Tech Design.....
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Room Response with Monopole, Dipole and Cardioid Woofers,
Part II: Examination of the in room response.

John Kreskovsky
Music 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 ro, may be expressed as
Here k is the wave number, ω/c, Z is the characteristic impedance of the medium into which sound is
radiating, S
ω is the source strength and gω(r|r0) is the free space Green's function.
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,
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
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
r0. Or, if you like, Green's function is the free space
transfer function between the observation point at
r and the source at r0. 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,
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 G
ω(r|ro) satisfies the same equation as gω(r|ro)
but with the addition of boundary conditions which limit the extent of the region enclosing the source.

So how do we find G
ω(r|ro)? 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
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 Λ
n 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 Λ
n and Kn ) the pressure at r from a source at ro may be found as
In the case of a rectangular room with rigid wall
and ψn (ω,ro) 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 K
n become
imaginary. A resonance mode exists when the real part of  K
n  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 K
n 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 K
n
is close to k the frequency is very close to a resonant  mode and a large peak in the response may
occur. When K
n 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 K
n 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 ro. 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
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
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.
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.
References:

Theoretical Acoustics, P. M. Morse and K. U. Ingard, Princeton University Press, 1968.

Formulas of Acoustics, F. P. Mechel, Springer - Verlag, 2002, 2004.