Acoustical Properties of Cellular Materials

Abstract

While the acoustical behaviour of standard homogeneous materials like sheet metal can be described by a few basic statements, cellular materials show a more complex behaviour. Depending on the structure of the cells (open cells or closed cells) different acoustical behaviour can be seen. Transmission, absorption and reflection appear and vary in quantity depending on the shape and geometrical dimensions of the cells. Of course, the basic material of a single cell as well as the type of interconnection between the cells also have a strong influence on the acoustical properties. Within this contribution the acoustical properties of cellular materials are investigated by simulation and experiments. Hollow sphere structures serve as examples and plastic foams serve as references. The absorption coefficients as well as the complex wave number and the characteristic impedance are presented. The theoretical description is based on the theory of sound in porous media.

Appendix

Simple Rayleigh theory:

$$ Z\left( \omega \right) = - j\rho_{0} c\frac{\sqrt \chi }{\sigma }\sqrt {1 - j\frac{{{{\Upxi}}\sigma }}{{\omega \varrho_{0} \chi }}} *{\text{ctg}}(k_{\text{a}} d) $$

(21)

$$ Z_{a} \left( \omega \right) = \rho_{0} c\sqrt \chi \sqrt {1 - j\frac{{{{\Upxi}}\sigma }}{{\omega \varrho_{0} \chi }}} $$

(22)

$$ k_{\text{a}} = k\sqrt \chi \sqrt {1 - j\frac{{{{\Upxi}}\sigma }}{{\omega \varrho_{0} \chi }}} $$

(23)

Definition of the parameters with units. For some parameters a typical range is added in round brackets:

d:

thickness of the absorber [m]

k_a :

wave number in the absorber [1/m]

ρ ₀ :

density of air [kg/m³]

c:

speed of sound in air [m/s]

σ:

porosity [%], (σ < 1)

χ :

structure form factor [.], (χ > 1)

ω:

frequency [Hz]

Ξ:

flow resistance [Ns/m⁴], (5,000 Ns/m⁴ < Ξ < 1,00,000 Ns/m⁴)

Z_a(ω):

characteristic Impedance [kg/m²/s]

According to Champoux and Allard [12], who introduced the concept of characteristic dimensions, it is possible to calculate the bulk modulus and the effective density for porous media. Additional parameters arise

The meaning of the additional symbols in the following formulas is

μ:

dynamic viscosity of air [Ns/m]

c_v :

shape factor of the tubes; viscous effects [.]

c_t :

shape factor of the tubes; thermal effects [.]

Pr:

Prandtl number [.]

γ:

adiabatic exponent for air [.]

Characteristic dimensions: Viscous effects:

$$ {{\Uplambda}}_{{{\upupsilon}}} = \frac{1}{{c_{\upsilon } }}\sqrt {\frac{8\chi \mu }{{\sigma {{\Upxi}}}}} $$

(24a)

thermal effects

$$ {{\Uplambda}}_{\text{t}} = \frac{1}{{c_{\text{t}} }}\sqrt {\frac{8\chi \mu }{{\sigma {{\Upxi}}}}} $$

(24b)

$$ G_{{{\upupsilon}}} \left( \omega \right) = \sqrt {1 + \frac{{j4\chi^{2} \mu \rho_{0} \omega }}{{\sigma^{2} {{\Uplambda}}_{{{\upupsilon}}}^{2} {{\Upxi}}^{2} }}} $$

(25a)

$$ G_{t} \left( \omega \right) = \sqrt {1 + \frac{{j4\chi^{2} \mu \rho_{0} \omega Pr}}{{c_{\text{t}}^{4} \sigma^{2} {{\Uplambda}}_{\text{t}}^{2} {{\Upxi}}^{2} }}} $$

(25b)

$$ {{\uplambda}}_{{{\upupsilon}}} = c_{{{\upupsilon}}} \sqrt {\frac{{8\chi \rho_{0} \omega }}{{\sigma {{\Upxi}}}}} $$

(26a)

$$ {{\uplambda}}_{\text{t}} = \frac{1}{{c_{\text{t}} }}\sqrt {\frac{{8\chi \rho_{0} \omega }}{{\sigma {{\Upxi}}}}} $$

(26b)

Bulk modulus:

$$ K\left( \omega \right) = \frac{{\gamma P_{0} }}{{\gamma - (\gamma - 1)\left[ {1 + \frac{{8G_{\text{t}} (\omega )}}{{jPr\lambda_{\text{t}}^{2} }}} \right]^{ - 1} }} $$

(27)

Effective density:

$$ \rho_{eff} \left( \omega \right) = \rho_{0} \chi \left[ {1 + \frac{{8c_{\upsilon }^{2} G_{{{\upupsilon}}} (\omega )}}{{j\lambda_{{{\upupsilon}}}^{2} }}} \right] $$

(28)

Impedance of the medium:

$$ Z_{c} = \sqrt {\rho_{eff} (\omega )K(\omega )} $$

(29)

Impedance of sample with thickness d:

$$ Z = \frac{{ - jZ_{a} { \cot }(k_{a} d)}}{\sigma } $$

(30)

Wave number in the porous medium:

$$ k_{a} = \omega \sqrt {\frac{{\rho_{eff} (\omega )}}{K(\omega )}} $$

(31)

Coefficient of absorption:

$$ \alpha = \frac{{4{\text{Re}}(\frac{Z}{\rho c})}}{{\begin{array}{*{20}c} {\left[ { {\text{Re}}\left( {\frac{Z}{\rho c}} \right) + 1} \right]^{2} + \left[ {{\text{Im}}\left( {\frac{Z}{\rho c}} \right)} \right]^{2} } \\ {} \\ \end{array} }} $$

(32)