Reflection from a Finite-Sized Surface

**Strutt|Auditorium Acoustics|Reflection from Finite Surface** calculates the reflection strength from a finite-sized reflecting surface taking into account the reflector geometry and surface finish.

The overall strength of the reflected level (relative to the direct sound) is calculated as follows:

`Delta L_("total")=Delta L_("distance") + Delta L_("diffraction") + Delta L_("absorption") + Delta L_("curvature")`

The **distance loss** is calculated as:

`Delta L_("distance") = -20log((d_1+d_2)/d)`

where:

`d_1` is the distance from the source to the reflector.

`d_2` is the distance from the reflector to the receiver.

`d` is the distance from the source to the receiver.

The **diffraction loss** is based on the Rindel methods as implemented in *Odeon* and is calculated based on the major and minor dimensions `a` and `b` of the reflector.

The overall diffraction loss is given by:

`DeltaL_("diffraction")=10log(K_a*K_b)`

`K_a` and `K_b` are factors related to the cut-off frequencies `f_a` and `f_b` below which the reflector does not reflect efficiently.

`K_a = {(1 \ " for " \ f>f_a

\ ),(f/f_a \ " for " f<=f_a \ ):}`

`K_b = {(1 \ " for " \ f>f_b

\ ),(f/f_b \ " for " f<=f_b \ ):}`

This results in a diffraction loss of 0 dB above both cut-off frequencies.

Between `f_a` and `f_b` the diffraction loss has a slope of 3 dB/octave.

Below both cut-off frequencies the diffraction loss has a slope of 6 dB/octave.

The cutoff frequencies `f_a` and `f_b` are calculated from the panel dimensions, angles of incidence and the source-reflector-receiver geometry as follows:

`f_a = (c d"*")/(2(a cos(theta))^2`

`f_b = (c d"*")/(2(b cos(phi))^2`

where:

`a,b` are the dimensions of the reflector.

`theta` is the angle of incidence relative to reflector dimension `a`

`phi` is the angle of incidence relative to reflector dimension `b`

`d"*"=(d_1 d_2)/(d_1+d_2)` is the characteristic distance for diffraction from the reflector

`c` is the wave speed of sound (343 m/s)

The (optional) **absorption loss** is calculated as follows:

`DeltaL_"absorption"=10log(1-alpha)`

where:

`alpha` is the absorption coefficient of the reflector at the given frequency.

In cases where `alpha>=1`, the maximum value is set to `alpha=0.99`

Note that this term can also include the effect of scattering (by the surface finish of the reflector) by inputting an effective absorption coefficent:

`alpha_"eff"=alpha+s`, where `s` is the scattering coefficient of the reflector at the given frequency.

The (optional) **change in level due to curvature** is calculated as follows:

`DeltaL_"curvature" = -10log(1+(d"*")/(R_a cos(theta)))-10log(1+(d"*")/(R_b cos(phi)))`

where:

`R_a//R_b` is the curvature of the reflector surface in the `a,b` directions

Note that focussing (concave) surfaces have a negative sign of `R_a,R_b`; flat surfaces have `R_a,R_b=0`; dispersive (convex) surfaces have a positive sign of `R_a,R_b`

References:

*Odeon* technical manual, p6-76

AAc Tech Note: Attenuation of Sound Reflections from Plane and Curved Surfaces (After Rindel)

Comments or suggestions to strutt@arup.com