Optimal Position of a Loudspeaker in a Rectangular Room

Final student paper of Fabrice Ducomble, electronics engineer

Introduction

The position of a loudspeaker in a room, rectangular or not, influences in a sensitive way the quality of a sound to such a point that you can get better results with a bad loudspeaker judiciously placed than with a good speaker placed without planning. In fact, the imperfection appearing in the loudspeaker response curve can be attenuated by the resonance created in the room or, on the contrary, be amplified by them. It should be noted that the position of the listener is almost as important as the source of the sound.

The problem then is to find the optimum
positions for both the source and the listener.

We are particularly interested in the behavior of the low frequencies in the room with the presence of the loudspeaker. It is, in fact, in this range of frequencies that the interaction between these two components is the most noticed. So, a study in depth modal behavior has been carried out, from their formation to their influence on the distribution of the sound field.

From this study we devised a model that permitted us to predict the behavior of low frequencies in a rectangular room. This model takes into account another very important phenomenon : The influence of the distance from the walls on the power of the emitted sound.

The program carried out is based on this model. It calculates the response curves relative to all possible positions regarding the source-listener, and chooses, with the help of two tests, the best among them. The optimum position of the loudspeaker, and of the listener, is then the positions corresponding to the best response curve.

This research can be divided into three parts :
1. The study of the influence on the proximity of the walls on the sound power emitted.
2. The study of modal behavior and the establishment of a model permitting the determination of the influence on the sound field.
3. The presention of the results.

First Part - The influence of the distance of the boundaries on the emitted power

According to hypotheses used by K.O.Ballagh [3], a rectangular room can be modelled with the three walls closest to the sound source. These hypotheses impose a frequency limitation and mean we must be careful with the results obtained for the low frequencies using the formula derived from this model.

This formula, developped by Waterhouse, presents the power emitted by a source placed within the proximity of one, two or three walls in function of the power emitted in the free field by this same source. In naming the distances from walls as X, Y and Z, the Waterhouse formulas for one (1), two (2) and three walls (3) are :

(1)

(2)

(3)

where
(4)

These three functions are represented in figure 1 respectively by the curves A, B, and C, for a source placed symmetricaly to the walls.

The formula relative to the three boundaries presents the advantage of being easily put to use, but has a major inconvenience : it doesn't take into account the Fundamental phenomenon of resonance. Concerning this subject, Morse shows in his work [2] that resonance has a strong influence on the power emitted by a source. This phenomenom is shown in figure 2. We can observe peaks with each resonance frequency.

Fig. 1 - Representation of the
three Waterhouse formulas

Fig. 2 - Radiation impedance as a function of frequency

Conclusion

The Waterhouse formula gives good results as long as the hypotheses are respected. We must therefore, expect some inaccuracies from results obtained using this formula for low frequencies. During the presentation we will see from the results that these inaccuracies are the main source of limitations to the program's being carried out.

Second Part - Modal Behavior and Establishment of a Model permitting the Determination of the Influence on the Sound Field

Eigenmodes (or simply "modes") are resonance phenomenoms : discrete phenomenoms consisting in the amplification of an initial periodical phenomenom. Every resonance phenomenom is caracterized by an establishment duration. So the amplitude of a particular mode depends on the time that the source radiates. The establishment duration depends on the absorbtion factor of the room's walls. The more reverberant a room is, the longer this duration is (til 10 sec. for a very reverberant room). Figure 3 presents the establishment of resonance in the theoretical case of two parallel walls.

Fig.3 - Establishment of resonance in the theoretical case of two parallel walls

Eigenvalues and eigenfunction

A mode is caracterized by an eigenvalue and an eigenfunction. With the eigenvalue, we can obtain the phenomenom's eigenfrequency. The eigenfunction gives the spatial distribution of the mode's amplitude. These two expressions can be found by solving the following eigenvalues problem :

(5)

where has to satisfy the boundary conditions imposed by the walls.

In the general case (absorbing walls), the boundary condition imposes that the vibration speed of the walls' particles equals the vibration speed of air :

(6)

So, by solving the problem, we obtain :

(7)

(8)

(9)

with

(10)

(11)

The resonance pulse is given by the following equation :

(12)

where corresponds to the resonance pulse and represents the damping factor.


Then the resonance frequency is expressed by equ. (13).

(13)

In the case of perfectly reflecting walls, this formula corresponds to the Rayleigh's well-known formula :

(14)

For an absorbing room, there can be differences up to 3 Hz between Rayleigh's formula and (13). Thus the formula (13) is really useful for determining exactly the resonance frequencies of a real-world room.

Determination of a mode's maximum value

The eigenfunction determines the spatial distribution of the pressure relative to the maximum value of the mode. Four methods have been tried. All four give similar results. The first three formulas assume that the energy is uniformely distributed throughout the room. This is generally the case when the energy is distributed through a great number of modes. The fourth method is less restrictive, because it takes the energy distribution inside a mode to directly determine the maximum pressure, without using established formulas in the case of a uniform energy distribution. Here are the formulas for the two most interesting methods (2 & 4) :

(15)

(16)

where is the term in square brackets of the next formula (17).

(17)

None of these formulas perfectly reflect reality. The values measured values are sometimes higher, sometimes lower than the ones obtained mathematically. While we obtain similar results for all 4 methods, the imprecision must come from the estimation of the power emitted by the loudspeaker. In fact, Waterhouse's formula conditions are difficult to fulfill in practice. It would be better to calculate the exact radiation impedance of our loudspeaker in a rectangular room to obtain more accurate results, but this is outside of the scope of this paper.

Frequency distribution of modes

Though modes are discrete phenomenoms in the frequency space, they look like steep peaks instead of vertical lines. A mode can be modeled by the following formula :

(18)

Figure 4 compares the calculated and the measured characterisitics of a mode.

Fig.4 - Calculated and measured characterisitics of a mode

The final model

By combining formulas (7), (15) (or (16)) and (18), we obtain the following model :

(19)

where represents the pressure's maximum value in the mode and "fact_pond" is the ponderal factor (formula (18)).

So to obtain the pressure in one point at one frequency, we must add the contribution of all modes. In practice, the amount of calculation involved would be much too high. The method to solve this is simple : we first calculate the mode's contribution from the nearest mode to the furthest one, and we stop the calculation when the contribution becomes negligeable.
Remarks :
- To obtain the total pressure, the pressure due to the source direct field has to be added.
- this model can be used practically only at low frequencies.

Conclusion

This model allows to determine the spatial distribution of the pressure in a rectangular room, at any point and any frequency. Part 3 will show how the optimal position is determined.

Third Part - Results

The program's goal is to determine the optimum positions for the loudspeaker and the listeners. From equation (19), the loudspeaker position plays the same role as the listener position. So the perceived sound quality depends from the loudspeaker position and also from the listener's position.

Principles of the program

- For each loudspeaker-listener couple : calculate the frequency response with formula (19) ;
- Compare those frequency responses and keep the best ones.

To calculate the frequency responses, several parameters are taken into account : walls characteristics, loudpseaker response, spatial and frequency steps. The optimum position is caracterized by the most linear frequency response. So the best way to determine this is to calculate the standard deviation of the curve. The lower the value, more linear the curve is. The program also calculates a standard of the deviations to eliminate the curves that present big peaks.
To optimize the calculation duration, room symetry is taken into account and preselected regions are determined both for the loudspeaker positions and the listener positions. A few particular cases have been chosen to achieve this : listener's position known in advance, loudspeaker position known in advance, calculation for a specific loudspeaker-listener couple, and so on.

A simulation using the "OptiSpeak" software (which is based on the techniques developped here)

The dimensions of the simulated room are :
- Length : 7 meters ;
- Width : 4 meters ;
- Heigth : 3 meters.

The room parameters are illustrated in figure 5. The results are shown in figures 6, 7 and 8.

Fig. 5 - Room parameters




Fig. 6 - Information on the simulation case




Fig. 7 - Results obtained (first position highlighted in blue)




Fig. 8 - Results obtained (third position highlighted in red)

References

[1] ROY F. ALLISON, The influence of room boundaries on loudspeaker output, Journal of the audio enginneering society (June 1974).
[2] MORSE & INGARD, Theoretical acoustics, Mc Graw-Hill
[3] K.O.BALLAGH, Optimum loudspeaker placement near reflecting planes, Journal of the audio enginneering society (December 1983).

可惜图片和公式没法同时贴上.

看下面原文吧:

http://www.far-audio.com/popt1.html

Loudspeaker System Modelisation Final student paper of Luc Lemaire, electronics engineer Introduction This paper attempts to model a complete loudspeaker system, e.g.: a two or three way system with the filters' action taken into account. The calculation is based on two different models for each type of driver. The first one is the theory of analogous circuits and the second one uses mathematical integration. The work will be presented in three parts. The first one deals with the analogous circuits, the second one with the mathematical modelisation and the third one will present the results obtained. First Part - Modelisation with the analogous circuits The fundamentals of the analogous circuit theory can be found in many books. Beranek [1] used this theory to model the different enclosures encountered in practice. First Thiele [2], then Small [3 & 4] presented their analysis with their own parameters. Today every manufacturer refers to these parameters for the drivers. The theory presented here is based on the papers from Thiele-Small [2-4]. Low frequency model

The two most used enclosures are the closed and the bass-reflex boxes. But the only one implemented in the final program for the low frequency driver within the enclosure is the bass-reflex model. The acoustical equivalent circuit is presented in figure 1a. The left part represents the voice coil. The centre part represents the volume of the enclosure and the right part represents the vent. The transfer function (1) of such an enclosure is a fourth order high pass filter (slope of 24 dB per octave). Thiele presented a few alignments (Butterworth, Chebyshev,...) in [2]. The frequency responses are given in figure 1b for the most used alignments.

(a) (b) Fig. 1 - Bass-reflex systems: a) acoustical analogous circuit b) frequency response

(1)
Mid and high frequency model
The mid-range and high-range drivers into the enclosure can be modelled by the closed box. In fact, mid-range drivers are generally inserted in a small closed volume into the enclosure, and behind the membrane's high-range drivers, there is often a low volume, so in all cases it is very similar to a closed box. The acoustical equivalent circuit is the one presented in figure 2a. The transfer function (2) is a second order high pass filter (slope of 12 dB per octave). The frequency responses are presented in figure 2b. (2) These two models are all limited to the piston range of the drivers, so it will be necessary to find another modelisation. This one is presented in the second part.	(a) (b) Fig. 2 - Closed-box systems: a) acoustical analogous circuit b) frequency response

Second Part - Mathematical Modelisation Mechanical modelisation of loudspeaker cones To model the cone of a loudspeaker mechanically, the model described by Frankort in [5] will be used. The goal of this modelisation is to calculate the axial impedance Za of the cone. Once this axial impedance is known, it is possible to integrate the radiated sound by the loudspeaker's cone. As explained in [5], a few simplifications are made. The alternative current supplying the loudspeaker voice coil is supposed to be independant of the frequency. The study is limited to the frequency range above the fundamental resonance frequency : the behavior under this frequency is well known and can be modelled with the analoguous circuits as explained in Modelisation with the analogous circuits. A real cone is far from being rigid in all the frequency range. Above a given frequency, resonance phenomenoms known as nodal lines (fig. 3a) and nodal circles (fig. 3b) appear on the cone's surface. Only the latter are taken into account in the following modelisation. In fact, asymetrical waves which produce the nodal lines don't influence the radiated sound as much as the axisymetrical waves (nodal circles) do. The cone is considered to be a set of conical rings and its outer edge is assumed to be free (a greater loss factor is used in the calculations to take this into account). In [5], it is also shown that the fact that the cone is not rigid increases its bandwidth. (a) (b) Fig. 3 - Resonance phenomenoms in loudspeaker cones : a) nodal lines b) nodal circles Two kinds of waves appear on the cone's surface : longitudinal and bending waves. These waves are dependant on each other as opposed to the case of a plate where longitudinal and bending waves may exist independantly. Due to this dependancy, standing waves appear only above a given frequency which is called the antiresonant frequency. Below the antiresonant frequency, the cone motion is nearly uniform. For further details, see [5]. So the cone geometry is presented in figure 4. Fig. 4 - The cone geometry The equation (3) presents the system for the axisymetrical waves without losses. With losses taken into account, the system's size doubles (12x12). (3) Behavior of the cone in the frequency range

The frequency range can be divided in three parts. The first part (fig. 5a) is caracterized by proportional longitudinal and transverse displacements below the antiresonant frequency. The second part is caracterized by the presence of a point called the transition point (x). This point moves from the outer edge of the cone to the inner edge. Between the cone's base and the transition point, the transverse displacement is determined by a longitudinal wave of high wavelength, and between the transition point and the outer edge, there is only a bending wave of short wavelength. The cone is submitted to a succession of tranverse resonance phenomenoms. As illustrated in figure 5c, the cone is entirely covered by bending waves in the third part. The cone behaves like a semi-infinite plate.

(a) (b) (c) Fig. 5 - Tansverse (left) and longitudinal (right) displacements at : a) low frequencies b) medium frequencies c) high frequencies

Mathematical modelisation of loudspeaker domes The integration of the radiated sound produced by loudspeaker domes is based only on the diffusor's shape. The simple model described in [6] is used.

Third Part - The results

The results are presented in three categories :
- modelisation with the analoguous circuits ;
- mechanical modelisation of loudspeaker cones ;
- a complete two ways loudspeaker system.

Modelisation with the analoguous circuits

The first curve (figure 6) presents the frequency response of a woofer placed in a bass-reflex box. A value of 3 has been used for the quality factor Ql (eq. 1).

Fig. 6 - Frequency response of the neoflex 5" woofer placed in the CR-10S enclosure

Figure 7 presents the frequency response of a tweeter dome with the closed box model. The loss factor in eq. 2 has been neglected.

Fig. 7 - Frequency response of the vifa 1" dome tweeter placed in the CR-10S enclosure

Mechanical modelisation of loudspeaker cones

The mechanical parameters of the loudspeaker cone had to be measured before the simulation. A value of 0.15 has been used for the loss factor. The frequency characteristic of the reduced axial admittance of the cone is illustrated in figure 8. From this graph, the three regions can be clearly found. The first region is situated below 2.5 kHz. The second one between 2.5 kHz and 10 kHz, and the third one, above 10 kHz.

Fig. 8 - Reduced axial admittance of the neoflex 5" woofer

Figure 9 presents the frequency response of the same woofer as in figure 6, but this time, based on the mechanical modelisation.

Fig. 9 - Frequency response of the neoflex 5" woofer based on mechanical modelisation

A complete two ways louspeaker system

Finally, complete loudspeaker systems have been simulated. One is presented below. The system is composed of the two preceding components : the neoflex 5" woofer and the vifa 1". The simulated curve is the "0 dB" labelled one. The real curve is well simulated. The deep at 2 kHz found in the measurements is also present in the simulation. The only difference is that the simulated curve starts to decrease before the measured one. This is mainly due to the fact that asymetrical waves (nodal lines) have been neglected. The slope of 24 dB per octave in the low frequencies isn't visible in the measured curve due to non-perfect conditions during the measurement session in that frequency range.

Fig. 10 - Calculated frequency curve of the CR-10S two ways system

Fig. 11 - Measured frequency curve of the CR-10S two ways system

Conclusion

The final program is a mathematical modelisation of the entire behavior of a complete loudspeaker system (woofer, medium, and tweeter placed in an enclosure with filters). The program permits visualization and therefore to predict the behavior of future developments in loudspeaker systems. It is then easier to determine which driver and also which kind of filters have to be used. One possibility of improvement would be to use a more accurate model for louspeaker domes (medium and tweeter).

References

[1] L. L. BERANEK, Acoustics (McGraw-Hill, New York, 1954).
[2] A. N. THIELE, Loudspeakers in Vented Boxes, Journal of the Audio Engineering Society.(Loudspeakers Vol.1 pg.281)
[3] R.H. SMALL, Closed-Box Loudspeaker Systems, Journal of the Audio Engineering Society (Loudspeakers Vol.1 pg.285).
[4] R.H. SMALL, Vented-Box Loudspeaker Systems, Journal of the Audio Engineering Society (Loudspeakers Vol.1 pg.316).
[5] F.J.M. FRANKORT, Vibration and sound radiation of Loudspeakers cones Thesis, Philips Res. Repts Suppl. 1975, No.2.
[6] J.M. KATES, Radiation from a Dome, Journal of the Audio Engineering Society.(Loudspeakers Vol.1 pg.413)