Acoustics - superposition of waves Watch

JackF
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#1
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Hi all,

I have a question regarding the combination of sound wave pressures.

My lecture notes inform me that:

For coherent soundfields (e.g., tones at the same frequency, reflections at constant time delay, correlated sources): Pressure (total) = Sum of soundwave pressures at that point (generally RMS values used) i.e. Ptot = P1 + P2 + P3...

For incoherent soundfields, (e.g., tones at different frequencies, diffuse
sound, uncorrelated sources): Ptot^2 = P1^2 + P2^2 +P3^2....
(generally RMS values used)

Now the definition of a coherent soundfield given states reflections at constant time delay are counted as being coherent, however my coursework where I assumed this, I was told that I should have used the incoherent case.

The situation was a square room, with a singular sound source near the perimeter. The phase difference between the direct and reflected waves was PI/2.

Why is this not a coherent soundfield? it would seem to be a reflection at constant time delay. Also from wiki, the description of a coherent field would seem to apply: Two waves are said to be coherent if they have a constant relative phase.

I can see that the way in which the pressures add is more sensibly described by the latter method - like vectors as they have different phase. But I'm confused as the situation does not seem to fit the description of an incoherent soundfield.

If anyone can demistify this for me it would be much appreciated,

Cheers.

EDIT: To be coherent, do the waves need to be of the same frequency and in phase (no phase difference at all)? This is the only situation that I can visualise simply adding as scalars works.
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Stonebridge
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The soundfield in the room is made up of thousands of reflections. Each one has travelled a different distance to and from different points on the walls. Each returning sound wave will have travelled a different path length, and each will have a different phase difference. The wave are not coherent in this case because they do not have a "constant time delay".
Coherence requires the waves to have a constant phase difference (angle) at a particular point. They don't have to be in phase. Having the same frequency is a condition, but the phase difference is also important.
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JackF
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I should've explained that in the example it says to consider this reflection as the only reflection occurring - the first early reflection or something of the likes.

From what I understand though, the only time that the wave amplitudes can be added as scalars is when in phase as with any wave, am I right in saying that just because a wave is coherent it does not by any means imply it can be summed in this manner (unless in phase: amplitude1 + amplitude2 = amplitude(total) )?

EDIT: If the waves are coherent but not in phase, do they add like vectors/phasors?
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Stonebridge
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Point 1.
I'm still not sure what you mean by "only one reflection". That would be impossible in a room with 4 walls and a sound source near the perimeter. There would be many "early" (single) reflections before you start considering multiple reflections.

Point 2
The superposition of waves always uses the vector sum, no matter what the phase angle.
It just so happens that if the waves are either perfectly in phase or 180 deg out of phase, you can add them simply using a scalar sum. It's just like adding two velocities. If they are both in the same direction it's just v1 + v2 and if in opposite directions it's v1 - v2.
If they are at an angle to each other its vector addition and v² = v1² + v2²
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JackF
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I know that the situation is physically impossible but the question is made that way for simplicity - it later considers more reflections.

I am aware of the relationship between scalars/vectors, but was unsure of the application to this scenario as in the incoherent case the sum is of powers - it is not a vector sum despite looking like one for the case of two or three pressures I think.

I think I am right in saying that as they are pressures (analogous to voltage), you can consider each pressure as a complex number and hence find the magnitude of the resultant pressure by summing the complex numbers then taking the modulus. This is the case for waves of the same frequency (similar to AC voltages) and hence is applicable to finding the magnitude of superimposed coherent waves (for any phase).

Does that sound right?
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Stonebridge
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(Original post by JackF)
Hi all,

I have a question regarding the combination of sound wave pressures.
Yes. I think that was the point I was trying to make, that you can add sound waves in the same way as for example the voltages in an AC circuit, using a phasor diagram (or complex numbers). You are correct, that it is normally used when the waves are of the same frequency but can be of any phase angle to each other, ie coherent (and as many waves as you want).

But surely, the resultant amplitude at a point in a medium is always the sum of the amplitudes of all the component waves. It doesn't even need the waves to be of the same frequency.
I'm thinking of a wave being, for example, the sum of a fundamental and various harmonics or overtones.

I need have another think about this one. It's an interesting question.

Edit:
Returning to this after sleeping on it...
Sound pressure, in the everyday sense of "intensity" and "loudness" is the sum of the pressure of the component sound waves.
Adding these components together is done when considering, for example, the overall sound intensity due to a number of different sources.
These sources would normally be incoherent. The instantaneous sound pressure is a wave function and would ultimately be the vector sum of all the component waves, whether or not they are coherent. Obviously, for multiple sources the maths would be impossible to add these together, but then what point would that serve?
The effective sound pressure is taken as a "mean" value over a given time interval. This is a root mean square value.
When you have the much simpler case of two sound waves added together, if they are coherent then you can get interference effects.
What this means in practice is that if you do the effective pressure calculation and sum the waves, you will not be taking into account the fact that two sound waves could actually add to produce zero displacement (and therefore intensity/pressure) at a particular point. This is demonstrated in the case of standing waves.
What I'm getting at is that in your question, the final analysis of the sound in the room will involve the sum of the pressure of thousands of waves that are incoherent. The resultant wave's effective pressure will be the sum as given here
http://en.wikipedia.org/wiki/Sound_p...ltiple_sources
When you sum over a given time interval (effective pressure), rather than taking an instantaneous value, you lose the phase relationship (and the vector nature of the waves.)
I'm guessing that, to simplify the problem, you were told initially to consider just one wave and its reflection. Normally, in this case you could look at a phasor diagram and add the amplitudes as vectors. That works fine for two waves that are coherent, which would be the case here.
However, that approach could not be extended to what happens when you consider all reflections, because of the fact that the multiple reflections are incoherent. As you would then go on to sum many incoherehnt waves, you were told to consider the first, singe, refelection in the same way.
Hope this helps.
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JackF
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Hey, yeah that makes a lot of sense, the vector relationship is only considerable for a small number of coherent sources whereas in the case of incoherent waves (numerous reflections) you would take into account the effective pressure:
Ptot-rms = sqrt(P1rms^2 + P2rms^2...)

Thanks for your time, I'm happy with the concept now.
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Stonebridge
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(Original post by JackF)
Hey, yeah that makes a lot of sense, the vector relationship is only considerable for a small number of coherent sources whereas in the case of incoherent waves (numerous reflections) you would take into account the effective pressure:
Ptot-rms = sqrt(P1rms^2 + P2rms^2...)

Thanks for your time, I'm happy with the concept now.
It made me think a lot too. Good luck.
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JackF
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By the way, just in case you are interested, the case considered was based on the impulse response for Early Reflections (a few strong reflections after the direct sound) and the case for many reflections that was not considered is Late Reflections (reverberation - many weak reflections after the Early reflection period). So I guess the reason for considering the case may be due to the larger effect of these early reflections on the instantaneous amplitude of the sound pressure.
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