## Wave Questions

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jewish_scientist
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### Wave Questions

I have to make this brief, so sorry in advance.

In an ideal system, N identical white noise generators start at the same time.

A = instantaneous amplitude of 1 generator, a = average amplitude of 1 generator
F = instantaneous frequency of 1 generator, f = average frequency of 1 generator
I = instantaneous intensity of 1 generator and i = average intensity of 1 generator

AN = N*A
aN = N*a
FN = F
fN= f.

Is this correct? If no, would it be correct if the generators where musical instruments? What about the rest of the variables

Completely unrelated question below.
What is the relationship between intensity and amplitude? What is the relationship between intensity and frequency?

cyanyoshi
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### Re: Wave Questions

Could you go into more detail what kind of scenario you are imagining? I am not sure it makes sense to talk about the amplitude of white noise because it doesn't have a well-defined waveform; it's more-or-less a random number at each time instant. White noise also doesn't have a frequency in the usual sense. (You can however talk about its power spectrum, which says how much of the signal is at each frequency. White noise has the same relative power at all frequencies.)

jewish_scientist
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### Re: Wave Questions

I give you the time vs air pressure graph for 1 white noise generator and ask you to make the graph for 2 identical white noise generators. If you doubled the y value of every point on the given graph, would the resulting graph be correct?

Assuming constant frequency, what is the mathematical relationship between amplitude and intensity?

Assuming constant amplitude, what is the mathematical relationship between frequency and intensity?

gmalivuk
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### Re: Wave Questions

Are you talking about white noise or are you talking about a sound with a single pitch?
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jewish_scientist
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### Re: Wave Questions

For the first question, white noise. For the second and third, a single pitch.

doogly
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### Re: Wave Questions

1: If they are so identical that they are in perfect phase at all times, then yes. That is not a real situation though.

2,3: I ~ A^2 w^2. See wikipedia, "Sound Intensity"
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wumpus
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### Re: Wave Questions

Looks like someone needs to take ENEE201 (or whatever is the pre-req for DSP work and control theory). Start with a quick check of Fourier transforms and start again... I'll try to look at it from an "instantenous" aspect, then analyze it over time for the real answer. [clickbait]The one that is physically meaningful will surprise you![/clickbait]

It isn't quite certain what a "white noise generator is". I'd assume that an ideal white noise generator has a Fourier transform that has a flat amplitute throughout its bandwidth and a randomized phase (you need the randomized phase if you are starting multiple white noise generators at the same time. As mentioned below you want a finite bandwidth to avoid mathematically nasty outputs).

Such a system doesn't have an "instantanous frequency" that is anything other than an infintitely long fourier transform (assuming it plays a finite length of time). Quite literally *nothing* has an instantenous scaler frequency that isn't an eternally playing sinewave (which may be a problem in universes that start with a big bang). This type of thing gets even weirder when dealing with discrete sampling (i.e. everything done via digital signal processing). The stuff looks amazingly like Heisenberg's uncertainty principle and is probably due such matter being fundamentally a wave. Waves simply have such uncertainty and don't have such properties as "instantaneous frequency". (This problem gets worse in digital signal processing and is handled mostly under "windowing theory").

Instantenous amplitude. Presumably you get that, but with infinite bandwiths it can be discontinous. You probably don't want discontinous instantanous amplitude in your signal. The best example of discontaneous amplitude (in an otherwise reasonable function) is the Gibbs phenonmemon. Any repeating signal can be reproduced arbitrarily well by a sum of fourier series, however producing a square wave that way will contain an 18% overshoot (that is arbitrarily narrow as you increase the amount of frequencies summed) during each discontinous change.

Instantenous intensity: This doesn't appear to be well definied. I assume it is some sort of power measurement, but I can't imagine the instantenous definition (largely because it would require instaneous frequencies). You might want to look up "power spectral density" as that is something pretty important in this type of work. Total intensity (over time) will be defined as the intergral of the amplitude of the fourirer transform (I suspect that there are supposed to be various correction constants. I'll ignore them).

Combinations:
There are two issues for combining amplitude. First is the math problem, and I'm guessing that N*A is correct (I keep wanting to sum squares, but your "instantanous" makes that unlikely to be true). The second is the physics problem (and finally back to the topic of the board): you have wave generators. Of *course* they will have wave interference problems (we'll ignore the inverse square distance law). This is the physics forum after all. So assuming that each generator is seperated such that the speed of sound is one unit between generators we get:

total amplitude=A1(m)+A2(m+1)+A3(m+2)+...+AN(m+N-1)
total frequency: wait a bit.
total instensity ?

For a single (eternal) pitch:
total frequency: easily computed as a fourier transform with each frequency noted and each phase adjusted by distance from speaker. For each frequency, sum all frequencies as vectors (amplitude*phase) in your summed Fourier tranform. That's your "frequency".
Total amplitude: invert the fourier transform, that is a function of amplitude over time.
Total intensity: presumably a sum of the amplitudes of the fourier transforms. Not an instantenous value (if you assume it has a "frequency" than the minimum legnth of time that intensity can be calculated would be the "wavelength").

And now for the payoff. The total frequency of the white noise generators is the vector sum each of the fourier functions of the generator. Note that we are assuming each to have a uniform amplitute across each frequency, so all we need to do is add the phases up. But the phases are equally likely to subtract from each other as to add to each other, so the end result will be a white noise generator with the highest amplitute of all white noise generators (or more specifically Integral(phi,phi+epsilon) F(theta)=integral(phi,phi+epsilon) max(F(theta) for all white noise generators)). The amplitude of this new intergral will vary randomly between 0 and N, but will average 1 over any non-zero distance.

total intensity = Integral of Fourier Transform = 1 (or the intensity of a single (maximum) white noise generator)
intensity (for non-zero lengths of time) - also the same as intensity(max white noise generator(for same non-zero length of time)

Yes, its weird. But that's the math I get (I'm not so sure its right). Just forget about "instantenous" in terms of waves. It isn't a property that they have.

PS: on "adding up to 1". Basically I'm assuming a random summation of bound real numbers with equal probability and bounding points +/- the same numbers will sum to zero with sufficient values chosen. The summation of random phases (of equal value) can be seen as two independent sumations with a probability function equal to a sine function from -pi/2 to pi/2, and both of these should also tend to zero with enough points chosen. Note that in our example "enough chosen" is from infinite real frequencies, not white noise generators. Hopefully, I got that right. Also, the correction constant will be the same for the inverse of the summed fourier transform as the inverse for the max transform, so they wouldn't change anything.

PPS. Sorry about any percieved hostility to inaccurate terminology. It's just that the whole topic is pretty weird and after going through the basic class mentioned and a further DSP course I though I understood frequencies. Not true, long after I graduated I was trying to compress some audio and really had a nasty reintroduction to all the windowing issues I thought I understood.

Zamfir
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### Re: Wave Questions

Wumpus, I can't fully follow your argument, but it feels off. I think you're missing a square somewhere - destructive interference decreases magnitude of the Fourier coefficient, constructive interference increases it, each with equal odds. But power is proportional to magnitude squared, so the expected value of the power is increased.

Let's look at discrete noise for simplicity ( should be OK, we're not interested in the high-freq behaviour of a physical sound noise source). Say, our signal consists of samples of the gauge air pressure at a unit distance from 1 noise source.

For white noise, these samples are independent draws from a distribution with zero mean and some non-zero variance. Variance has squared units of the signal. Sound intensity at the measurement point in [W/m2] is proportional to the variance, and the source power output in [W] is intensity multiplied by some surface. A sphere in simplest case. Or more precisely, the expected values of intensity and source power are proportional to the variance of the distribution.

If we have two or more sources, then each sample is the sum of random variables. For simplicity, we can assume noise sources with Gaussian distribution. Then the new distribution is again a Gaussian with mean zero, and variance equal to sum of variances of the sources.

If I did this right, then (expected) sound intensity at the measurement point is simply the sum of intensities of the sources separately.

I would say that the closest to an 'amplitude' of the noise is the standard deviation of the distribution. That's basically what electro people call the RMS amplitude.

wumpus
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### Re: Wave Questions

Zamfir, I think you found my mistake.

If you define the f(t) as a function of time that recieves all audio input in a single place, then I'd be fine claiming that f(t) is the amplitude (I won't call it the intensity). While I'm still sure that f(t) will *average* 1 (or the maximum output of any white noise generator), I see that if you average (f(t))**2, you will get something other than 1 (I still doubt they will add linearly. I'd expect to sum squares due to interference).

It certainly feels like the intensity should involve a square, but I don't know where the second value comes from. In electronics, the power will follow the square of the voltage (or current), and browsing through wiki implies a square when looking at "sound energy".

doogly
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### Re: Wave Questions

yeah, if you have f as the displacement function, the intensity is pressure * volume, and that winds up with a df/dt * df/dx, so you get an ~f^2

but then f(t,x) should average 0, so maybe the way you're thinking of f is to already include some squaring somewhere? what do you mean "receives all audio input" ?
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wumpus
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### Re: Wave Questions

My f(t) is the sound received by some listener/microphone in a location. I assume f(t,x) would be the same for all audio passing through the x-axis. I'd expect the end result of any x for f(t,x) to be the same as my f(t), assuming random phases and ignoring distance losses. Obviously ignoring the distance losses make things a bit fake, but obviously you should get easier/smaller results from adding smaller and smaller levels of input.

While f(t) obviously averages to zero, what you want is to average the absolute value (obviously for intensity, but also amplitude values typically only measure absolute values from positive to negative, or twice the average absolute value). Also I haven't been looking at f(t) all that often, because white noise isn't really directly defined for f(t) but for F(theta) [where F is the fourier transform of f]. This is going to be a complex value, and more obvious that you want the absolute value. The math should work the same, but is much more clear with F(theta) as |F(theta)| = 1 for the white noise generators in the bandwidth and 0 for all other theta.

So |F(theta)| should average to 1 (it is essentially a random walk, so the variance will grow the average should remain). But as the variance grows, average((F(theta))**2) should grow as well.

PS. RMS (root mean square) is abused a lot in audio circles (mostly in terms of amplifier output power. It is technically incorrect to use the term there, but vastly superior to other values marketing may choose). I suspect it is abused here, but is probably still what we want.

Zamfir
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### Re: Wave Questions

the intensity is pressure * volume

I assume you meant to write velocity, not volume?

@wumpus: there are several physical quantities in a sound field that are proportional to each other. Particle velocity, pressure (relative to average pressure), density (relative to average density), displacement. Depending on the type of wave (travelling, standing), there is a phase difference between them. These quantities all have a zero mean by construction, since they are defined relative to an average.

So if you talk about a "sound signal", it could refer to any of these. For signal people, it doesn't really matter which one, since the conversion to a signal implies an arbitrary constant anyway.

Intensity is not like those. The same goes for other energy-related quantities, like source power. Intensity is pressure times velocity, so it's proportional to pressure squared, or velocity squared, etc. These energy-related terms typically stay positive everywhere (except perhaps for a direction-related choice of sign), with a positive mean.

Since signal people care little for constants of proportionality, they just square whatever value they have and call it "power". The word suggests physical power (Watts), and sometimes it is indeed a physical power. But it really just means "one of those quantities that involve squaring".

Edit: the confusion gets worse when people use "levels", the logarithmic versions of quantities. For the levels of "field" quantities, they sneak in a factor 2 in the definition, making them behave like the levels of "power" quantities. So if sound power level of a source goes up by 20dB, it means that power is multiplied by 100. However if sound pressure level or line voltage level go up by 20dB, pressure or voltage go up by a factor 10...

doogly
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### Re: Wave Questions

i most definitely did
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