[sdiy] ot: law for high frequency damping in air? law for high frequency damping on reflections?

[Magnus Danielson]

From: "Czech Martin" <Martin.Czech at micronas.com>
Subject: [sdiy] ot: law for high frequency damping in air? law for high frequency damping on reflections?
Date: Wed, 21 Jan 2004 09:32:39 +0100
Message-ID: <D9D56E8FA1A73542BE9A5EC7E35D37FFF38FD0 at EXCHANGE2.Micronas.com>

Martin,

Nice to hear that you are continuing to think about sound ;O)

> I'm now glad that I understood the 1/r law for far field sound pressure
> level. (You can laugh about me).

Why? How would it help you to have me laught at you? It wouln't!

> Now, this is not the only thing that happens.

Oh, no! ;O)

> It must be SPL(r,f).
> 
> As sound travels in air, there are losses, sound waves will
> heat up the air a little bit. This means:
> the 1/r law assumes that no energy is dissipated, but that
> the soundwave expands like a bubble, stating that energy is constant,
> but the surface it travels through is expanding.

Exactly. It is a measure relating to the wavefront theory in an otherwise ideal
medium, just like if it would have been a electromagnetic wavefront in vacuum.

> If we now introduce energy dissipation, the higher frequencies will
> be damped more, how will the SPL law be: SPL(r,f)?

That is actually quite complex. As you noted, it relates to heat-transfer.
This is quite natural when you think about it... when you compress a gas the
temperature goes up, and if you decompress it the temperature goes down.
That is all general gas law equations. The frequency comes into the equations
by the wavelength and rate of pressure changes. If we could stay there, we
would have a fairly short expression, something like this

         my - 1 4*pi²*f²*M*k
alpha  = ------ ------------
     H     my   3*rho0*c³*Cv

where

my = Cp/Cv

and

Cp = molar specific heat at constant pressure
Cv = molar specific heat at constant volume
f  = frequency
M  = mass per mole or the molecular weight
rho0 = equilibrium density
k  = thermal conductivity
c  = velocity of sound

This trivial little thing follows the

       -alpha r
I = I e      H
     0

expression naturally.

Also, it is not valid for all that high frequencies, because more stuff is
happening in air than you first suspect. There is also the viscosity
contribution of damping:

         16*pi²*f²*eta
alpha  = -------------
     V      3*rho0*c³

where all the variables is the same as above, except for the contribution of

eta = viscosity

Now, with viscosity and thermal radiation covered, we need to look at another
effect, that of molecular relaxation. Dominant to the responce of air in audio
range is N2 and O2, but also H20 contributes and is naturally an effect of the
amount of water in the air.

If you look more carefull you will find more....

> And: do you know how the reflection from a surface can be descibed?
> Like a filter (plus traveling time)?

Traveling time is indeed "part" of a filter's impulse-responce. The reflection
of a surface is a filter in which the zeroes is certainly a function of where
the source and destination of the sound is, and (to make this more complex)
which the responce is in different directions of both the source and the
destination, since the total responce is not a line between source-surface-
destination but rather an integration over the space the destination "sees"
(really an angular area) over the surface and which comes from the source -
which have different responces in all different directions. If we just defined
all that, THEN you can view it as a pure impulse-responce and all the other
properties you can derive out of that will tell you alot. Start to change the
3D properties of position, orientation or waveform properties and you got a
different case.

A sound-source have an impulse-responce which depends on your 3D orientation
in relation to the sound-source. Reciprocally the same goes for a sound-sensor.
Difficult sound-sources is harder to handle and predict (ported bas-speakers!).

Naturally, the distance from the source/sensor makes a difference, but things
average out at a long distance (far-field) while it can be very drastic in the
near-field. Now, many of the early measures and ways of analysing assumed the
far-field responce with its approximatively spherically wavefront. These days
we can even put the whole audience in a ourdoor venue in the near-field.

Cheers,
Magnus