Radiative Transfer Equation for Vibroacoustics

Statistical Energy Analysis (SEA) is a statistical theory of vibroacoustics, similar to Sabine's formula in room acoustics. It is based on the idea that the vibrational field is diffuse within each sub-system and that these sub-systems exchange energy.

The general principles of radiative transfer theory in vibroacoustics are presented in references [1, 3, 4, 6, 11].

Energy Balance

The first step is to determine the vibrational energy field (acoustic or structural) emitted by a point source s radiating in free field in a linear, isotropic and homogeneous medium.

Figure 1. Free field sound level. The energy received at point r depends only on the source-receiver distance R.

If the receiver point r is located at a distance R from s (see figure 1), then by symmetry the energy G depends only on the distance R. The exact expression of G depends only on the dimension of space and not on the nature of the propagating wave,

\[ G(R) = \left\{ \begin{array}{ll} \frac{e^{-mR}}{2 c_\text{g}} & \text{dim}=1 \\ \frac{e^{-mR}}{2\pi c_\text{g} R} & \text{dim}=2 \\ \frac{e^{-mR}}{4\pi c_\text{g} R^2} & \text{dim}=3 \end{array} \right. \]

For the same reason of symmetry, the intensity vector H also depends only on the distance R. It is carried by the unit radial vector u directed from s to r.

\[ \mathbf{H}(R) = \left\{ \begin{array}{ll} \frac{e^{-mR}}{2} \mathbf{u} & \text{dim}=1 \\ \frac{e^{-mR}}{2\pi R} \mathbf{u} & \text{dim}=2 \\ \frac{e^{-mR}}{4\pi R^2} \mathbf{u} & \text{dim}=3 \end{array} \right. \]

where c_g is the energy propagation velocity or group velocity of the wave and m is the attenuation factor of the wave. In the case of a structural wave, m is related to the hysteresis damping factor tau by,

\[ M_i = \omega_i \eta_i n_i >> 1 \]

While for a fluid like air, its values are provided in tables. They depend on temperature, frequency and the humidity tau of air.

Sound level in the presence of obstacles. The energy received at point r is the sum of contributions from sources rho, reflection sources sigma and diffracting sources lambda.

\[ W(R,t) = \int_\Omega \rho G \,d\Omega + \int_\Gamma \sigma G \,d\Gamma + \int_\Delta \lambda G \,d\Delta \]

\[ \mathbf{I}(R,t) = \int_\Omega \rho \mathbf{H} \,d\Omega + \int_\Gamma \sigma \mathbf{H} \,d\Gamma + \int_\Delta \lambda \mathbf{H} \,d\Delta \]

These energy fields W and I satisfy a local energy balance equation,

\[ \mathbf{div}\cdot \mathbf{I} + \eta \omega W + \frac{\partial W}{\partial t} = 0 \]

1. A. Le Bot. Vibroacoustic model for high frequency analysis, Journal of Sound and Vibration, vol. 211(4), p. 537-654, 1998.

2. A. Le Bot and A. Bocquillet. Comparison of an integral equation on energy and the ray-tracing technique for room acoustics, Journal of Acoustical Society of America, vol. 108(4), p. 1732-1740, 2000.

3. V. Cotoni and A. Le Bot. Radiation of plane structures at high frequency using an energy method, International Journal of Acoustics and Vibration, vol. 6(4), p. 209-214, 2001.

4. A. Le Bot. Energy transfer for high frequencies in built-up structures, Journal of Sound and Vibration, vol. 250(2), p. 247-275, 2002.

5. V. Cotoni, A. Le Bot and L. Jezequel. High frequency radiation of L-shaped plate by a local energy flow approach, Journal of Sound and Vibration, vol. 250(3), p. 431-444, 2002.

6. A. Le Bot. A functional equation for the specular reflection of rays, Journal of Acoustical Society of America, vol. 112(4), p. 1276-1287, 2002.

7. V. Cotoni, A. Le Bot and L. Jezequel. Sound transmission through a plate by an energy flow approach, Acustica with Acta Acustica, vol. 88(6), p. 827-836, 2002.

8. E. Reboul, A. Le Bot and J. Perret-Liaudet. Introduction of acoustical diffraction in the radiative transfer method, Comptes Rendus Mécanique, vol. 332(7), p. 505-511, 2004.

9. A. Le Bot. Comparison of vibrational conductivity and radiative energy transfer methods, Journal of Sound and Vibration, vol. 283, p. 135-151, 2005.

10. E. Reboul, A. Le Bot and J. Perret-Liaudet. Radiative transfer equation for multiple diffraction, Journal of Acoustical Society of America, vol. 118(3), p. 1326-1334, 2005.

11. A. Le Bot. Energy exchange in uncorrelated ray fields of vibro-acoustics, Journal of Acoustical Society of America, vol. 120(3), p. 1194-1208, 2006.

12. A. Le Bot. Derivation of Statistical Energy Analysis from radiative exchanges, Journal of Sound and Vibration, vol. 300, p. 763-779, 2007.