Acoustics/Fundamentals of Room Acoustics

Introduction

Three theories are used to understand room acoustics :

1. The modal theory
2. The geometric theory
3. The theory of Sabine

The modal theory

This theory comes from the homogeneous Helmoltz equation ${\displaystyle \nabla ^{2}{\hat {\Phi }}+k^{2}{\hat {\Phi }}=0}$. Considering a simple geometry of a parallelepiped (L1,L2,L3), the solution of this problem is with separated variables :

${\displaystyle P(x,y,z)=X(x)Y(y)Z(z)}$

Hence each function X, Y and Z has this form :

${\displaystyle X(x)=Ae^{-ikx}+Be^{ikx}}$

With the boundary condition ${\displaystyle {\frac {\partial P}{\partial x}}=0}$, for ${\displaystyle x=0}$ and ${\displaystyle x=L1}$ (idem in the other directions), the expression of pressure is :

${\displaystyle P\left({x,y,z}\right)=C\cos \left({\frac {m\pi x}{L1}}\right)\cos \left({\frac {n\pi y}{L2}}\right)\cos \left({\frac {p\pi z}{L3}}\right)}$

${\displaystyle k^{2}=\left({\frac {m\pi }{L1}}\right)^{2}+\left({\frac {n\pi }{L2}}\right)^{2}+\left({\frac {p\pi }{L3}}\right)^{2}}$

where ${\displaystyle m}$,${\displaystyle n}$,${\displaystyle p}$ are whole numbers

It is a three-dimensional stationary wave. Acoustic modes appear with their modal frequencies and their modal forms. With a non-homogeneous problem, a problem with an acoustic source ${\displaystyle Q}$ in ${\displaystyle r_{0}}$, the final pressure in ${\displaystyle r}$ is the sum of the contribution of all the modes described above.

The modal density ${\displaystyle {\frac {dN}{df}}}$ is the number of modal frequencies contained in a range of 1 Hz. It depends on the frequency ${\displaystyle f}$, the volume of the room ${\displaystyle V}$ and the speed of sound ${\displaystyle c_{0}}$ :

${\displaystyle {\frac {dN}{df}}\simeq {\frac {4\pi V}{c_{0}^{3}}}f^{2}}$

The modal density depends on the square frequency, so it increase rapidly with the frequency. At a certain level of frequency, the modes are not distinguished and the modal theory is no longer relevant.

The geometry theory

For rooms of high volume or with a complex geometry, the theory of acoustical geometry is critical and can be applied. The waves are modelised with rays carrying acoustical energy. This energy decrease with the reflection of the rays on the walls of the room. The reason of this phenomenon is the absorption of the walls.

The problem is this theory needs a very high power of calculation and that is why the theory of Sabine is often chosen because it is easier.

The theory of Sabine

Description of the theory

This theory uses the hypothesis of the diffuse field, the acoustical field is homogeneous and isotropic. In order to obtain this field, the room has to be sufficiently reverberant and the frequencies have to be high enough to avoid the effects of predominating modes.

The variation of the acoustical energy E in the room can be written as :

${\displaystyle {\frac {dE}{dt}}=W_{s}-W_{abs}}$

Where ${\displaystyle W_{s}}$ and ${\displaystyle W_{abs}}$ are respectively the power generated by the acoustical source and the power absorbed by the walls.

The power absorbed is related to the voluminal energy in the room e :

${\displaystyle W_{abs}={\frac {ec_{0}}{4}}a}$

Where a is the equivalent absorption area defined by the sum of the product of the absorption coefficient and the area of each material in the room :

${\displaystyle a=\sum \limits _{i}{\alpha _{i}S_{i}}}$

The final equation is : ${\displaystyle V{\frac {de}{dt}}=W_{s}-{\frac {ec_{0}}{4}}a}$

The level of stationary energy is : ${\displaystyle e_{sat}=4{\frac {W_{abs}}{ac_{0}}}}$

Reverberation time

With this theory described, the reverberation time can be defined. It is the time for the level of energy to decrease of 60 dB. It depends on the volume of the room V and the equivalent absorption area a :

${\displaystyle T_{60}={\frac {0.16V}{a}}}$ Sabine formula

This reverberation time is the fundamental parameter in room acoustics and depends trough the equivalent absorption area and the absorption coefficients on the frequency. It is used for several measurement :

• Measurement of an absorption coefficient of a material
• Measurement of the power of a source
• Measurement of the transmission of a wall