ = 343.6/20000 = 0.01718 m = 17.18 mm.

      3.3.1 Sound Pressure

      With sound waves in a fluid such as air, the sound pressure at any point is the difference between the total pressure and normal atmospheric pressure. The sound pressure fluctuates with time and can be positive or negative with respect to the normal atmospheric pressure.

      Sound varies in magnitude and frequency and it is normally convenient to give a single number measure of the sound by determining its time‐averaged value. The time average of the sound pressure at any point in space, over a sufficiently long time, is zero and is of no interest or use. The time average of the square of the sound pressure, known as the mean square pressure, however, is not zero. If the sound pressure at any instant t is p(t), then the mean square pressure, 〈p²(t)〉_t, is the time average of the square of the sound pressure over the time interval T:

      (3.8)

      where 〈〉_t denotes a time average.

      It is usually convenient to use the square root of the mean square pressure:

      which is known as the root mean square (rms) sound pressure. This result is true for all cases of continuous sound time histories including noise and pure tones. For the special case of a pure tone sound, which is simple harmonic in time, given by p = P cos(ωt), the rms sound pressure is

(3.9)

      where P is the sound pressure amplitude.

      3.3.2 Particle Velocity

      As the piston vibrates, the gas immediately next to the piston must have the same velocity as the piston. A small element of fluid is known as a particle, and its velocity, which can be positive or negative, is known as the particle velocity. For waves traveling away from the piston in the positive x‐direction, it can be shown that the particle velocity, u, is given by

(3.10)

      where ρ = fluid density (kg/m³) and c = speed of sound (m/s).

      If a wave is reflected by an obstacle, so that it is traveling in the negative x-direction, then

(3.11)

      The negative sign results from the fact that the sound pressure is a scalar quantity, while the particle velocity is a vector quantity. These results are true for any type of plane sound waves, not only for sinusoidal waves.

      3.3.3 Impedance and Sound Intensity

      We see that for the one‐dimensional propagation considered, the sound wave disturbances travel with a constant wave speed c, although there is no net, time‐averaged movement of the air particles. The air