Sound Reflection, Absorption, and Transmission
4. Sound reflections in a duct
4.2 Sound Absorption
4.2.6 Generalization; Impedance Boundary Condition
The analysis of reflection and absorption will now be generalized to a boundary which is specified acoustically by a complex normal impedance z(ω), i.e., the ratio of the
complex amplitudes of sound pressure and the normal component of velocity at the boundary. It is assumed that this impedance is known from experiments or has been calculated from known properties of the boundary, as was the case for the resonator example given in the previous section. The generalization also involves considering
Figure 4.3:Obliquely incident wave on a boundary.
sound at oblique incidence and we start by discussing the description of such a wave, shown schematically in Fig. 4.3.
The wave is incident on the plane boundary at x = 0 (yz-plane) under an angle φ with the x-axis which is normal to the boundary. Let the coordinate along the direction of propagation be r. The corresponding vector is r. We also introduce the propagation vector k with the magnitude k= ω/c = 2π/λ and direction along the line of propagation, i.e., along r. Thus, kr = k · r = kxx + kyy, where we have expressed the scalar product k· r in terms of the components kx, ky and x, y of k and r.
The complex amplitude of the incident plane wave can then be expressed as pi(ω)= |pi|eikr = |pi|eikxxeikyy, (4.45) where kx = k cos(φ) and ky = k sin φ. With k = 2π/λ it follows that kx = 2π/λx, where the geometrical meaning of λx = λ/ cos φ is shown in the figure. It is the spatial period of the wave in the x-direction (i.e., the distance between two adjacent wave crests).
The reflected wave from the boundary has a propagation vector with the compo-nents−kxand kyso that the complex amplitude of the reflected wave is
pr(ω)= |pr|e−ikxxeikyy. (4.46) The factor exp(ikyy)is of little interest in what we are going to do here so that in what follows it can be considered included in|pi| and |pr|.
The total sound pressure field on the left side of the boundary is then
p(x)= pi(x)+ pr(x), (4.47)
where p stands for the total complex pressure amplitude.
As discussed in detail in Chapter 5, the velocity field follows from the equation of motion ρ∂ux/∂t = −∂p/∂x with the complex amplitude version −iωρux(ω) =
−∂p(ω)/∂x.5
We have−iωρui = −ikxpi = −ik cos φpi with a similar expression for the re-flected wave. Thus, the total velocity field that corresponds to the pressure field in Eq. 4.47 is
ρc ux= cos φ[pi(x)− pr(x)], (4.48) where we have made use of kx = k cos φ and k = ω/c.
The complex normal impedance of the boundary (at x= 0) is z, and this condition requires that p(0)/u(0)= z, i.e.,
pi(0)+ pr(0)
pi(0)− pr(0) = ζ cos φ, (4.49) where ζ = z/ρc is the normalized impedance. It follows then that the pressure reflection coefficient is
Pressure reflection coefficient
Rp(ω)= pr(0)/pi(0)= (ζ cos φ − 1)/(ζ cos φ + 1) (4.50) [pi(0), pr(0): Incident and reflected complex pressure amplitudes at the boundary.
ζ: Normalized impedance of the boundary. φ: Angle of incidence].
The ratio of the incident and reflected intensities is Ii/Ir = |R|2and the ratio of the absorbed and incident intensity is Ia/Ii = (Ii−Ir)/Ii = 1−|R|2. In other words, the absorption coefficient Ia/Ii is
α= 1 − |R|2. (4.51)
If the impedance is expressed in terms of a real and imaginary part, ζ = θ + iχ, it follows from Eq. 4.50 that
Absorption coefficient
α(φ)= 4θ cos φ/[(1 + θ cos φ)2+ (χ cos φ)2] (4.52) [θ, χ: Real and imaginary parts of the normalized boundary impedance. φ: Angle of incidence].
As will be discussed in Chapter 6, the sound field in a room often can be approxi-mated as diffuse, which means that if the field is regarded as a superposition of plane waves traveling in different directions, the probability of wave travel is the same in all directions. In regard to the absorption by a plane boundary in such a field, we have to average the absorption coefficient in Eq. 4.52 over all angles of incidence.
There are many directions of propagation that correspond to an angle of incidence φand these directions are accounted for in the following way. The probability that acoustic intensity will strike an element of the wall at an angle between φ and φ+ dφ is proportional to the solid angle 2π sin φ dφ, i.e., the ring-like surface element on a
Figure 4.4: The probability of a wave having an angle of incidence φ in a diffuse field is proportional to the solid angle element (shaded) 2π sin φ dφ.
unit sphere centered at the wall element, as indicated schematically in Fig. 4.4. The power that strikes a wall element of unit area is the product of the intensity i(φ)= I and the projection cos φ of this area is perpendicular to the incident wave direction.
Thus, a factor cos φ has to be included in calculating the average absorption coefficient which then becomes
αd =
π/2
0 α(φ)2π sin φ cos φ , dφ
π/2
0 2π sin φ cos φdφ = 2
π/2
0
α(φ)sin φ cos φdφ, (4.53)
where α(φ) is obtained from Eq. 4.52. The denominator expresses the total intensity striking the wall element. The coefficient αdwill be called the diffuse field absorption coefficient, sometimes also called the statistical average. The results in Eqs. 4.50 and 4.52 are valid even if the impedance ζ depends on the angle of incidence. For some boundaries, called locally reacting, the impedance is independent of the angle and thus equals the value for normal incidence. The impedance can then be measured with relatively simple experiments in which the sample is placed at the end of a tube and exposed to a plane wave of sound, as described in Section 4.2.7. An example of a locally reacting boundary is a honeycomb structure backed by a rigid wall, in which the cell size is much smaller than a wavelength. The oscillatory velocity in each of the cells then depends only on the local pressure at the entrance to the cell and there is no coupling between the cells, preventing wave propagation along the boundary within the absorber.
With ζ independent of φ, the diffuse field absorption coefficient in Eq. 4.53 can be expressed in closed form (see Problem 8).
For a nonlocally reacting boundary or boundary with an extended reaction, the impedance is angle dependent and the experimental data of it are normally not
avail-5An element of thickness x has the mass ρx. With the pressure being a function of x, the pressures at the two surfaces of the elements are p(x) and p(x+ x) so that the net force on the element in the x-direction is p(x)− p(x + x) = −∂p/∂x x and the equation of motion, Newton’s law, is ρ∂ux/∂t=
−∂p/∂x. For further details, see Chapter 5
able. For relatively simply types of boundaries, however, the impedance can be cal-culated, but αdgenerally has to be determined by numerical integration in Eq. 4.52.
An example of a nonlocally reacting boundary is a uniform porous layer backed by a rigid wall.
Sheet Absorber
As an example of a resonator absorber, we have chosen to analyze an absorber which is frequently used in practice. It consists of a porous sheet or wire mesh screen backed by an air layer and a rigid wall, as illustrated schematically in Fig. 4.5. Two configurations are shown, one with and the other without a honeycomb structure in the air layer. The honeycomb has a cell size assumed to be much smaller than a wavelength and it forces the fluid velocity in the air layer to be normal to the wall, regardless of the angle of incidence of the sound. The first configuration is a locally and the second a nonlocally reacting absorber, as indicated.
As we shall see, either configuration can be considered to be a form of acoustic resonator but unlike the resonator absorber in the previous example, it has multiple resonances. In the present context, the relevant property of a sheet or screen that
Figure 4.5:Porous sheet-cavity absorber. Left: Locally reacting. Right: Nonlocally reacting.
can readily be measured is the steady flow resistance. A pressure drop P across the sheet produces a velocity U through the sheet and the flow resistance is defined as r = P /U. The same resistance is approximately valid also for the oscillatory flow in a sound wave.
In the locally reacting absorber, the fluid velocity in the air layer is forced by the partitions to be in the x-direction, normal to the boundary, so that kx = k. The normal impedance is simply the sum of the sheet resistance θ and the impedance of the air column in a cell which we have found earlier to be i cot(kL) (see Eq. 3.64), both normalized with respect to ρc, where k = ω/c and L the thickness of the air layer. Thus, the absorption coefficient is obtained by inserting the impedance
ζ = θ + i cot(kL) into Eq. 4.52.
α(φ)= 4θ cos φ
(1+ θ cos φ)2+ cos2(φ)cot2(kL) (4.54) For the sheet absorber without partitions, the fluid velocity in the air layer is no longer forced to be in the x-direction and the normal impedance of the air layer has to be modified. One obvious modification is that we have to use kxL = kL cos φ rather than kL. Furthermore, since the normal impedance is the ratio of the complex amplitude of the pressure and the normal component ux = u cos φ of the fluid velocity, the normalized impedance of the air layer will be i(1/ cos φ) cos(kL cos φ). Thus, the absorption coefficient for the nonlocally reacting sheet absorber becomes (see Eq. 4.52)
α(φ)= 4θ cos φ
(1+ θ cos φ)2+ cot2(kLcos φ). (4.55) The corresponding diffuse field absorption coefficient is obtained from Eq. 4.53.
Figure 4.6: Absorption spectra of sheet absorber. (a) Normal incidence. (b) Diffuse field, locally reacting. (c) Diffuse field, nonlocally reacting.
Fig. 4.6 shows the computed frequency dependence of the absorption coefficient of a sheet absorber in which the frequency parameter is the ratio of the thickness L and the wavelength λ. On the left, the flow resistance of the sheet is r = ρc, i.e., θ= 1, and on the right, θ = 2. In each graph, three curves are shown; one for normal incidence and two for diffuse fields corresponding to an air backing with and without a honeycomb.
When the thickness is an odd number of quarter wavelengths in the locally reacting absorber, the absorption coefficient will have a maximum. The standing wave in the air cavity then has a pressure anti-node at the sheet so that there will be no back pressure on it. Then, if r = ρc, the impedance of the absorber is matched to the wave impedance so that no reflection occurs and all incident sound is absorbed.
On the other hand, when the thickness is an integer number of half wavelengths, the velocity is zero at the screen so that there will be no absorption (anti-resonance). This is true for both normal incidence and diffuse field. For nonlocal reaction, however, the standing wave pattern depends on the angle of incidence and zero absorption cannot be obtained at all angles of incidence at a given frequency.