Descripción de variables en participantes rellamadas

The combination of Eqs. 3.19 and 3.20 yields the following expression for the speed of sound

Speed of sound; ideal gas c=√

1/(κρ)=√

γ P /ρ=√

γ RT /M (3.21)

[κ: Compressibility. ρ: Density. γ : Specific heat ratio, ≈ 1.4 for air. R ≈ 8.3 joule/K, universal gas constant (per mol). T : Absolute temperature (K). M: Molar mass,≈ 0.029 kg for air. At 20^◦C(T = 293 K), c ≈ 342.6 m/s for air.]

We have assumed isentropic compression which turns out to be appropriate in most cases of sound propagation in free field. Within a porous material, on the other hand, the large heat conduction and heat capacity of the solid material prevent temperature fluctuations from occurring, and the compressibility becomes closer to isothermal, at least at sufficiently low frequencies.

The last step in Eq. 3.21 refers to an ideal gas. In that case, the sound speed depends only on temperature. This is consistent with the molecular model of sound propagation according to which the sound speed is expected to be approximately equal to the average thermal speed of the molecules which is known to be proportional to√

T.

With R = 8.3 joule·K⁻¹, M = 0.029 kg, γ = 1.4 and a temperature of 20^◦C (68^◦F , T = 293K) the sound speed in air becomes 343 m/sec (1125 ft/sec). At a temperature of 1000^◦F , it is 570.6 m/sec (1872.5 ft/sec). The isothermal wave speed in normal air is smaller than the isentropic by a factor of 1/√γ ≈ 0.845 and is ≈ 290 m/sec. The experimental evidence for sound waves over a wide range of frequencies is in overwhelming favor of the isentropic value.

In the linear approximation, u << c, the sound speed is independent of the strength of the wave, i.e., independent of the fluid velocity u. However, had we not assumed u << c in Eqs. 3.19 and 3.20, we would have found the wave speed to be c+ u, where c is the sound speed at the slightly elevated temperature in the wave due to the compression.

It is sometimes convenient to express the compressibility in terms of the sound speed. Thus, with c=√

1/κρ, we get

κ= 1/(γ P ) = 1/(ρc²). (3.22)

The motion thus described is a sound wave. In the linear approximation, u << c, the sound pressure p is proportional to the fluid velocity u, p= (ρc)u, as obtained from Eq. 3.20. Actually, as derived in Eqs. 3.19 and 3.20, this relation is valid for a wave traveling in the positive x-direction, with u counted positive in this direction. The pressure p does not depend on the direction, and for a wave traveling in the negative x-direction, the fluid velocity becomes negative, and we have to put p= −(ρc)u. Thus, for a plane traveling wave, the relation between sound pressure and fluid velocity is

Pressure–velocity relation in plane wave p= (ρc)u for wave in positive x-direction p= −(ρc)u for wave in negative x-direction

. (3.23)

The constant of proportionality ρc is called the wave impedance of the fluid, Wave impedance

ρc=√

ρ/κ (3.24)

[ρ: Density. c: Sound speed. For air at 1 atm and 20^◦C(T = 293 K), ρ ≈ 1.27 kg/m³, c≈ 342.6 m/s, ρc ≈ 435 MKS. At 0^◦C(273 K) the value is≈ 420 MKS.]

For water vapor at 1000^◦F (811 K) and a pressure of 1000 psi (≈ 6.8 · 10⁶N/m²), ρc ≈ 12700 MKS (i.e., about 32 times greater than for normal air). This latter condition is typical for the steam in a nuclear power plant.

The derivation of the relation between p and u was based on the analysis of a positive displacement of the piston of duration
t generating a compressional wave.

If the piston is moved in the negative direction, a rarefaction (expansion) wave is generated in which the perturbations of density, pressure, and fluid velocity will be negative. A succession of positive and negative pulses can be used to build up an arbitrary time dependence. Thus, for an infinitely extended tube or a tube with an absorber at the end so that no reflected sound is present, the relation p= ρcu is valid for any time dependence of a wave traveling in the positive x-direction.

If the tube extends to the left of the piston along the negative x-direction, a negative displacement (and velocity) of the piston gives rise to a compression wave in the negative x-direction but the velocity in this wave is in the negative x-direction and the relation between pressure and velocity is still p= −ρcu. In the rarefaction wave generated in the positive x-direction, both pressure and velocity are negative so that p= ρcu is still valid.

Of particular interest is the harmonic time dependence. Then, for a wave traveling in the positive x-direction the pressure and velocity waves take the form

p(x, t )= |p| cos(ωt − kx)

u(x, t )= (|p|/ρc) cos(ωt − kx), (3.25) where k = ω/c = 2π/λ, as mentioned in Eq. 3.1. For a wave traveling in the negative x-direction,−kx is replaced by kx and u by −u. Quantity |p| is the pressure amplitude.

If the piston is located at x= x^rather than at x= 0, the time of wave travel to the observation point x will be (x−x^)/cso that kx in Eq. 3.25 will be replaced k(x−x^). We can incorporate both directions of wave travel by replacing ωt = k(x − x^)by ωt− k|x − x^|.

Rms value.

The mean square value of the sound pressure is

p²(t ) = (1/T )

_T

0

p²(t )dt (3.26)

and for a harmonic pressure wave, this becomes|p|²/2. The square root of this quantity is the rms-value of the pressure which for the harmonic wave is

p_rms= |p|/√

2 (harmonic wave). (3.27)

It is this value that is usually indicated on instruments that measure sound pressure and we shall often use the symbol p for it if there is no risk of misunderstanding.

Density and Temperature Fluctuation in Sound

With an isentropic compressibility in the change of state that occurs in a sound wave we have dP /P = γ dρ/ρ. Then, with dP = p follows that the density fluctuation that is caused by a sound pressure dP = p becomes dρ = p/(γ P /ρ) = p/c².

There is also a temperature fluctuation. From the equation of state P = rρT follows dP /P = dρ/ρ + dT /T , or dT /T = (γ − 1)dρ/ρ = (γ − 1)p/c². The acoustic perturbation in temperature is then

dT = (γ − 1)T p/c²= γ− 1

γ (p/P )T . (3.28)

In a plane wave, u= p/ρc so that dT = T (γ − 1)(u/c).

Intensity

The work per unit area done by the piston in Fig. 3.3 as it moves forward during the time
t is pu
t; it is pu per unit of time. Conservation of energy requires that this energy must be carried by the wave. Thus, the wave energy per second and unit area is I = pu; it is called the acoustic intensity. Since p = ρcu, it follows that

I (x, t )= p(x, t)u(x, t) = ρcu²(x, t )= (p²(x, t )/ρc). (3.29) In the case of a single traveling wave with harmonic time dependence, p(x, t)=

|p| cos(ωt−kx), the intensity is I (x, t) = ρc|u|²cos²(ωt−kx) = |p||u| cos²(ωt−kx).

We are generally interested in the time average of the intensity which is I= |p||u|/2.

The same notation, I , will be used for both, but when time is involved, it is shown explicitly as an argument, I (t); without this argument, time average is implied. If rms values p and u are used for the amplitudes, the intensity is simply I = pu = ρcu²= p²/ρc. For the traveling wave, it is independent of x.

At the threshold of hearing, with pr = 2 × 10⁻⁵N/m², the threshold intensity is I_r ≈ 10⁻¹²w/m².

Acoustic Energy Density

The energy density in a wave is the sum of the kinetic energy density ρu²(t )/2 and the potential or compressional energy density which can be expressed as κp²(t )/2, where κ = 1/(γ P ) = 1/ρc²is the compressibility. In a single traveling wave, with p = ρcu, these quantities are the same and if the total energy density is denoted W = ρu²/2+ κp²/2, it follows that

I = Wc

W = ρu²/2+ κp²/2. (3.30)

In a single traveling wave, the pressure is ρcu, and the corresponding reaction force on the piston that drives the wave is Aρcu, proportional to the velocity like a viscous friction force. If the piston is part of a harmonic oscillator, the power transferred to the wave results in damping of the oscillator, usually referred to as radiation damping and the wave impedance ρc is often called wave resistance.

Complex Amplitude Description

Suppose a problem has been solved for the complex pressure amplitude p(ω) and the corresponding velocity u(ω). How do we use these amplitudes to express the intensity in the sound field? To find out, we go back to the corresponding real quantities, p(t) and u(t), and express these quantities in terms of the complex amplitudes. This is facilitated with the aid of complex conjugate quantities. With reference to Appendix B, we are reminded that the complex conjugate of a complex number z= r + ix is z^∗= r − ix, so that r = (z + z^∗)/2.

Thus, we express p(t) as p(t) = (1/2)(p(ω) exp(−iωt) + p^∗(ω)exp(iωt) and u(t ) = (1/2)(u(ω) exp(−iωt) + u^∗(ω)exp(iωt). The time average intensity

I = p(t)u(t) then becomes I = (pu^∗ + p^∗u)/4. We note that pu^∗ is the complex conjugate of p^∗u so that the sum is twice the real part of pu^∗. Thus, with p = |p| exp(iφ1) and u = |u| exp(iφ2), we have u^∗ = |u| exp(−iφ2 and pu^∗= |p||u| exp(iφ), where φ = φ1− φ2. Thus,

I = (1/2){pu^∗} = (1/2)|p||u| cos(φ), (3.31) where φ is the phase difference between pressure and velocity. If the amplitudes are rms values, the factor 1/2 has to be eliminated.

Intensity Probe

An intensity probe consists of two closely spaced microphones in combination with a two-channel FFT (Fast Fourier Transform) analyzer. The sum of the output signals is the average sound pressure between the microphones and the difference represents the gradient of the pressure, respectively. The particle velocity is proportional to the gradient and the product of these quantities yields the intensity. In terms of the signals from the two microphones, this turns out to proportional to the cross spectrum density of these signals, which is automatically determined by the analyzer. All that remains is a constant of proportionality which can be incorporated in the signal processing program.

The formal derivation of this result is given below. It is based on the Fourier trans-forms of the pressure and the velocity and is quite similar to the derivation of the energy spectrum density discussed in Chapter 2.

Derivation

The sound pressure p(x, t) is expressed in terms of its Fourier amplitude p(ν), i.e.,

p(x, t )= §p(x, ν)e^−i2πνtdν (3.32) and the particle velocity u(x, t) in the x-direction in terms of its Fourier amplitude is u(x, ν).

Then, from the momentum equation ρdu/dt= −∂p/∂x it follows that u(x, ν) = (1/iωρ)∂p/∂x.

The intensity in the x-direction is

I (t )= p(x, t)u(x, t) = §p(x, ν)e^−i2πνtdν§(1/iωρ)∂p(x, ν^)/∂xe^−i2πνdν^

= (1/iωρ)§§e^{−i2π(ν+ν)t}dνdν^. (3.33) Integrating I (t) over all times produces δ(ν+ν^)and integration over ν^yields a contribution only if ν^= −ν and we obtain

§I (t)dt= (1/iωρ)§p(x, ν)∂p(x, −ν)/∂x dν. (3.34) The microphones are located at x− d/2 and x + d/2 at which points the pressures are p₁ and p₂. We put p(x)= (p₁+ p₂)/2 and express the gradient as ∂p(x)/∂x = (p₂− p₁)/d.

With p(−ν) = p^∗(ν), the integrand in Eq. A.9 becomes (p₁+ p₂)(p^∗₂− p₁^∗). Neglecting the term|p₂|²− |p₁|²and realizing that p₂p^∗₁is the complex conjugate of p₁p₂^∗, the remaining p₁p^∗₂− p₂p₁^∗is twice the imaginary part of p₁p^∗₂. Thus, we obtain

§I (t)dt= (1/iωρd)§2{p1(ν)p₂^∗(ν)} dν ≡ §I (ν)dν, (3.35)

where the intensity spectrum is

I (ν)= (2/ωρd){p₁(ν)p^∗₂(ν)}. (3.36) With the signals from the two microphones analyzed with a two-channel analyzer, the quan-tity p₁(ν)p₂^∗(ν), the cross spectrum density, is obtained directly from the two-channel FFT analyzer.