- The 'transfer function' applies to any measuring/listening point at a distance "r" from the line. Since the line is infinitely long, cylindrical symmetry dictates that there is no difference for any points along the 'z' (vertical) axis ... and all points at a distance "r" from the line in the 'x-y plane' will measure identically. So the only relevant info (spatially), for any point in 3-D space, is the distance from the infinite line ("r"). Finally, the response from all points that form the line itself are included in the calculation.
- The concept of 'transfer function' is really independent of what the source excitation may be. If the source is "impulsive" in time, and therefore flat in frequency, then the 'transfer function' is a direct indicator of the measured frequency response. If the source is not impulsive in time, and therefore not flat in frequency, then the measured response can be found by multiplying the frequency response of the source by the 'transfer function'.
Hope that helps?
The line source is infinitely long (along the vertical axis).
How much of the line is above you, and below you, if you are 10 feet above the 'x-y plane' where z=0?
How much of the line is above you, and below you, if you are 100 feet above the 'x-y plane' where z=0?
How can a line that's infinitely long, possibly know (or care) where you are positioned along the line?
Correct! If the line is infinitely long, and uniformly excited, there can't be any variation measured along it's length. That's nothing but a statement of symmetry (as i see it).
When we "truncate" the line, to make a finite-length line source, things get MUCH more interesting along the 'z-axis' ...
Finally, although an infinitely-long line seems like a theoretical abstraction, there are ways to build "reasonable approximations" in practice. We'll discuss some, before this thread is done
Not a sidebar, but an important point since no real source can be infinite in length.
I will have more to say about this aspect when it is discussed, because we normally don't want floor and ceiling reflections so making them happen (to simulate an infinite length) has always seemed, to me, to be the wrong thing to do. And finite length line sources behave quite different than infinite length ones.
Earl already explained it, but here's my shot ...
We "formed" a continuous line-source by connecting infinitesimally small point-sources, placed infinitesimally small distances from each other ... and then we let that line extend without bound.
What does that mean? Pick a small, but finite, size for a point source ... no matter what size you pick, these point sources are smaller Then pick a small, but finite, distance between the point sources on the line ... no matter what distance you pick, these distances are smaller Finally, pick a huge, but finite, number for how many point-sources form our line ... no matter what number you pick, we have more of them
This is the essence of the integral that originally defined the so-called 'transfer function'.
We "formed" a continuous line-source by connecting infinitesimally small point-sources, placed infinitesimally small distances from each other ... and then we let that line extend without bound.
What does that mean? Pick a small, but finite, size for a point source ... no matter what size you pick, these point sources are smaller
Then pick a small, but finite, distance between the point sources on the line ... no matter what distance you pick, these distances are smaller Finally, pick a huge, but finite, number for how many point-sources form our line ... no matter what number you pick, we have more of them This is the essence of the integral that originally defined the so-called 'transfer function'.
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We started the analysis by placing the x-y plane at "z=0" on our infinite line (quite typical). Certainly, there are other x-y planes to be imagined ... one at z=1, perhaps another at z=73, and yet another at z=-145.
What's the material difference between any of these planes, if the line source is infinitely long? Which plane is "closer" to the "end" of the infinite line? How will the response on the plane at z=73, differ from the response on the plane at z=-145?
Human brains can't visualize infinity. The idea so far is that a simplified mathematical model of line source array is being developed. Simplified means it is somewhat simplified compared to the exact physical reality, but as you can see the math is not so simple even to work on developing a simplified model of what is actually complicated physical line array. So, to keep the development manageable it is being done is stages. Mathematically it's easy to represent an infinitely long cylinder even if they don't exist physically and humans can't visualize one. It's also easy mathematically to add a line down one side that vibrates, again not physical possible and can't be exactly visualized. You probably just need to follow along as best you can for the ride and see where it leads. Over time the modeling well get closer to a real line array, and then it may be useful to ask questions about closely the final model represents a real array.
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POST #15
D. Infinite Line Source: time-domain impulse response
... and finally we arrive at the essence of the thread well, the main reason for me starting it, anyway!
How do we find the time-domain impulse response of a system, given the 'transfer function'? We've already done it, for the point-source we simply take the Inverse Fourier Transform of the frequency-domain transfer function
Remember that our 'transfer function' for the Infinite Line Source is :
H(jw) = (rho/2pi)*INT[(1/R)*exp[-j*(w/c)*R]]dz, from z = zero to +infinity
where i'm using the 'direct' frequency variable "w", instead of "k" (k=w/c), and :
R = sqrt[r^2 + z^2]
The definition of the Inverse Fourier Transform is :
h(t) = (1/2pi)*INT[H(jw)*exp[jwt]]dw, from w = -infinity to +infinity
So all we gotta do to find the time-domain impulse response, is plug the 'transfer function' into this expression for the Inverse Fourier Transform, and solve the integral
good news ... even though we'll be taking the integral of an integral, this problem actually does have a neat & tidy solution
D. Infinite Line Source: time-domain impulse response
... and finally we arrive at the essence of the thread
well, the main reason for me starting it, anyway! How do we find the time-domain impulse response of a system, given the 'transfer function'? We've already done it, for the point-source
we simply take the Inverse Fourier Transform of the frequency-domain transfer function Remember that our 'transfer function' for the Infinite Line Source is :
H(jw) = (rho/2pi)*INT[(1/R)*exp[-j*(w/c)*R]]dz, from z = zero to +infinity
where i'm using the 'direct' frequency variable "w", instead of "k" (k=w/c), and :
R = sqrt[r^2 + z^2]
The definition of the Inverse Fourier Transform is :
h(t) = (1/2pi)*INT[H(jw)*exp[jwt]]dw, from w = -infinity to +infinity
So all we gotta do to find the time-domain impulse response, is plug the 'transfer function' into this expression for the Inverse Fourier Transform, and solve the integral
good news ... even though we'll be taking the integral of an integral, this problem actually does have a neat & tidy solution
What do you mean by, "removing the vertical axis's effect?"
Also, the time of arrival can't be "jumbled" if there are an infinite number of source points, if you mean what I think you mean by using the word "jumbled." The effect of them has to add up smoothly, since there is an infinite number of them.
If you mean an infinite line source is not all that much like a real line source, maybe not. There are some very important differences, but we haven't gotten to those yet. At this point we are trying to see what an idealized line source might be like. In engineering we used idealized models a lot. Models of capacitors are ideal, for example. Then we make corrections for how they differ from ideal, such as by giving them a voltage coefficient. If you want to understand engineering, you have to get used to working with idealized models, and then adjusting as needed for reality.
Also, the time of arrival can't be "jumbled" if there are an infinite number of source points, if you mean what I think you mean by using the word "jumbled." The effect of them has to add up smoothly, since there is an infinite number of them.
If you mean an infinite line source is not all that much like a real line source, maybe not. There are some very important differences, but we haven't gotten to those yet. At this point we are trying to see what an idealized line source might be like. In engineering we used idealized models a lot. Models of capacitors are ideal, for example. Then we make corrections for how they differ from ideal, such as by giving them a voltage coefficient. If you want to understand engineering, you have to get used to working with idealized models, and then adjusting as needed for reality.
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