# Acoustics/Fundamentals of Room Acoustics

![](Acoustics_fundamentals_of_room_acoustics.JPG "Acoustics_fundamentals_of_room_acoustics.JPG")

## 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
$\nabla ^2 \hat \Phi  + k^2 \hat \Phi  = 0$. Considering a simple
geometry of a parallelepiped (L1,L2,L3), the solution of this problem is
with separated variables :

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$P(x,y,z)=X(x)Y(y)Z(z)$

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

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

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

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

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

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where $m$,$n$,$p$ 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 $Q$ in $r_0$, the final
pressure in $r$ is the sum of the contribution of all the modes
described above.

The modal density $\frac{{dN}}{{df}}$ is the number of modal frequencies
contained in a range of 1 Hz. It depends on the frequency $f$, the
volume of the room $V$ and the speed of sound $c_0$ :

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$\frac{{dN}}{{df}} \simeq \frac{{4\pi V}}{{c_0^3 }}f^2$

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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 :

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$\frac{{dE}}{{dt}} = W_s  - W_{abs}$

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Where $W_s$ and $W_{abs}$ 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 :

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$W_{abs}  = \frac{{ec_0 }}{4}a$

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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 :

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$a = \sum\limits_i {\alpha _i S_i }$

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The final equation is : $V\frac{{de}}{{dt}} = W_s  - \frac{{ec_0 }}{4}a$

The level of stationary energy is :
$e_{sat}  = 4\frac{{W_{abs} }}{{ac_0 }}$

### 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 :

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

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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

