When sound travels through solids rather than air, the speed of propagation varies with the frequency (I believe acoustics folks talk about “bending” as the mechanism for propagation). Does it mean if 2 frequencies are mixed together and coupled into a solid, will the phase relationship between the two frequencies vary as the pickup point is moved along the surface of the solid.

Does this frequency dependent speed of prorogation imply non linearity thereby generating new frequencies when several frequencies are mixed together. Will the distortion also exist with single frequency excitation. Can this be modeled or characterized.

If the speed changes fairly slowly with frequency, just the

speed vs. frequency curve will be enough.

Consider that optical materials also change speed with frequency,

but away from resonances they can usually be considered linear.

Non-linear optics is done at high amplitude where things

do go non-linear. The displacement of the electrons is large

enough that non-linear effects become significant.

I think the effect of the sound speed as a function of frequency is

dispersion. I don’t think it implies nonlinearity at low amplitudes.

Here are a couple of documents that discuss measurements.

Improved Surface Wave Dispersion Models and Amplitude Measurements

ADA422916

http://stinet.dtic.mil/cgi-bin/GetTRDoc?AD=ADA422916&Location=U2&doc=…

Discrimination, Detection, Depth, Location, and Wave Propagation

Studies Using Intermediate Period Surface Waves in the Middles East,

Central Asia, and the Far East

ADA417753

http://stinet.dtic.mil/cgi-bin/GetTRDoc?AD=ADA417753&Location=U2&doc=…

Even if the topics aren’t an exact match to your interest, take a look

at the introductory sections.

Direct use of non-linearity at high drive levels is used in

“parametric” oscillators and amplifiers. Take a look at the discussion

of the non-linearity in this document as an example.

http://www.sea-acustica.es/Sevilla02/ult04021.pdf

Hi Joe,

I don’t think that this implies non-linearity or the creation of sum and

difference frequencies. Between any two locations the audio transmission

should be defined by an impulse response and a frequency response, just as

any linear system. As you mention, the frequency response will be a

constant amplitude and a changing phase, with respect to frequency. Off

hand, I don’t see any reason that this cannot be linear (someone smarter

might correct me!).

Propagation velocity being a function of wavelength is called

“dispersion” whether in optics or acoustics. Snell’s law applies in

either case. Acoustic lens structures exist and have been used to focus

and disperse sounds. Non-linearity, if involved at all, is a side effect

and not fundamental to the phenomenon.

My problem is that if I think of a linear mechanical system, I should

be able to define a spatial R-L-C lumped approximation, and once I

have done that I don’t see how propagation speed could depend on

wavelength (propagation speed in not the same as group delay at a

given spatial point; I am assuming no effects from rolloffs, the

system is flat at all frequencies that are injected). Try to think of

a R-L-C transmission line model where the propogation speed varies

with frequency; I can’t think of how this could happen in a linear

system.

Dispersion occurs on transmission lines also. They can be modeled (below

some critical frequency) with R-L-C components and no others. Instead of

my writing a treatise on transmission lines, try Google. Start with

http://www.ece.uci.edu/docs/hspice/hspice_2001_2-269.html

Waveguides are also dispersive. Unlike with transmission lines, it is

not even theoretically possible to avoid it. All optical fibers show

dispersion. The propagation velocity is the inverse of the refractive

index, and refractive index is a function of wavelength. No distortion

is involved.

I think this is different. Consider the following experiment. I drive

a narrow and thin strip of plastic at one end with a transducer of

some sort. I dial the frequency to 1KHz, and then I measure the

PHYSICAL wavelength somewhere near the end of the strip by moving 2

pickup microphones around until their outputs are in-phase. The

propogation speed is then calculated by dividing the physical

wavelength by 1ms. Let’s say the wavelength turns out to be 1 cm.

Now I double the input frequency to 2KHz, and repeat the experiment.

This time, though, the physical wavelength comes out to be .4cm

instead of the expected .5cm. indicating that the propogation speed

has changed.

I could run the exact same experiment with a transmission line, and I

would argue that no matter how much dispersion you get, the measured

wavelength will always be equal to the propogation velocity divided by

the frequency.

—-

> wave travels through a solid rather than air, the speed of propagation

> varies with the frequency

Is it TRULY frequency dispersion, or is it amplitude dispersion? I

find brief mention of the difference here: “http://

imamat.oxfordjournals.org/cgi/content/abstract/1/3/269”. I seem to

recall from my SONAR work, for example, that water exhibits amplitude

dispersion, which causes sinusoidal waveforms to turn into sawtooth

waveforms. That does not, however, address frequency dispersion for

non-sinusoidal waveforms in that medium.

> is frequency-dependant also imply non-linearity, and therefore have I

> generated new frequencies when I mix several frequencies together? Do

> I get distortion with only a single frequency excitation?

Again referring to the abstract mentioned above, media that are linear

but dispersive seem to be possible. If a medium is linear but

dispersive, I think that it can be treated like a linear filter with

nonlinear phase response — the kinds of filters that we encounter

routinely.

> propagation speed and the frequency (for single frequencies), how

> would I characterize such a system mathematically?

Perhaps characterize the nonlinearities and the dispersion separately;

nonlinearities as distortion products, and dispersion as filters (as

above)?

Greg

————————–

>> not even theoretically possible to avoid it. All optical fibers show

>> dispersion. The propagation velocity is the inverse of the refractive

>> index, and refractive index is a function of wavelength. No distortion

>> is involved.

I should add that transmission lines need some dissipation in order to

exhibit dispersion, but waveguides don’t. They *always* exhibit

dispersion, which (to first order) is independent of loss.

> a narrow and thin strip of plastic at one end with a transducer of

> some sort. I dial the frequency to 1KHz, and then I measure the

> PHYSICAL wavelength somewhere near the end of the strip by moving 2

> pickup microphones around until their outputs are in-phase. The

> propogation speed is then calculated by dividing the physical

> wavelength by 1ms. Let’s say the wavelength turns out to be 1 cm.

> Now I double the input frequency to 2KHz, and repeat the experiment.

> This time, though, the physical wavelength comes out to be .4cm

> instead of the expected .5cm. indicating that the propogation speed

> has changed.

Propagation velocity is wavelength times frequency by definition.

> would argue that no matter how much dispersion you get, the measured

> wavelength will always be equal to the propogation velocity divided by

> the frequency.

That’s true by definition. How do you conclude from it that the

propagation velocity is independent of frequency?

—

In the real world, dispersion is the expected state. Glass exhibits

chromatic aberration, water-wave velocities depend not only on the

water’s depth but also on the wavelength, waveguides have group

velocities that are asymptotically free-space at very high frequencies

and go to to zero at cut-off. (The phase velocity goes infinite at

cut-off.) Glass fibers are waveguides built from dispersive material.

Even their dispersion is dispersive!

Simple electrical transmissions are often thought of as dispersion free.

That’s a useful treatment because we try to keep losses small and it

turns out that such lines — including circuit-board traces — will be

free of dispersion if they’re lossless *AND* their component materials

have salient properties (e.g., dielectric constant) that are themselves

independent of frequency. Dielectric constant is the square of

refractive index. The dielectric constant of water measured at audio

frequencies is a little over 80, which should make its refractive index

about 9. That’s what it is at ELF, but it varies about 1.23 to 1.33 for

visible light. (Propagation velocity is c/n, where c is free-space

velocity and n is refractive index.)

We make simple models — models that allow us to ignore dispersion for

the most part. That makes it easy for us to get the mindset that

dispersion is rare and anomalous.* In fact, lack of dispersion is rare.

Jerry

____________________________________

* There is anomalous dispersion, but that’s a different subject! 🙂

—

(snip)

> That’s a useful treatment because we try to keep losses small and it

> turns out that such lines — including circuit-board traces — will be

> free of dispersion if they’re lossless *AND* their component materials

> have salient properties (e.g., dielectric constant) that are themselves

> independent of frequency. Dielectric constant is the square of

> refractive index. The dielectric constant of water measured at audio

> frequencies is a little over 80, which should make its refractive index

> about 9. That’s what it is at ELF, but it varies about 1.23 to 1.33 for

> visible light. (Propagation velocity is c/n, where c is free-space

> velocity and n is refractive index.)

In addition, note that both the dielectric constant and index of

refraction are complex. The imaginary part of the index of refraction

turns into absorption when it goes into exp(iwt). Ellipsometry:

http://en.wikipedia.org/wiki/Ellipsometry

is pretty much the science of complex index of refraction.

> the most part. That makes it easy for us to get the mindset that

> dispersion is rare and anomalous.* In fact, lack of dispersion is rare.

Far enough from resonance, (and for fields much less than those

inside atoms) dielectrics are pretty close to linear

Heaviside worked out the conditions needed to allow transmission lines

to be nondispersive. A lot of people would not believe him and the

early trans-Atlantic cables were horribly dispersive and in fact one

of them got destroyed when one engineer tried to overcome the

dispersion by increasing the source voltage to the point of arcing out

the cable. Imagine having to explain you just destroyed the cable just

after it had been run across the ocean! Do the cable’s owners dock

your paycheck in this case?

In addition, all real materials are non-linear, though often they

are used where linear is a very good approximation. (In the case

of both mechanical and electromagnetic waves.)

There is the field of non-linear optics, studying materials where

the light intensity (electric field) is high enough that materials

are non-linear.

http://en.wikipedia.org/wiki/Non-linear_optics

Now, can a (non-physical) material be linear in amplitude,

but non-linear in frequency? Assuming that the response to

electric or mechanical waves (P vs. E, for example) is an

analytic function then I believe the answer is no.

Can it be close enough that you can’t measure the difference?

Maybe so.

http://groups.google.com/group/comp.dsp/browse_thread/thread/68931bdcb7210c69

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