Figure 1. Prediction models compared to a measurement in a listening room.

Prediction model critique
Both the modal decomposition and image source models offer a better representation of the sound field in the space than the simple modal frequency equation. This is primarily because the modal decomposition and image source models allow for absorption, but also because it is possible to calculate a quantity - the modal response - that is easier to relate to the listener experience. Both models, however, are not completely accurate. Figure 1 compares measurements in a listening room to the modal decomposition and image source models. The listening room has dimensions 6.9 x 4.6 x 2.8m. All the walls were smooth plastered concrete except the back wall, which was covered with diffusers, some diffusers were on the ceiling and the floor which was covered with carpet.

Below 100 Hz, good agreement between the models and the measurement are shown. The agreement diverges above 100Hz - see below. Slightly better agreement can be achieved by taking more terms in modeling equation. The models deliberately used a reduced number of terms in the infinite sums to enable calculations to be quick enough for subsequent optimisation. (Read more)

Great care to normalize for loudspeaker resonance is required for these measurements, in this case the loudspeaker resonance was about 80Hz. The sound power of the loudspeaker is difficult to measure, as anechoic conditions are not achieved at 20Hz in normal test chambers. The cone acceleration was used as a reference for the frequency response normalization. This was measured by an accelerometer attached near the centre of the loudspeaker cone. If the cone radiates as a piston at such low frequencies, the free-field pressure should be omni-directional and proportional to the cone acceleration. Thus, the cone acceleration provides a convenient means of normalization. This worked well over most of the frequencies from 30 to 100Hz. At the lower and upper ends the normalization might still be affected by poor signal to noise ratio and directional radiation form the cone. Indeed, it is assumed that the divergence between the measurement and predictions above 100Hz is due to cone directivity.

Consequently, the method for choosing room dimensions is based on a better prediction model than previous methods. There is, however, scope for future improvement, by including better prediction models when they are developed. There are some basic problems with both the modal decomposition and image source models, and currently there are no established solutions to deal with these difficulties. For example, while absorption coefficients for surfaces are widely available, the surface impedances, which includes both phase and magnitude information, are not. Indeed, given that room surfaces at low frequencies will often not behave as isolated local reacting surfaces, defining a surface impedance can be problematical. Consequently, for this work an assumption of no phase change on reflection has to be made, which means that the models are more accurate for walls that are more rigid. It might be envisaged that a finite element model could overcome some of these difficulties, but currently the calculation time would be too slow for optimization. During an optimization process, many hundreds or thousands of room configurations have to be calculated, consequently the prediction time for a single calculation must be kept small.

For the results presented here the image source model was favored over the modal decomposition model. This is because the image source model is considerably faster. For the modal decomposition model, all modes within the frequency range of interest must be considered, plus corrections for those outside the range must be done. In the image source model, all images contribute to the impulse response in a cuboid room. Consequently, using the image source model reduces the optimization time. (Furthermore, the methodology outlined has been previously applied to finding the best location for loudspeakers and listeners in rooms and using a time based approach for that problem enables the early arriving sound to be examined as well as the modal response). The relationship between the modal decomposition and the image solutions for a loss-less room has been derived and shown to be equivalent for a rigid boundary (Read more).