# Reduction Index and Boundary Conditions for a Wall Between Two Rectangular Rooms. Part I: Theoretical Results

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The sound transmission through a wall between two rooms depends on various boundary conditions.

Cremer's fundamental theories for transmission losses for a panel were derived for an infinite wall and for a diffuse incident field. It is therefore of interest to find how a finite model modifies the results. Here it is assumed that two rooms are separated by a wall. All flanking walls are supposed to be acoustically hard and also the energy transmission between test panel and adjacent walls are neglected. In room 1 there is a sound source with a certain power output. The sound source will in turn excite the fields in room 1, in the panel and in room 2. The rooms and the wall constitute a coupled system. If the source is assumed to be known there are three unknown functions, namely for the sound fields in the rooms and for the panel displacement. By means of the three governing wave equations the three unknown fields can be solved as functions of the terms describing the source. These source terms depend not only on the power output but also on source position. Exact expressions – in the form of infinite series – can be derived for the various wavefields. When the velocity potentials are known then the space averages of the absolute values of the pressure squared follow directly. Also frequency averages can be derived.

For two cases, a point source in a corner and for a completely diffuse primary field, it is particularly simple to derive explicit expressions describing the SPL difference between the rooms. Finally, the expressions for the point source can be generalized to apply to an ordinary loudspeaker. Thus, the final result is the difference between the levels in the rooms averaged over space and frequency.

The boundary conditions for the wall determine the mode patterns for the flexural waves on the panel and also the wave field in room 2. It is found that the transmission loss is 3 dB higher for a simply supported panel than for a clamped panel when f<f c. For f>f c the transmission loss is independent of the boundary conditions.

The transmission loss also depends on the character of the primary field. For a nondiffuse field the resulting SPL difference depends on the losses in both rooms for all frequencies. This effect is particularly noticeable when the rooms are equal or when the length of one room is an even multiple of the length of the other room. When the sound fields are diffuse then the SPL difference only depends on the losses in room 2. The reduction indices are obtained as:

1) Diffuse field; R D = ΔL−10 log A 2/S = R,

2) Non-diffuse field; R S = ΔL−10 log A 2/S−10 log[A 1+A 2/2S]−6 = R (equal and empty rooms).

ΔL is the measured SPL difference, A is the equivalent absorption and S is the radiating area of the partition. For f>f c R is given by:

R = 20 log m+30 log f−10 log f c+10 log η−47+5 log(1−f c/f).

In Part II, the results of eqs. 1 and 2 are confirmed experimentally.
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Document Type: Research Article

Publication date: January 1, 1972

• Acta Acustica united with Acustica, published together with the European Acoustics Association (EAA), is an international, peer-reviewed journal on acoustics. It publishes original articles on all subjects in the field of acoustics, such as general linear acoustics, nonlinear acoustics, macrosonics, flow acoustics, atmospheric sound, underwater sound, ultrasonics, physical acoustics, structural acoustics, noise control, active control, environmental noise, building acoustics, room acoustics, acoustic materials, acoustic signal processing, computational and numerical acoustics, hearing, audiology and psychoacoustics, speech, musical acoustics, electroacoustics, auditory quality of systems. It reports on original scientific research in acoustics and on engineering applications. The journal considers scientific papers, technical and applied papers, book reviews, short communications, doctoral thesis abstracts, etc. In irregular intervals also special issues and review articles are published.
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