Natural Science Forum / Physics / Acoustics / December 2003

I have had an e-mail conversation with Gary Sokolich as a result of my
post, and I thought that the rest of the group might be interested in
sharing this as well. I have edited the emails to get a reasonably
understandable text. Any typing errors are mine, not Gary Sokolich's.

-----I posted:

If I have a sound field with a certain SPL at a point and I put a
helmholtz resonator there, how do I calculate how much the air in
the resonator's neck will move?

-----Gary Sokolich responded:

The volume velocity of air in the neck is equal to the free-field
pressure divided by the sum of the acoustic radiation impedance
(resistance plus mass reactance)  looking outward from the plane
of the neck opening and the acoustic input impedance (resistance
plus compliance and mass reactances) looking into the resonator
from the plane of the neck opening. All of these quantities can be
calculated using formulas in any introductory textbook on
acoustics.   If you only want to know the volume velocity at
resonance, you only need to calculate the input resistance of the
resonator, as the sum of the compliance and mass reactances equals
zero at resonance.  Finally, to get the particle velocity, divide
the calculated volume velocity by the neck area.

-----I responded:

Hmm, would not that give me the amplitude for the sound pressure
that can be measured outside the resonator, with the resonator in
place? I would like to know the neck velocity, given the sound
pressure WITHOUT the resonator in place. Or am I missing something?

-----Gary Sokolich responded:

The electrical analog of the situation is pressure<----->voltage,
current<----->volume velocity and electrical impedance<---->
acoustic impedance.  The impedance is the sum of the radiation
impedance and the input impedance.  The two impedances form a
voltage divider, with the applied voltage being analogous to the
free field pressure (in absence of the resonator) and the voltage at
the divide point being analogous to the pressure at the opening of
the resonator.  Therefore, the pressure at the entrance of the
resonator is obtained by the voltage/pressure division formula
Ps/Pl=(zl/(zs+zl)).  Likewise, because the current equals the input
voltage divided by the total impedance, or analogously the volume
velocity equals the free field pressure divided by the sum of the
two acoustic impedances, or U=Ps/(zs+zl).

-----I responded:

OK, now I understand how you think. But why would zs be the radiation
impedance? The radiation impedance of what, by the way. Do you mean
the radiation impedance of the resonator, or of the source?

I am pretty familiar with the analogies, I don't think my problem is
there. Hmm... At the point of the resonator there is a pressure
(~voltage) which, if we "load" it will drop, and the amount it drops
would be determined by the "internal impedance" of the source. Hmm...
And you say the that internal impedance would be the radiation
impedance? Hmm... I don't see why, but on the other hand I don't have
an alternative.

Would you comment on this?

-----Gary Sokolich responded:

The radiation impedance is that of the resonator.  

The amount the pressure drops will be determined by BOTH the impedance
of the resonator and the source impedance of the medium that is seen
by the resonator looking outward into the medium.

The slug of air in the neck moves with the same phase. The surface of
that air at the opening of the neck constitutes a radiator.  That
radiator is equivalent to a piston having the area of the neck which
is mounted in an unflanged tube (the neck itself).  That radiator,
like a circular piston in an infinite baffle (eg a loudspeaker cone)
has a radiation impedance.   That radiation impedance is the impedance
that the slug of air sees looking outward into the medium and is
essentially the source impedance of the medium seen by the slug of
air. It is that source impedance which forms a pressure divider with
the input impedance of the Helmholtz resonator.

-----Gary Sokolich responded a second time:

Perhaps the following analogy will help.  You have a voltage source
which has some (internal) source impedance.  You have a load impedance
that you connect to it.  The current that flows will be the open
circuit voltage divided by the sum of the source and load impedances.
The imedance of the source is the impedance looking into the source at
the source terminals.  The source terminals are the terminals to which
you connect the load impedance.

Similarly, you have an open-circuit pressure (free-field) source which
has some source impedance. You have a load impedance which is the
input input impedance of the Helmholtz resonator.  The impedance of
the source is the impedance that you see looking outward at the source
from the the spatial interface with the load. When you look from the
neck opening outward at the source, you see the radiation impedance of
a piston that is mounted in an unflanged tube whose area is the same
as the area of the neck opening    The volume velocity of sound at the
neck opening, as well as in the neck, is equal to free field pressure
(analogous to open circuit voltage) divided by the sum of the source
impedance (radiation impedance of the piston) ) and the load impedance
(input impedance of the Helmholtz resonator).

This is a very simple problem once you see the analogy.

-----I responded: (Now I finally got it...)

Darn! That IS a good explanation! Of course, I wish I could have
figured that out. It kind of reminds me about the best way to measure
the output impedance of an audio amplifier is to apply a current to
the output and no input signal to the amplifier, and then measure the
voltage across the output terminals.

-----Gary Sokolich responded:

Actually, that is exactly analogous to the way the radiation impedance
is calculated.  The analytical problem to be solved is to calculate
the force/pressure that the medium exerts on the piston when the
piston moves with a specified volume velocity.

-----I responded:

Would you mind if I post your responses back to the newsgroup, so that
others could read them too? If so, should I exclude your email adress
(you will definitely receive some extra spam if I post it)?

-----Gary Sokolich responded:

You have my permision to post my responses under the following
conditions: 1) Credit me personally as Gary Sokolich, 2) exclude my
email address, and 3) diligently review and correct the numerous typo
and other non-technical errors (eg double word) in my emails to you.

-----End

Thank you, Gary!