Infinite Line Source: analysis

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Well it's been a long time since i offered something like a tutorial on this message board, so it's high time to maybe give a little something back to this wonderful community, once again :) and so, with your kind permission and indulgence, let's begin!

I'll highlight the posts where i'm advancing the tutorial (so to speak), to allow for plenty of discussion in the time and space "in between" ...

POST #1: OUTLINE

A. Point Source: frequency-domain pressure response
B. Point Source: time-domain impulse response
C. Infinite Line Source: frequency-domain pressure response
D. Infinite Line Source: time-domain impulse response

I thought we'd start with the idealized point source, to establish both a nomenclature and methodology. I'm an engineer (or ... was one, once upon a time), rather than a physicist, and that training tends to drive my perspective and analysis. The analysis will get a bit "heavy" in Fourier math ... it's rather unavoidable ... but i'll do my best to minimize detours and stay focused on the goal, which is to hopefully highlight some pros & cons of the idealized infinite line source, compared to the idealized point source.

Regarding practical implications, for right now i will only offer that which many already understand: the "image" theory of reflections quickly reveals that a floor-to-ceiling line array (with drivers of appropriate size & spacing) does a reasonable job of approximating the idealized infinite line source :)

Much of what i'll be presenting is already known. There may be some new info, particularly regarding the time-domain impulse response of an infinite line source (which i haven't seen developed), but at the very least i may offer a somewhat concise "repository" of idealized infinite line source info. I hope it's valuable on it's own merits, but i also hope this thread may be a good starting point, reference, or framework for future discussions regarding the analysis of line sources of finite length ... a much more difficult and fascinating topic!

Without further ado ...
 
POST #2

A. Point Source: Frequency Domain Pressure Response

A simple radiating point source, an "elemental monopole", radiating into "4pi" space, with a constant volume acceleration = Ao. It's been analyzed to death :( but cherished forever for it's beauty, simplicity, building-block utility and sonic virtue :)

Our point source radiates uniformly in ALL directions in space (4pi radiation) ... so, as a function of "space", symmetry dictates that the pressure can only be dependent on distance "r" from the source, with no regard for "direction". As a function of both time and space, the pressure is often written as :

p(r,t) = [Ao*(rho)/(4pi*r)]*exp[j(w*t-k*r)]
(eqn 1)

where:

rho = density of the medium (air)
exp[x] = classic complex exponential
w = radian frequency
t = time
r = distance from source
k = wavenumber = w/c, where c= speed of sound

Personally, i hate this form of the expression. HATE IT.


This is MUCH better :

p(r,t) = {Ao*exp[jwt]} * {[rho/(4pi*r)]*exp[-jwTd]}
(eqn 2)

where:

Td = time delay = r/c


Let's take a moment to understand (eqn 2), and explain why i LOVE it:

The first term in brackets, {Ao*exp[jwt]}, is the excitation. In fact, it's nothing more than a complex sinusoidal excitation :) feel free to think of it as a complex form of sin[wt] (it's really equal to cos[wt] + j*sin[wt]). It's our little elemental monopole vibrating away at a certain frequency, "w", with an Amplitude of Ao :)

(Aside : we use complex exponentials a lot in signal processing, rather than more simple, readily recognizable, "real" sinusoids, for one reason : they are so-called eigenfunctions of linear, time invariant systems :) ... and wonderfully simple things happen when linear, time invariant systems are excited by eingenfunctions. i can elaborate, if anyone wishes ...)

The second term in brackets, {[rho/(4pi*r)]*exp[-jwTd]}, has no time-dependency, and represents what we'll call the transfer function of our little "system". It simply represents an attenuation factor of "1/r", along with the Fourier representation of "time delay", Td. More on this, in short order ...


Being an engineer, I really like separating the time-dependent excitation from the time-independent transfer function ... even though (eqn 1) and (eqn 2) are identical. The reason is that the concise identification of the transfer function allows us to inverse-Fourier-Transform it, to reveal the valuable time-domain impulse response :)

But alas, this post is already long enough :( i'll stop for now ....
 
Elaborate away! :)
(this post will be a short detour, so it won't get a post number)

The processing of signals by linear, time invariant systems ALWAYS begins with this thing called "convolution". Let me state right now ... the concept of convolution is much, MUCH easier to understand with discrete-time signals :) In fact, it's not too crazy to suggest that we should all BEGIN the study of signal processing with DSP, so that we can all understand this crazy thing called "convolution" much better.

Anyway ... "convolution" is, at its heart, simply the time-domain formulation of how a linear, time-invariant system (analog or digital) responds to a signal :) But a few definitions first:

Linear System: a linear system is simply a system whose output will 'double' in size (or amplitude), if you 'double' the input size. Of course, you may pick other scaling factors rather than 'two' (one of our favorites is zero ... if you put NOTHING into a system, and still get an output, then that system is NOT linear). You may also ADD a few inputs, and a linear system will respond with the SUM of the individual outputs. Over a wide range of signals, circuits composed of passive elements like resistors, capacitors and inductors are "linear", whereas most of our friendly active elements like transistors tend to be "non-linear" (but, even these elements tend to approach acceptably linear behavior with small signals, which allows clever circuit designers to use feedback to create surprisingly linear-like circuits with non-linear elements).

Time Invariant System: a system is said to be time-invariant if delaying the input simply results in a delayed ... but otherwise unchanged ... output. Most systems behave this way, but a great example of a NON time-invariant system is the 'sampling' process ... yes, that process that turns analog signals into digital ones ;)


So what is convolution?

First, convolution simply recognizes that we can "consider" ANY arbitrary input signal to be a summation of SCALED and DELAYED signals we call "impulses" (very short duration blips or bursts). This is where DSP really shines :) impulses in the analog world are a bit more abstract, and we'll return to them "anon".

Second, convolution recognizes that a linear, time-invariant system must respond to these scaled and delayed impulses with ... you guessed it ... scaled and delayed "impulse responses" :) :) an LTI system simply has no choice :) and we immediately begin to appreciate why the so-called time-domain "impulse response" is so important in system analysis and design.

Here's the (analog, rather than digital) mathematical formulation for what i just wrote :

y(t) = INT[h(b)*x(t - b)]db

where

x(t) = time domain input signal
h(t) = time domain 'impulse response' of our LTI (linear, time-invariant) system
y(t) = time domain output signal
INT[f(b)]db = integral of function f(b), where "b" may be a dummy integration variable

The whole study of linear, time-invariant systems could end, right there. After all, we have a mathematical way to figure the OUTPUT of any LTI system, given the INPUT. All we need, is the time-domain "impulse response" of the system ... h(t).

But alas, engineers are a LAZY lot. We don't like integration, if we can avoid it. And that "convolution integral" (above) is kinda messy ... doesn't offer much "insight" into system behavior. But that clever little integral does lend itself to an interesting observation ... up next (yes this short detour will take one more post)
 
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Let's see what happens when we apply a time-domain "exponential" signal to our LTI system :

x(t) = exp[s*t]

where "s" can be any factor, real or complex (in the most general case).

We find the output signal, y(t), from our convolution integral :

y(t) = INT[h(b)*x(t - b)]db
y(t) = INT[h(b)*exp[s*(t-b)]]db
y(t) = INT[h(b)*exp[s*t]*exp[-s*b]]db

notice : that exp[s*t] factor inside the integral is independent of the integration variable "b", so it can be pulled out of the integral :

y(t) = {exp[s*t]} * {INT[h(b)*exp[-s*b]]db}

Note that the first term above is exactly the input signal, and the second term is independent of the variable "t" (time). Let's re-write the above:

y(t) = {exp[s*t]} * H(s)


viola !!!!!

We've just discovered something VERY interesting, and EXTRAORDINARILY powerful :

When an LTI system is excited by an exponential signal, the output is the exact same signal, multiplied by a (complex-looking) scaling factor :) The ugly "integration" of convolution has become relatively simple "multiplication".

For this reason, we give exponential signals a special name : "eigenfunctions" of LTI systems :)


OK, ok ... you should be asking ... "So what? How often do we excite LTI systems with so-called exponentials?"

Great question! Turns out we don't often excite LTI systems with single exponentials ... but remember, LTI systems behave quite well when we "sum" input signals ;)

OK, ok ... so now you should be asking ... "So what? How many input signals can be represented as a sum of (possibly complex) exponentials?"

Great question! The answer is ...

ALL OF THEM :) :) :) :)


/end detour

time to return to Point Sources and Infinte Line Sources
 
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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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:up: 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.
 
Superb from every aspect....Thanks.

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.
 
POST #3



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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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.
 
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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 :)
 
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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