Re: Thermal Waves ?

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

Ludwik Kowalski wrote:

> I was not comfortable with the concept of a "temperature
> wave" because, for some reason, I usually associate waves
> with restoring forces (for example, due to elasticity) and
> with inertia. Mechanical waves propagate in a medium
> which can be modeled as a sequence of linked harmonic
> oscillators.

I agree with that reasoning and that terminology.

> But I can not imagine a thermal harmonic oscillator.

There are some highly specialized situations where it
is possible to have such a thing.  This is where "second
sound" comes from.

Imagine a mixture of 3He and 4He.

1) We all know that ordinary sound involves a local
increase in density of both components, and a corresponding
increase in pressure.  This propagates as a wave.

2) In contrast, now imagine a local increase in the partial
pressure of one component, exactly balanced by a decrease
in the other component, so that there is no change in
total pressure.

2a) If we are talking about 3He/4He gas at room temperature,
scenario (2) would not be oscillatory or wavelike.  Calling
it a wave would be quite a misnomer, like calling an
RC circuit an oscillator.  Yechhh.  The excursion in
density would diffuse away in a highly overdamped way.
There's nothing driving the mixture back to uniformity
except for random-walk diffusion.

2b) But now suppose we are talking about 3He/4He liquid
at low temperature.  The story is different now, because
of degeneracy.  There is what chemists call a "molecular
field" or "chemical potential".  In physics terms it's
just the kinetic energy of the atoms, which is increased
by the presence of like atoms via the exclusion principle.
This provides enough of a restoring force that you can
actually get oscillations and wavelike propagation.

This can, if you want, be called a "thermal" wave since
the entropy of one component is different from the other,
so there is indeed a wavelike temperature excursion.

Of course it's not "just" a temperature excursion;  the
trick is that the temperature is coupled to something
else that has inertia and a nontrivial restoring force.

> Suppose the resulting temperature fluctuations at
> the surface are between 19 C and 21 C. In a layer below the
> surface the temperature oscillations are repeated with
> delays proportional to x.

For the diffusion equation, as opposed to wave
equations, the delays are not "proportional" to x,
as you can verify by plugging f(x-ct) into the
equation and getting a contradiction.

> The temperature wave amplitude
> decreases rapidly with depth.

Verrry rapidly.

> The speed of propagation of heat is

There is no well-defined "speed of propagation"
in the diffusion equation.

This posting is the position of the writer, not that of Lois Steem, Liza
Round, or Mike Easter.

This posting is the position of the writer, not that of SUNY-BSC, NAU or the AAPT.