Speed of Sound and Incompressible Media
Sound needs a compressible medium to be transmitted
through, and it travels faster through a medium of higher density
(lower compressibility) versus one of lower density (higher
compressibility). Vacuum has no compressible medium so sound cannot
be transmitted. So, would sound transmit through a (theoretically)
incompressible medium at an infinite rate of speed? If so, how can
sound even travel through a medium that cannot be compressed? A
classmate posed this to me and I am stumped.
You pose an interesting question. If you think of a transmission
medium as a matrix of ball bearings all connected to each other with
springs, I think you can visualize how a disturbance (initial sound)
at one end of the medium can propagate through the matrix to the other
end. The stiffer the springs, the faster the disturbance would
propagate since there is less of a delay from one ball bearing to
another - the medium is less spongy. If the medium is completely solid
with no wiggle room (a realistic impossibility), a push on the near
end will result with a corresponding instant push on the far end.
Unfortunately, this results in an instantaneous transmission of
information, which runs contrary to the ultimate speed limit, the
speed of light. What you will find is that the signal can propagate
through the medium just as a light wave can propagate through empty
space, the speed of which is 3 x 10^8 meters per second. Two books you
might want to check out are Einstein's Theory of Relativity by Max
Born and Spacetime Physics by Edwin Taylor and John Wheeler. They both
have well presented conceptual discussions of what I described, and
the included math is not overwhelming, or even necessary to make the
point. Hope this helps.
As you decrease the compressibility, you increase the rate at which sound propagates.
But, simultaneously, you make it more difficult to get the sound propagating in the
first place. Take the example of granite. Sound travels very fast in granite, but it
is difficult to yell through a wall of granite, since most of the sound reflects off.
I would interpret the limit of zero compressibility as follows: if you could get sound
to propagate through an incompressible object, it would travel at "infinite" speed. But,
it is "infinitely" difficult to get sound waves to propagate through the medium in the
first place. In other words, a large block of incompressible stuff just reflects sound
waves back perfectly. So, sound will not travel through an incompressible medium.
There is another point here, though. What if the block of incompressible stuff is small
and free to move? Then the sound waves cause the whole thing to vibrate as a unit. This
means that the sound is transmitted instantaneously from one side to the other. You can
infer, then, that everything must be a little compressible, or else we could use such a
such a system to send signals faster than light.
A fun idea, to say the least.
Since sound waves transmit energy and information, their absolute
limit would be the speed of light.
Long before that however, the shear (no pun intended) mechanics of
moving matter puts a limit on things. The acceleration of particles
in a medium is limited by many factors hence the absolute velocity.
Perhaps, an even more intriguing idea is this: transiting
compressional waves cause pressure gradients. A completely
incompressible material would not accommodate internal pressure
gradients and, would instead be the prefect sound insulator.
R. W. "Bob" Avakian
B.S. Earth Sciences; M.S. Geophysics
Oklahoma State Univ. Inst. of Technology
First, I am not an expert on the mechanics of acoustics; however,
there are some factors that are basic physics that (I hope) can shed some
light (sound) on the problem. Second, sound usually, but not always
requires a compressible medium; or put another way, the detector of the
sound may not recognize that sound is being transmitted. The classical
example is a high frequency sound wave (ultrasonic sound wave) whose
frequency greatly exceeds the ability of the receiver to respond -- for
humans that is greater than about 20,000 cycles/sec and that is being
generous. So that "sound" is acoustically "invisible". Third, at very low
frequencies, let's say 0.01 cycle/sec, the detector (our ear) would not
sense this as "sound". An example here is waves on a beach. We hear the
"noise" made by the water breaking up, but we don't hear the fundamental
frequency which may only be 1 cycle/sec. Fifth, there is no actual
material that is totally incompressible. If there were, there would be no
transmission of sound. It would all be reflected or diffracted/scattered.
A silly, but not irrelevant example would be the coyote on a Road Runner
cartoon. Sixth, is the propagation of "sound" in a vacuum. The
conventional wisdom is that no sound should be produced. However, there
are classes of stars (pulsars and others) that create "sound" by a totally
different mechanism. These stars eject matter into "empty" space, some at
acoustic frequencies. So they create their own medium. A detector of
matter of the appropriate design would "hear" the on/off oscillations of
matter waves. So the energy of the oscillating star in a sense becomes
"sound". So in this case "sound" travels through a vacuum. The closest
analog I can think of in our environment is the bass drum in a marching
band. You hear the "boom, boom, boom" of the drum long before you hear the
brass and piccolo, but the analogy is weak.
Let us clarify by stipulating your medium is a perfectly incompressible liquid, not an incompressible solid.
And don not forget mass. Sound travels through a "spring-and-mass" transmission medium.
Without either one it probably gets relatively meaningless.
Without both you're really lost.
"Sound" normally has variations in local density, velocity, and pressure.
Pressure, applied to one limited area of an ideally incompressible liquid,
might fan out and get smaller at a distance,
but otherwise be transmitted there instantaneously.
Your fist pushing on one spot would be pushing against the mass of the liquid,
and the acceleration required to get out of the way would set the pressure
and enforce equal distribution of displacement and pressure in all directions,
assuming the liquid was free to move outward.
Since it was free at its boundaries, the pressure there would always be zero.
Only inside would inertia provide some temporary confinement for pressure.
However, suppose there was an ideally immovable wall
surrounding the liquid on all sides at some distance,
Any pressure put on a flexible membrane over a hole in that container
would be instantly equal throughout the whole container, I suppose.
The squirmy extrusion of parts of the membrane between your fingers
would be the only remaining "yield" in the system.
If you thunked on a point on the membrane there would be some ringing vibration
simply because of the mass of the liquid and the elasticity of at least two free parts of the membrane.
I think that could work, as a method of thinking about such a non-real abstraction.
Notice how some kind of yield keeps working its way into my scenarios.
Pressure is force per unit area. Force is, uh, ...
Notice that to define the amount of force you are applying _will_ require:
a) some elasticity somewhere, to be stretched by the force, or
b) some free mass somewhere, to be accelerated by the force.
It is not quite feasible to "squeeze" these both out of a pseudo-physical scenario.
Even if it is an abstract discussion, you'd need to define what you mean by "force".
And notice that a "medium" is an infinite body of fluid.
So your incompressible medium may really incorporate two infinities.
If you fill infinite space with an infinite mass of incompressible liquid,
remind me what motion can still happen? or what "events"?
Quite likely to turn meaningless. Handle only one infinity at a time, I would guess.
(To be fair, "swirls" could still happen...)
In my mind, this pressure-in-a-stiff-liquid would not quite be "sound",
because sound is almost by definition a wave,
a travelling wave, not just a time-varying pressure.
If infinite speed, it is everywhere already, no "travelling" to be done.
And travelling only happens if there is springiness and massiveness throughout the medium.
Which there always is, to some quantifiable extent, in any physically real medium.
So I tend to condition my debates to remain meaningful even when
mass-density and elasticity are introduced with some finite value,
starting from zero or from infinity as needed to gradually start affecting the system.
I.e., suppose your infinitely incompressible liquid was actually extremely slightly compressible,
and had mass, but the speed of sound was so fast that the time to cross the finite container
was, say, 100 times faster than the particular observations of finite speed you are imagining....
Thus, a real glass fishbowl full of ordinary water, covered by plastic-wrap,
would seem to act about the same in your hands as the aforementioned container of incompressible.
This is a sign that your abstract problem is well-defined, fully understandable.
This "introduction" is the opposite process to imagining
the limiting behavior as stiffness gradually tends towards infinity, but is never yet "infinite".
Which is a common academically reasonable discussion.
If you want to imagine sound in a solid,
well, there are about two distinct elasticities in solids:
compressibility, and bending elasticity (also called "shear" stiffness).
You will have to work to keep those two from confusing your problem,
maybe only stipulate long narrow rods of the solid, pushed from one end.
Or consider the whole solid body finite in extent but perfectly rigid in all dimensions,
which is self-consistent. A discrete object, not a medium.
A medium that cannot be compressed or bent :
is an ideally rigid wall to anything approaching from outside its surface.
Of course no motion can get through a finite distance of this stuff,
so no "sound" can get through it.
There is no reason to suppose that sound can get through such a medium.
If it was pressured, it would not bend, so it would not move or push anything on the other side.
If there is no specified effect, there is no nothing to discuss, nothing to be called "sound".
But a vacuum does not block motion...
PS- vacuum is missing two things: the compressible medium and the mass-density.
Also missing is "coupling" to material object, and to us:
what can you even measure about it, this point in a vacuum?
You cannot grab it and put a mechanical force on it,
you cannot stick a pin in it and measure how much it is displaced...
Such a fully "transparent" medium is a different idea,
which does not really include sound.
Gravity, electric fields, and magnetic fields
are about the only properties I can think of that a point in a vacuum really has.
And perhaps distances from various material objects occupying nearby points.
You might want to learn about fields, because they are the only
ideas we have that vaguely resemble "pushing on a vacuum".
Light-waves and gravity waves are your "sound-in-a-vacuum".
There are multiple kinds of waves in some real fluids.
In the temperature range from 1-3 Kelvin where Helium is partly normal liquid and partly super-fluid,
there is another wave called "second sound" which carries heat energy, not compression.
Unlike heat in a normal solid, it can reflect off mirrors and resonate in cavities.
It's not just sluggish old diffusion, it really is a wave, that travels.
It gradually attenuates with distance traveled, turning into heat,
just like normal sound always eventually does.
"Normal sound" also travels through the same helium, at a different speed than the second-sound.
Hope those give you a new perspective.
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Update: June 2012