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Infinite Line Source: analysis
Infinite Line Source: analysis
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Old 18th November 2017, 03:07 AM   #11
werewolf is offline werewolf  United States
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POST #3

A. Point Source: frequency-domain pressure response (continued)

Our little point source has what i've called a frequency-domain transfer function :

H(w) = [(rho)/(4pi*r)]*exp[-jwTd]

or, alternately,

H(k) = [(rho)/(4pi*r)]*exp[-jkr]

where:

r = distance from point source
c = speed of sound
w = radian frequency = 2pi*f
Td = time delay = r/c
k = wavenumber = w/c

In a nutshell, H(w) tells us what the pressure will be at any distance "r" from the point source, given a point source excitation at frequency "w".

Let's dig a little deeper ...

In engineering parlance, so-called "transfer functions" are written as :

H(w) = mag(w)*exp[j*phase(w)]

Transfer functions are complex functions of frequency (w), which simply means that they are functions of frequency with both magnitude and phase (or, real part and imaginary part) components. If the input signal is a sinewave, the output signal will also be a sinewave ... at the same frequency (w) ... but with a different magnitude and a different phase, as "modified" by the system's transfer function.

We can compare this general formulation of a transfer function to our point source transfer function, to reveal, for a point source :

mag(w) = rho/(4pi*r)
phase(w) = -w*Td

From which we conclude :

The magnitude response of the point-source transfer function (radiating into 4pi space) is FLAT, independent of frequency. BUT, it does drop at a rate of 1/r = -6dB for every doubling of distance.

The phase response of the point-source transfer function (radiating into 4pi space) is linear, corresponding to a pure time delay = Td.

At this point, we can also introduce another property of transfer functions called "group delay", which is the negative of the first derivative of phase wrt frequency. For our little point source, the group delay associated with our transfer function is simply Td.


Editorial : I think it may be a bit uncommon to describe radiation patterns of point sources and line sources (or any sources) as radiating elements with spatial (and, of course, frequency-dependent) "transfer functions". But that's because i'm an electrical engineer, rather than a physicist and it's supportive of our ultimate goal, which is to develop and compare both frequency domain and time domain responses for these sources.


Summary: The magnitude response of the point source transfer function is wonderfully "flat" with frequency ... meaning that ALL frequencies will be measured (pressure) with equal amplitude (at the same point in space). The magnitude of ALL frequencies will, however, drop at a rate of 6dB for every doubling of distance from the source. The phase response of point source transfer functions is linear, representing a simple time delay = Td.

next up : the time-domain impulse response of the point source transfer function.
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Old 18th November 2017, 03:37 AM   #12
BYRTT is offline BYRTT  Denmark
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Infinite Line Source: analysis
having a seat this thread and thanks language werewolf explaining into details in logic way, all the math is mystery to me but can relate stuff to own diy speaker work using Rephase and REW to help make complicated overviews and corrections to systems.
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Old 18th November 2017, 12:29 PM   #13
Overkill Audio is offline Overkill Audio  United Kingdom
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Default Superb from every aspect....Thanks.

Quote:
Originally Posted by werewolf View Post
POST #3

Editorial : I think it may be a bit uncommon to describe radiation patterns of point sources and line sources (or any sources) as radiating elements with spatial (and, of course, frequency-dependent) "transfer functions". But that's because i'm an electrical engineer, rather than a physicist and it's supportive of our ultimate goal, which is to develop and compare both frequency domain and time domain responses for these sources.
Thanks Werewolf, I find my self reading your explanations with a smile that says " Now I get it" .... Finally someone who has the power of explanation as well as the power of knowledge..."

Loving all of this big time.
Cheers
Derek.
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Old 18th November 2017, 03:22 PM   #14
gkh is offline gkh  Austria
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Old 18th November 2017, 04:47 PM   #15
Dr1v3n is offline Dr1v3n  United States
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Old 18th November 2017, 05:44 PM   #16
GM is offline GM  United States
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Originally Posted by wesayso View Post
I'm getting out the popcorn and drinks for this one...
Ditto!

GM
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Old 18th November 2017, 06:46 PM   #17
gedlee is offline gedlee  United States
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Quote:
Originally Posted by werewolf View Post
POST #3

[B]

Summary: The magnitude response of the point source transfer function is wonderfully "flat" with frequency ... meaning that ALL frequencies will be measured (pressure) with equal amplitude (at the same point in space). The magnitude of ALL frequencies will, however, drop at a rate of 6dB for every doubling of distance from the source. The phase response of point source transfer functions is linear, representing a simple time delay = Td.
What you describe could be a little misleading. I will use my physicist jargon rather than the EE terminology that you are using.

In physics what you are describing is what we call the Green's function it is similar to a EE impulse response except that it is multi-dimension in space-time i.e. 4 dimensions (non-relativistic of course). The Green's function relates the pressure at any point in space due to a source at another point in space. The part that you are missing above is the source strength which is normally given in terms of the volume velocity. For a Green's function that relates a volume velocity to a pressure the response to a frequency independent volume velocity would go as omega, i.e. it would rise as the frequency goes up at +6 dB/oct.

A speaker is basically flat because its volume velocity is falling (above resonance) at exactly the same rate of -6 dB/oct, hence a net flat response.

The subject of line arrays is covered in my book in Chapter 9 of my book, which is free at my web site. One reviewer said about the book that if you thought that you understood line arrays then you needed to read this chapter.
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Old 18th November 2017, 08:05 PM   #18
werewolf is offline werewolf  United States
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Thank you, good sir!

In my analysis ... yes, engineering vs physics ... I associate the strength of the source with the source itself, rather than with the so-called "transfer function". In Post #2, I've called the strength of the source "Ao", and identified it as a constant volume acceleration, rather than a constant volume velocity. In this scenario, the measured pressure will drop at 6dB for every doubling of distance from the point source, but the measured pressure at any point in space will be flat (constant) with frequency.

Last edited by werewolf; 18th November 2017 at 08:08 PM.
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Old 18th November 2017, 08:17 PM   #19
werewolf is offline werewolf  United States
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Quote:
Originally Posted by Overkill Audio View Post
Thanks Werewolf, I find my self reading your explanations with a smile that says " Now I get it" .... Finally someone who has the power of explanation as well as the power of knowledge..."

Loving all of this big time.
Cheers
Derek.
Thanks for the encouragement! It's a bit challenging sometimes, to offer mathematically-oriented tutorials on message boards ... but it's well worth it, if we all (including yours truly!) learn something
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Old 18th November 2017, 08:20 PM   #20
gedlee is offline gedlee  United States
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Quote:
Originally Posted by werewolf View Post
Thank you, good sir!

In my analysis ... yes, engineering vs physics ... I associate the strength of the source with the source itself, rather than with the so-called "transfer function". In Post #2, I've called the strength of the source "Ao", and identified it as a constant volume acceleration, rather than a constant volume velocity. In this scenario, the measured pressure will drop at 6dB for every doubling of distance from the point source, but the measured pressure at any point in space will be flat (constant) with frequency.
Sorry, I missed where you had defined the source strength that way. Normally in acoustics we use volume velocity.
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