Re: Speed of Mechanical Wave

["John S. Denker" <jsd@MONMOUTH.COM>]

David Abineri wrote:
> I understand that the speed of a sound wave in air is inversely
> proportional to the density (possibly the square root of the density?)
> of the air, all other factors being the same.
>
> Can anyone supply a metaphor or way of seeing that this is true

Before asking why it is true, we ought to
ask _whether_ it is true.

And it is _never_ possible to assume "all" other
factors are the same.  The density figures into
various laws (notably the ideal gas law) so if
the density changes, at _least_ one other factor
must change also.

In particular, suppose you pump half the air out
of a resonant cavity and let it come to thermal
equilibrium.  The density will have gone down,
but the compressibility will have gone up, and
the result is that the speed of sound is unchanged!

In fact, if you know the temperature of the air,
you know the speed of sound, over a huuuge range
of variation in pressure and density.

If you could change the density without changing
the compressibility, perhaps by comparing 3He to 4He,
you would find that the speed of sound went down
like sqrt(rho).  It's a simple mass-on-a-spring
problem:  heavier mass means slower response.

Similarly, at constant density, the speed of sound
depends on the sqrt of compressibility.  Again,
it's a mass-on-a-spring problem.  Floppier spring
means slower response.

I get lots of static from students who have a memoroid
(i.e. a pseudo-memory) of some textbook formula, and
who think the speed of sound in the atmosphere "must"
decrease in proportion to sqrt(rho), even in the
stratosphere (which is isothermal).  But it just
ain't true.  Not even close.

Feynman volume I chapter 47.