Tech Design.....

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Room Response with Monopole, Dipole and Cardioid Woofers,

Part II: Examination of the in room response.

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Room Response with Monopole, Dipole and Cardioid Woofers,

Part II: Examination of the in room response.

John KreskovskyMusic and Design © May, 2008 |

In Part I the mechanics of room pressurization for a source radiating in a close room was developed for

monopole, cardioid and dipole woofers. It was shown that, when operating below a room fundamental

resonance, monopole and [fully equalized] cardioid woofers pressurize a room in the same fashion. As

such, the response of monopole and cardioid woofers system can, in theory, be augmented by room

pressurization to extend the in room response well below the nominal free field roll off. Conversely, it

was shown that a dipole is incapable of pressurizing a room and as a consequence the dipole

response can not be augmented in a similar manner. When the frequency range moves above the

room fundamental we enter another range, the modal region of room response. This region extends

from somewhat below the room fundamental to the frequency called the Schroeder frequency. The

Schroeder frequency depends on the room volume, the total surface area of the walls, ceiling and

floor, and the absorptivity of the surfaces. Below this frequency individual modes are easily identified

by peaks and dips in the response and the in room response must be determined by modal analysis.

Above this frequency the individual modes are difficult to identify. Geometrically acoustic (ray tracing) is

typically useful in this region. Here we shall concern ourselves with modal analysis.

**Modal analysis**

To begin the discussion we first consider a simple source radiating in free space. The pressure at any

point**r **for a periodic source located at a point **r**o, may be expressed as

monopole, cardioid and dipole woofers. It was shown that, when operating below a room fundamental

resonance, monopole and [fully equalized] cardioid woofers pressurize a room in the same fashion. As

such, the response of monopole and cardioid woofers system can, in theory, be augmented by room

pressurization to extend the in room response well below the nominal free field roll off. Conversely, it

was shown that a dipole is incapable of pressurizing a room and as a consequence the dipole

response can not be augmented in a similar manner. When the frequency range moves above the

room fundamental we enter another range, the modal region of room response. This region extends

from somewhat below the room fundamental to the frequency called the Schroeder frequency. The

Schroeder frequency depends on the room volume, the total surface area of the walls, ceiling and

floor, and the absorptivity of the surfaces. Below this frequency individual modes are easily identified

by peaks and dips in the response and the in room response must be determined by modal analysis.

Above this frequency the individual modes are difficult to identify. Geometrically acoustic (ray tracing) is

typically useful in this region. Here we shall concern ourselves with modal analysis.

point

Here k is the wave number, ω/c, Z is the characteristic impedance of the medium into which sound is

radiating, Sω is the source strength and gω(**r**|**r**0) is the free space Green's function.

radiating, Sω is the source strength and gω(

where

Dipole consists of two simple sources separated by a distance, **d.** The pressure at any point, r, for a

dipole my likewise be expressed as,

dipole my likewise be expressed as,

or, since exp(-i π) = -1,

Lastly the pressure at r from a cardioid can be expressed as

which is similar to a dipole except that the second source is also delayed by Td where Td = d/c, (ωTd =

ωd/c = kd). Since all the sources have the factor exp(-iωt) in common it will be convenient to drop that

factor understanding that it is always present.

The expressions presented above look complicated but in reality they are simple to understand. We

begin by noting that Green's function satisfies the equation

ωd/c = kd). Since all the sources have the factor exp(-iωt) in common it will be convenient to drop that

factor understanding that it is always present.

The expressions presented above look complicated but in reality they are simple to understand. We

begin by noting that Green's function satisfies the equation

in an unbounded medium. The result is that Green's function is the spatial form of a wave emanating

from a harmonic point source positioned at**r**0. Or, if you like, Green's function is the free space

transfer function between the observation point at**r** and the source at **r**0. When the source is place in

a bounded region of space such as a room we must Green's function to a more general form,

from a harmonic point source positioned at

transfer function between the observation point at

a bounded region of space such as a room we must Green's function to a more general form,

where **χ(r)** satisfies the homogeneous equation

This additional term allows us to satisfy the boundary conditions which must be imposed at any

surfaces which enclose the source. It follows that Gω(**r**|**r****o**) satisfies the same equation as gω(**r**|**r****o**)

surfaces which enclose the source. It follows that Gω(

but with the addition of boundary conditions which limit the extent of the region enclosing the source.

So how do we find Gω(**r**|**r****o**)? When we place the source in a room its Green's function can be

expanded in a series of normal modes and can be expressed as

So how do we find Gω(

expanded in a series of normal modes and can be expressed as

where ψn (ω,**r**) and Kn are the eigenfunctions and eigenvalues associated with the room modes, and V is

the room volume. In general, when the walls of the room have finite, non zero admittance which may be

frequency dependent, these modes, as well as Λn and Kn, may also be frequency dependent. This may

require that the room modes be computed using numerical methods such as finite elements. However,

given knowledge of the modes (and Λn and Kn ) the pressure at**r** from a source at **r****o** may be found as

the room volume. In general, when the walls of the room have finite, non zero admittance which may be

frequency dependent, these modes, as well as Λn and Kn, may also be frequency dependent. This may

require that the room modes be computed using numerical methods such as finite elements. However,

given knowledge of the modes (and Λn and Kn ) the pressure at

In the case of a rectangular room with rigid wall

and ψn (ω,**r****o**) reduce to the simple room eigenfunctions,

Thus for a rectangular room with rigid walls all that need be done is to compute the eigenfunctions and

eigenvalues for the room and the pressure at any point in the room or on the room boundary at any

given frequency can be computed by evaluating Green's function for the transfer function between the

source and observation point. When the wall have finite, nonzero admittance the Kn become

imaginary. A resonance mode exists when the real part of Kn is equal to k with the amplitude of the

mode given by the imaginary part. In the case where multiple sources are present, such as multiple

woofers or dipole and cardioid woofers, the principle of supper position applies and the pressure at

any give point in the room is just the sum of the pressures from each source. To find the frequency

response we must perform this calculation over the range of frequencies of interest.

Looking again at the equation for pressure we can make several observations.

eigenvalues for the room and the pressure at any point in the room or on the room boundary at any

given frequency can be computed by evaluating Green's function for the transfer function between the

source and observation point. When the wall have finite, nonzero admittance the Kn become

imaginary. A resonance mode exists when the real part of Kn is equal to k with the amplitude of the

mode given by the imaginary part. In the case where multiple sources are present, such as multiple

woofers or dipole and cardioid woofers, the principle of supper position applies and the pressure at

any give point in the room is just the sum of the pressures from each source. To find the frequency

response we must perform this calculation over the range of frequencies of interest.

Looking again at the equation for pressure we can make several observations.

First, recognize that the pressure at any frequency, as determined by the value of k = ω/c, is the sum

of contributions from all modes. Second, the term Kn squared - k squared in the denominator controls,

in part, the magnitude of the contribution of each mode to the pressure at a given frequency. When Kn

is close to k the frequency is very close to a resonant mode and a large peak in the response may

occur. When Kn is much greater than k the mode contributes little to the pressure at that frequency.

The form of the denominator indicates that each mode behaves as a 2nd order low pass filter with cut

off frequency defined by the real part of Kn and Q defined by the imaginary part.

Examining the numerator we see that the contribution from each mode is dependent on the product of

the magnitude of the eigenfunction for each mode at**r **and **r****o**. This tells us that the placement of the

source is no more or less important than the placement of the listening position. If the eigenfunction

value is zero at the listening position then it makes no difference whether the mode is excited by the

source or not. It will not contribute to the sound pressure at the listening position.

**Equalization of dipoles and cardioids:**

Dipoles and cardioids are 1st order gradient systems which have a free space on axis response that

rolls off at 6dB per octave below the on axis peak in the response. This response follow that of a 1st

order high pass filter with pole at the peak frequency and which asymptotes to a nearly constant 6db

slope about 1/2 octave below the pole. As such, the sources for the dipole and cardioid must be

equalized if the free space on axis response of the dipole or cardioid system is to the same as that of

the free space response of the individual monopoles which make up the system. The dipole and

cardioid also achieve a peak amplitude on axis which is 6dB greater than a monopole. The frequency

at which the unequalized dipole, with dipole moment (separation),**d**, will have the same on axis

amplitude as a single monopole source is given as

of contributions from all modes. Second, the term Kn squared - k squared in the denominator controls,

in part, the magnitude of the contribution of each mode to the pressure at a given frequency. When Kn

is close to k the frequency is very close to a resonant mode and a large peak in the response may

occur. When Kn is much greater than k the mode contributes little to the pressure at that frequency.

The form of the denominator indicates that each mode behaves as a 2nd order low pass filter with cut

off frequency defined by the real part of Kn and Q defined by the imaginary part.

Examining the numerator we see that the contribution from each mode is dependent on the product of

the magnitude of the eigenfunction for each mode at

source is no more or less important than the placement of the listening position. If the eigenfunction

value is zero at the listening position then it makes no difference whether the mode is excited by the

source or not. It will not contribute to the sound pressure at the listening position.

rolls off at 6dB per octave below the on axis peak in the response. This response follow that of a 1st

order high pass filter with pole at the peak frequency and which asymptotes to a nearly constant 6db

slope about 1/2 octave below the pole. As such, the sources for the dipole and cardioid must be

equalized if the free space on axis response of the dipole or cardioid system is to the same as that of

the free space response of the individual monopoles which make up the system. The dipole and

cardioid also achieve a peak amplitude on axis which is 6dB greater than a monopole. The frequency

at which the unequalized dipole, with dipole moment (separation),

amplitude as a single monopole source is given as

For cardioid with the same source separation, **d**, this frequency will be given as

due to the additional delay added to the rear source which makes the cardioid appear as if it the

separation were twice as great. Thus if we want to make direct comparisons between the monopole,

dipole and cardioid we must equalize the source strengths for the dipole and cardioid with a transfer

function, T, which has the following characteristic

separation were twice as great. Thus if we want to make direct comparisons between the monopole,

dipole and cardioid we must equalize the source strengths for the dipole and cardioid with a transfer

function, T, which has the following characteristic

This equalization is curtailed at a stop frequency given as

for the dipole and

for the cardioid. Above these frequencies the radiation patterns for the dipole and cardioid systems loose

there constant directivity characteristics and are not particularly useful. If we are interested in dipole and

cardioid woofers with a useful upper frequency limit of 100 to 150 Hz then a separation between 0.4 and 0.6

meters would be acceptable.

there constant directivity characteristics and are not particularly useful. If we are interested in dipole and

cardioid woofers with a useful upper frequency limit of 100 to 150 Hz then a separation between 0.4 and 0.6

meters would be acceptable.

To demonstrate the utility of this approach to estimating the behavior

of different woofer systems positioned differently in a rectangular room

several computations were made. The first calculations were made for

conditions which could be duplicated using the FEM room mode

component of SoundEasy. This was done to verify the procedure and

also to obtain estimates for the frequency dependent admittance to

use in the analysis presented above. The first comparison between

SoundEasy and the present analysis is shown directly to the left. This

computation was for a rectangular room with what is referred to in

SoundEasy as medium absorption by the walls. A monopole woofer

was used and the placement of the woofer and a listing position were

the same in both simulations. Adjustment of the admittance

parameters used in the present analysis was made until the

agreement shown to the left was obtained. As can be seen, the

agreement is very good. A second computation made for an

unequalized dipole using the same admittance parameters is shown at

the lower left. Again the agreement is very good.

These comparisons between the present simplified procedure and the

FEM results of SoundEasy gave confidence that the present analysis

would yield reasonable results for other room layouts as well. The

emphasis here is that it is not so much being able to predict the exact

result for a given room as it is being able to quickly look at how

different placements of different sources behave. The present

procedure takes only seconds to compute the response at the mic

position where as the FEM procedure can take hours. Additionally,

since the present procedure is based on the evaluation of analytical

expressions, the source and mic positions can be anywhere in the

room. With A FEM procedure the source and mic must be located at

FEM nodes limiting the placement to that of the spatial resolution of

the FEM mesh. Finally, while it could be modified by the developer, the

FEM procedure is presently limited to simple sources. Thus, while

monopole and dipole calculations can be performed, it is not possibly

to presently make calculations for cardioids which require the

additional delay.

The present analysis was then applied to look at the response of

monopole, dipole and cardioid woofers. The positions of the woofers

and the "mic" position were chosen arbitrarily but in a manner that one

might set up a home stereo system consisting of stereo, full range

speakers. In addition one computation was made for the woofers

places in a corner. A diagram depicting the placements is shown

below. The room was 3M high. The mic position was fixed at 2.5M from

the side wall, 5M from the wall behind the speakers and 1M off the

floor. The source locations are shown in the frequency response plots

for each case. The source position indicates the location of the front

source for the dipole and cardioid with the axis aligned at the listener.

In the case of corner placement, the rear woofer of the dipole/cardidid

systems was place in the corner.In all cases the dipole moment,**d,**

was set at 0.6M. The frequency response plots show the variation

between sources and with placement in this limited investigation. No

attempt was made to made to find optimum source positions. From the

frequency response results it would be difficult to conclude that any

type woofer has a specific advantage over another. The dip in the

response around 40 Hz seen in almost all the figure is a result of the

mic position being 1/2 way between the side walls. In the last figure the

mic position has been moved 1/2 meter to the left with the result that

this dip begins to fill in. Corner placement, (center figure on the right

below) however, clearly shows the effect of placing a dipole near a

velocity node. The output drops significantly.

Additional discussion of development of the in room response,

including discussion of transient effects, may be found here. A room

response simulation tool based on the analysis discussed above may

be downloaded here.

of different woofer systems positioned differently in a rectangular room

several computations were made. The first calculations were made for

conditions which could be duplicated using the FEM room mode

component of SoundEasy. This was done to verify the procedure and

also to obtain estimates for the frequency dependent admittance to

use in the analysis presented above. The first comparison between

SoundEasy and the present analysis is shown directly to the left. This

computation was for a rectangular room with what is referred to in

SoundEasy as medium absorption by the walls. A monopole woofer

was used and the placement of the woofer and a listing position were

the same in both simulations. Adjustment of the admittance

parameters used in the present analysis was made until the

agreement shown to the left was obtained. As can be seen, the

agreement is very good. A second computation made for an

unequalized dipole using the same admittance parameters is shown at

the lower left. Again the agreement is very good.

These comparisons between the present simplified procedure and the

FEM results of SoundEasy gave confidence that the present analysis

would yield reasonable results for other room layouts as well. The

emphasis here is that it is not so much being able to predict the exact

result for a given room as it is being able to quickly look at how

different placements of different sources behave. The present

procedure takes only seconds to compute the response at the mic

position where as the FEM procedure can take hours. Additionally,

since the present procedure is based on the evaluation of analytical

expressions, the source and mic positions can be anywhere in the

room. With A FEM procedure the source and mic must be located at

FEM nodes limiting the placement to that of the spatial resolution of

the FEM mesh. Finally, while it could be modified by the developer, the

FEM procedure is presently limited to simple sources. Thus, while

monopole and dipole calculations can be performed, it is not possibly

to presently make calculations for cardioids which require the

additional delay.

The present analysis was then applied to look at the response of

monopole, dipole and cardioid woofers. The positions of the woofers

and the "mic" position were chosen arbitrarily but in a manner that one

might set up a home stereo system consisting of stereo, full range

speakers. In addition one computation was made for the woofers

places in a corner. A diagram depicting the placements is shown

below. The room was 3M high. The mic position was fixed at 2.5M from

the side wall, 5M from the wall behind the speakers and 1M off the

floor. The source locations are shown in the frequency response plots

for each case. The source position indicates the location of the front

source for the dipole and cardioid with the axis aligned at the listener.

In the case of corner placement, the rear woofer of the dipole/cardidid

systems was place in the corner.In all cases the dipole moment,

was set at 0.6M. The frequency response plots show the variation

between sources and with placement in this limited investigation. No

attempt was made to made to find optimum source positions. From the

frequency response results it would be difficult to conclude that any

type woofer has a specific advantage over another. The dip in the

response around 40 Hz seen in almost all the figure is a result of the

mic position being 1/2 way between the side walls. In the last figure the

mic position has been moved 1/2 meter to the left with the result that

this dip begins to fill in. Corner placement, (center figure on the right

below) however, clearly shows the effect of placing a dipole near a

velocity node. The output drops significantly.

Additional discussion of development of the in room response,

including discussion of transient effects, may be found here. A room

response simulation tool based on the analysis discussed above may

be downloaded here.