CARACTERÍSTICAS DE LA ADOLESCENCIA

The most common acoustic absorber used as a wall treatment is a uniform porous layer of fiber glass, foam, porous metals, etc. Acoustically, it is equivalent to a collection of closely spaced porous screens; the absorption spectrum is broader than for the single screen absorber and the anti-resonances are absent. The flow resistance per unit thickness is obtained in the same way as for the resistive sheet. It is the most important material property as far as sound absorption is concerned. In this analysis, the porous frame will be assumed to be rigid.

The oscillatory air flow of the sound within the porous material is forced to follow an irregular path by the randomly oriented fibers and pores in the material. The corresponding repeated local changes in direction and speed of the flow results in a force on the porous material and a corresponding reaction force on the fluid which is proportional to acceleration and can be accounted for in terms of an induced mass density. In the mathematical analysis of sound absorption, only an average velocity is used and the irregular motion and the corresponding inertial reaction force on the material is accounted for by assigning a higher inertial mass density to the air, the sum of ordinary mass density and the induced mass density. The effect is analogous to the apparent increase of mass we experience when accelerating a body in water such as a leg or an arm. The (empirically determined) factor used to express the apparent increase in density of the air in a porous material is called the structure factor . It is typically 1.5-2.

The heat conduction and heat capacity of a solid material is much larger than for a gas. As was the case in the thermal boundary layer in Section 4.2.3 this makes the compressibility different than in free field, far from boundaries. The effect of heat conduction could be accounted for by means of a complex compressibility and the same is true in a porous material. Associated with it is a thermal relaxation time which expresses the time delay between the change in pressure caused by a change of volume. In harmonic time dependence this means a phase difference between the two. In a porous material, the relaxation time is related to the pore size which influences the flow resistance of the material. Consequently, there is a relation be-tween the thermal relaxation time and the flow resistance and bebe-tween the complex compressibility and the flow resistance. If the flow resistance per unit length in the material is denoted r, it is left as a problem to show that it is a good approximation to express the complex compressibility as frequency goes to zero, ˜κ goes to the isothermal value γ κ and in the high frequency limit it is κ. The transition frequency between the two regions is fv = ωv/(2π)= r/(2πρ). In terms of the normalized resistance θ = r/ρc, we get fv = cθ/2π.

Thus, for a (typical) material, θ = 0.5 per inch and with c ≈ 1120 · 12 inch/s, we get f_v≈ 1070 Hz. In other words, the compressibility will be approximately isothermal over a substantial frequency range.

Equations of Motion

In this section we shall outline how the absorption spectra of a uniform porous material can be calculated in terms of the physical properties of the material. It also serves as a practical example of the utility of complex amplitudes.

With H being the porosity, the amount of air per unit volume of the porous ma-terial is Hρ. We define the average fluid velocity in the sound field in such a way that ρu (rather than Hρu). We choose this definition since it will make the equa-tions and boundary condiequa-tions simpler. Under isentropic condiequa-tions, neglecting heat conduction, the relation between the density and pressure perturbations δ and p is δ/ρ = κp = 1/ρc², where κ (= (1/ρ)∂ρ/∂P ) is the compressibility of the fluid involved and c the ordinary (isentropic) speed of sound. The first term in the mass conservation equation ∂(Hρ)/∂t+ div u = 0 can then be written ρκ∂p/∂t and we get

H κ∂p

∂t = −div u. (4.61)

In the momentum equation we have to account for both the flow resistance and the induced mass. Thus, with the flow resistance per unit length of the material denoted rand the equivalent mass density ρ, where is the structure factor defined above, accounting for the induced mass, the momentum equation becomes

∂ ρu

∂t + ru = −grad p. (4.62)

The velocity can be eliminated between these equations by differentiating the first with respect to time and taking the divergence of the second. With div grad p= ∇²p, we then get

∇²p− (H /c²)∂²p

∂t² − (κrH )∂p

∂t = 0. (4.63)

If the flow resistance is small so that the third term can be neglected, we get an ordinary wave equation with a wave speed c/√

H . If the flow resistance is large so that the second term, representing inertia, can be neglected, we get instead a diffusion equation.

The assumption of an isentropic compressibility in the porous material is unrealistic because of the narrow channels in the material and the high heat conductivity of the solid material (compared to air). In harmonic time dependence we can account for heat conduction by using the making the compressibility complex and, as in Section 4.2.3, we denote it˜κ. Furthermore, in the momentum equation we combine the first and second term into one, (−iωρ + r)u ≡ ˜ρu, where ˜ρ is a complex density. The complex amplitude versions of Eq. 4.61 and 4.62 can then be expressed as

−iω ˜κ^p= −div u

−iω ˜ρu = −grad p, (4.64)

where ˜ρ = ρ( + ir/ωρ) and κ^ = H κ. Incidentally, the complex compressibility is analogous to the inverse of the complex spring constant which is used to account

for compressional losses in a spring in parallel with a dashpot damper. Similarly, the complex density corresponds to the complex mass in a mass-spring oscillator in which the forces due to inertia and friction are combined into one.

The complex density contains the flow resistance and the structure factor and on the basis of the results obtained from this analysis, experiments can be devised for the measurement of these quantities. For example, they can be obtained from the measurement of the phase velocity and the spatial decay rate of a sound wave in a porous material, assuming that the porosity has been determined from another experiment.

Propagation Constant and Wave Impedance

Eliminating u between the equations in Eq. 4.64, we obtain

∇²p+ ˜ρ ˜κ^p= 0. (4.65)

With a space dependence of the complex sound pressure amplitude∝ exp(iqxx+ iq_yy+ iqzz), we obtain from Eq. 4.65,

q²= qx²+ qy²+ qz²= k²(˜ρ/ρ)(˜κ^/κ), (4.66) where we have used for normalization the isentropic compressibility κ= 1/ρc²and k= ω/c.

The corresponding normalized propagation constant is Q≡ q/k ≡ Qr + iQi =

(˜ρ/ρ)(˜κ^/κ), (4.67) where ˜ρ and ˜κ^are given in Eqs. 4.64.

The front surface of the porous material is located in the yz-plane at x = 0 and a plane sound wave is incident on it. The complex pressure amplitude is expressed as p(x, y, z, ω)= A exp(ikxx+ ikyy+ ikzz), where, from the wave equation in free field, we get k²x+k²y+kz²= k²≡ (ω/c)². The direction of the wave is specified by the polar angle φ with respect to the x-axis and the azimuthal angle ψ, measured from the z-axis. In other words, the projection of the propagation vector on the yz-plane has the magnitude k sin φ and we have ky= k sin φ sin ψ and kz= k sin φ cos ψ.

Similarly, the wave function inside the material is exp(iqxx+ iqyy+ iqzz), where q² = qx²+ qy² + qz². The wave vector components in the y- and z-direction are continuous across the surface of the absorber so that qy = ky = k sin φ sin ψ and qz = kz = k sin φ cos ψ. This is equivalent to saying that the intersection of the incident wave front with the boundary and the corresponding intersection of the wave front in the porous material are always the same.

It follows then that q_x≡ (ω/c)Qx=

q²− q_y²− q_z²=

q²− k²sin²φ= (ω/c)

Q²− sin²φ, (4.68)

where Q= q/k, k = ω/c, and φ the angle of incidence. The velocity component in the x-direction is obtained from

u_x= (1/iω ˜ρ)∂p

∂x, (4.69)

where ˜ρ/ρ = s + izv/ωρ.

The wave admittance in the x-direction is the ratio ux/pfor a traveling wave in the x-direction for which ∂p/∂x = iqxp. It follows from the equations above that the normalized value of the wave admittance and the corresponding impedance are given by

η_w = 1/ζw = ρcux/p= Qx

˜ρ/ρ, (4.70)

where Qxis given in Eq. 4.68 and ˜ρ in Eq. 4.64.

We recall that the input impedance of an air layer of thickness L is z = i (ρc) cot(kL). The impedance of a uniform porous layer has the same general form but with ρc replaced by a complex wave impedance and k by a complex propagation constant qx, both containing the flow resistance per unit length and the structure factor. Once the input impedance of the layer has been expressed in this manner, the absorption coefficient can be computed from Eqs. 4.52 and 4.53. Examples of

Figure 4.7: Sound absorption spectra of a uniform porous layer of thickness L backed by a rigid wall. The frequency variable is L/λ and the parameter R, ranging from 1 to 32, is the normalized total flow resistance of the layer. λ is the free field wavelength. Left: Normal incidence. Right: Diffuse field.

computed absorption spectra thus obtained are shown in Fig. 4.7. The graphs on the left and on the right refer to normal incidence and diffuse field, respectively. The parameter which ranges from 1 to 32 is the normalized total flow resistance R of the layer. The absorber is assumed locally reacting. Under these conditions, R-values less than 1 yield a lower absorption than for R= 1 and generally are of little interest.

With the use of normalized values of the frequency parameter and the flow resis-tance in this figure, different curves for different layer thicknesses are not needed.

However, in practice, it is more convenient to have the frequency in Hz as a variable.

We leave it for Problem 9 to plot a spectrum or two in this manner, using the univer-sal curves in the figure as sources of data. For an R-value of 4 and a layer thickness larger than one-tenth of a wavelength, the absorption coefficient exceeds 80 percent for both normal incidence and diffuse fields.

4.2.9 Problems