Re: Speed of Waves

[John Denker <jsd@MONMOUTH.COM>]

At 10:32 AM 1/28/01 -0500, David Abineri wrote:
> We are discussing mechanical waves (sound) and some students feel
> (understandably at first) that the harder (faster) one 'hits' air
> molecules, the faster the disturbance will travel through the air. They
> are thinking that the air molecules are particles that simply move
> faster when hit harder thus making the disturbance move faster..
>
> What metaphor and explanation might help them to see that the medium
> dictates the velocity regardless of the 'hit'.

First of all, it is important to deal with the _unhelpful_ metaphors that
the students start with.  They have lots of experience with, say,
baseballs.  The harder you hit the baseball, the faster it propagates.  So
the first order of business is to explain why this analogy might not
apply.  For one thing, when A throws a baseball and B catches it, all the
molecules in the baseball move from A to B.  In contrast, when A launches a
sound wave and B detects it, no particular air molecules move from A to B.

A possibly helpful metaphor is the usual "lumped element"
approximation.  Rather than considering a continuum (which is for most
practical purposes the right model for air), for present purposes consider
instead discrete masses connected by discrete springs:

        sssssMsssssMsssssMsssssMsssssM...

Suppose the length of each spring is 5 inches, as represented by the 5 "s"
characters.  Now suppose at time t=0 you suddenly displace the left-hand
end of this array by one inch, shortening the spring from 5 inches to 4
inches.  It doesn't matter how suddenly you do this;  the first mass won't
move suddenly.  It will start _accelerating_ at time t=0, but it won't have
any appreciable velocity until later.  The motion of masses farther down
the line is even more strongly delayed.

Now for a sudden _large_ displacement the story might be different.  If you
shoot this mass/spring model with a high-velocity cannon shell, the shell
will sweep all the components before it, creating a nonlinear disturbance
that propagates faster than the normal wave-speed of the model.

Now, considering air, in a perverted sense the students are right;  if you
smack the air really really really hard you can create a shock wave that
travels faster than c, the nominal speed of sound.  So we should rephrase
the question:  why does the medium dictate the velocity of sound,
regardless of the excitation, for reasonable acoustic excitations?

An important idea is to visualize the air molecules as being in constant
motion.  Even in the absence of sound, the air molecules are whizzing
around at speeds on the order of the speed of sound, running into each
other.  When you hit the air with, say, a loudspeaker cone, in all
reasonable cases the cone velocity is very much smaller than the speed of
the air molecules.  So in some sense, the cone hardly hits the air at
all;  mostly the air hits it.

Now at this point you might be tempted to do the following superficial
argument:  calculate the cone-velocity as a percentage of c, and argue that
the propagation speed is only a small percentage greater than c.  But that
would be wrong, as we can see from the following:

The next piece of the explanation is to observe that a _negative_
displacement (i.e. lengthening the first spring by one inch) propagates in
the mass/spring model just as nicely as a positive displacement.  If you
make a superficial argument based on this observation, you might conclude
that the propagation is slightly slower than c.

So we see it is not helpful to think of sound as a sudden "hit" or as a
sudden "yank".  In reality it is a rather gentle fore-and-aft shaking.

To summarize:
  1) This is not a baseball or cannon scenario, where the substance moves
from point A to point B
  2) The correct view is not "hitting" or "yanking", but rather "shaking".
  3) The shaking involves velocities small compared to c.
  4) Now matter how frequently or how hard (within the reasonable range)
you shake one mass in the mass/spring array, the neighbors don't respond
immediately.