Helmholtz resonator with multiple necks--formula?

[EmilioL]

Does anybody happen to know the formula for resonant frequency of a Helmholtz resonator having N necks?

Physics is not my field and I'm a bit over my head. I need the formula for a computer program related to musical instruments--this is the only thing holding me up. It seems like it would be easy to derive from the standard formula, but it's not--at least not for me.

The only solutions I've been able to find in the literature are for special cases (2 identical necks, or 2 necks the same length), There is lots of "hand waving" about the analogy with electrical impedance, which I know how to calculate: R = 1 / ( 1/R1 + 1/R2 + ... 1/Rn) where R1...R are resistors connected in parallel. But I have not been able to find a formula for the general case in a publically-accessible paper.

Unfortunately, the term for acoustical impedance, (A * A) / m, has been factored out of the very simple equation for approximating the resonant frequency of a Helmholtz resonator:

F = (c / 2 * pi) * SQRT( A / (V * L'))

where c is the speed of sound, A is thecross-section area of neck, V is volume in the cavity, and L' is effective length of the neck.

Effective length is the fly in the ointment. It is give by L' = L + (k * a), where L is the physical length, a is the neck radius, and k is a empirically-determined constant. So even if all necks are the same physical length, their effective length will differ if their diameters differ. Somehow, effective length needs to be applied to area for each individual neck.

Backing up to the formula for angular frequency -- from which the standard formula is derived--makes for a messy equation with variables that are inconvenient to measure or not that signficant (i.e. changes in barometric pressure do not signficantly affect resonant frequency):

O = SQRT( g * (A * A) / m * P / V)

O (omega) is angular frequency in rad/s, g (gamma) is adiabatic index, m is mass of air in the neck, V is static pressure and A is as defined above. It seems to me if the single neck solution doesn't require
the adiabatic index or staic pressure, the multi-neck solution shouldn't either. From a practical standpoint, the only external variable significantly affecting the resonant frequency of a given Helmholtz resonator in air is the termperature (represented in the standard formula by the speed of sound).

Any help towards a practical formula would be greatly appreciated.

 

[EmilioL]

OK, let me try it this way: does anybody have an formula for the pitch of an ocarina?

An ocarina is a type of vessel flute that has been independently invented by many different cultures.

Ordinary flutes are air colums (standing wave), but the ocarina works on the principle of a Helmholtz resonator, but has a fipple (like a whstle or a recorder) that provides the energy to keep the oscillation going. The most familiar forms are the English ocarina (usually saucer shaped with four holes on top, holes are played in combinations) and the Italian ocarina or "sweet potato".

The formula for a Helmholtz resonator:

       c                     Area
F = ------  *   √  -------------------------
      2 π           Volume * EffectiveLength

is on the right track (it's proportional if all finger holes are closed). But it doesn't give the correct answer because:
(1) the ocarina is an *active* resonator;
(2) air from the mouthpiece blows over the fipple opening; changing its apparent length;
(3) the ocarina has finger holes in addition to the fipple opening.

Surely there is *one* person out there who knows something about acoustics and isn't put off by a few formulas?

 

[Tom.G]

It seems that the hole diameter and the wall thickness it penetrates are major determining factors for tuning, with hole positioning having a smaller effect.

Take a look at: https://pureocarinas.com/blog/
It's rather long with some useful info scattered around. I haven't looked at the rest of the site.

Cheers,
Tom

 

[EmilioL]

Thanks, Tom. That's an interesting blog. It has empircial data on how temperature and pressure affect pitch, but no formulas. Unfortunately, the author is not a physicist and doesn't try to explain his results: no math, no citations.

He plots pitch against temperature, and finds that the pitch goes up. No surprise there--temperature decreases air density, which raises the speed of sound (according to the Newton-Laplace equation): c = sqrt( K / r), where c is the speed of sound, K is the coefficient of stiffness and r (rho) is the density.

The speed of sound is a variable in the formula for the Helmholtz resonator (which I posted earlier in this thread). So temperature should affect pitch.
The surpirse is that the blog finds the relationship to be linear: he says temperature goes up 9 cents for ever 10 degrees. The range he plots is 0 - 25 C, wide enough so that it should look exponential--but he only plotted 8 points. Am I missing something?

There may be experimenter bias here because he blew into the ocarina to make the sound, and "the harder you blow, the sharper it goes"---he may be unconsciously adjusting the pitch. That's the problem with just making observations without (1) a hypothesis, (2) designing an experiment, (3) attempting to explain the results from the standpoint of physical principles, and (4) comparing them to published data. It's just anecdotes.

I've ordered *The Physics of Musical Instruments* by Neville H. Fletcher and Thomas Rossing (also the authors of *Principles of Vibration and Sound*). It has a section on the ocarina, which I'm hoping will contain an answer. The appearance of names like Newton, Laplace and Helmholtz indicates that we are not on the forefront of knowledge here. :-) I hope my $79 was well spent (surprisingly inexpensive for academic publishing).

The Internet is wondeful--as long as you just want to read opinions and watch dogs ride skateboards. There are a zillion "physics for dummies" sites, but real, quantitative science is surprisingly difficult to find if you don't have access to a big reserach library. I do--but I'd have to drive 400 miles to get there. Much of the USA is a "science desert": no big libraries, only junior colleges and technical schools. But you can find out anything about technology.

The public is supposed to fund the science priesthood--which keeps its secrets locked up in ridiculously expensive books and journals that only big university libraries buy. Well, we sure wouldn't want the public to have access to science--it might change how they vote! :-)

 

[Tom.G]

Well, we sure wouldn't want the public to have access to science--it might change how they vote! :-)

Nah, there are too many P.T. Barnums in politics for that to happen.

 

[EmilioL]

Got the book. It's *wonderful* -- a treasure trove of acoustics applied to musical instruments: real physics, no "hand waiving". I highly recommend it to anyone interested in the subject of how musical instruments make sound. Darned well worth it. The subject is developed from the basic concept of oscillation to the simplest physical systems, then to more complex systems. A model of how to approach complex systems (well, if you needed another model beside Newton's *Principia*). It goes like this:"hey, there are these diverse musical instruments, but guess what: they all embody a small number of physical principles!" (Which the authors proceed to develop *in detail*, so you can make real predictions. Yeah!)

The bad news: it doesn't have a section on the ocarina. The good news: it has a series of "network diagrams" showing energy flow though impedances for (1) the simple Helmholtz resonator; (2) the powered Helmholtz resonator (osciallation driven by a piston in the bottom).-- Section 8, "Pipes, Horns and Cavities", 8.15 "Network Analogs" (p. 227-230). I hope to post these diagrams soon--after I figure out how to apply them to my question.

Let's hear it for authors willing to put math in their books -- and publishers willing to publish them! Kudos to Springer for putting out this handsome and permanent book--loaded with real physics--at a moderate price. Nobody's going to ask "Where's the beef?:" It's all beef but the covers. :-)

The only downside is that it does not go into musical perception: pitch, consonance, beat tones, tense intervals, etc. It's strictly sound production.

Edit: fixed newlines and added last paragraph.

 

[EmilioL]

P.S. For anyone who thinks pitch is frequency, here's a link to a musical pitch that rises endlessly, but never gets any higher (the Shepard tone). Music is all about the ear, brain, and culture.
ttps://www.youtube.com/watch?v=BzNzgsAE4F0

Or maybe..."It's all about that bass-- fundamental bass, not thorough" (with apologies to Meghan Trainor
and a knod to Jean-phillippe Rameau):

 

[EmilioL]

This is going to take a while. *The Physics of Musical Instruments* is a 756 page reference book, but each section depends on earlier sections.. There is no index of symbols, and symbols are introduced (often without a formal defitiion) and then used 30 pages later. So you have to read the book like a novel, starting with page 1. And you have to keep a physics textbook open for key defintions and formula.

Moreover, the "systems" approach means that different physical resonators with very different properties--e.g., air columns and cavities--are treated in the same chapter. Yes, a bicycle and a motorcar both illustrate Newton's laws, but a Chilton's Guide doesn't cover both in the same book! (And it makes it clear why people hire mechanical engineers, not physicists: F = MA will not help you to build a car, drive one, or fix one. By the time you get to the level of detail that makes the science relevant, you are talking to an automotive engineer, not a physicist.)

But the failure to deal with musical *sound* (including perception) invalidates the entire approach.. Physics has no concept of musical pitch and it can't be derived from first principles, since It involves how the human ear, brain and culture work. A musical instrument isn't just a device that makes sound--the requirements of music determine the form that *every* instrument takes. Compare the construction and materials of the following two lists of sound production devices:

Devices List A: alarm bell, siren, buzzer, beeper, steam whistle. auto horn, starter's pistol, firecracker
Devices List B: violin, piano, guitar, marimba, harmonica, clarinet, timpani

Are they similar? Not very. Clearly, more just making a sound is at work in list B. Yet the same physics explains sound production in both lists. In music, the character of the sound (e.g. overtones) and the usability of the device are of paramount importance. Viva la difference!

The book mostly describes "ideal" instruments, not real ones, so it can't explain why a trombone is a musical instrument and a slide whistle is a toy. Perhaps the title should be "Musical Instruments as Examples of Textbook Physics" as there is no interest in musical instruments, per se. For example, there is little discussion of materials -- though it is rather hard to build an instrument without materials.

It is wonderful to see the physics of so many instruments described. But what is actually being descibed is a device that is similar to, say, a violin, but which is not a musical instrument. It need not, for example have f-holes or a fingerboard. It need not have tuning pegs. The bridge need not be curved. And it could be made out of, say, injection moulded plastic. The physics principles are the same. Anybody want to play a "violin" made like that? I'm pretty sure that Einstein wouldn't have touched it. :-)

 

[EmilioL]

Here's my understanding of Fletcher and Rossing's analysis of the simple Helmholtz resonator using the network analog. It eventually derives the familiar formula for resonant frequency. This is the *simplest* case!

Network analog:

The basic approach is similar to electrical network theory, with acoustic pressure p replacing electric potential, and acoustic volume flow U replacing electric current.

[Fletcher and Rossing 1998, p. 227]

Thus, instead of Ohm's Law
V = R I
or
V = Z I
we have the analogous
p = Z U
where
p is acoustic pressure
Z is acoustic impedance
U is acoustic current

Case 1. Classic Helmholtz Resonator
* Passive
* One neck
* No flange
* Tiny listening hole in bass ignored

Cross-sectional view

              .---------. 
             /           \ 
  ----------'             ` 
                          | 
                          =  <- listening hole 
  ----------.             | 
            \            / 
             `----------'

In the passive Helmholtz resonator, the source of sound is outside. So the resonator is a network composed of:
* A source of sound (analagous to an AC generator), and
* Three acoustic impedences:
Zpipe the "inductance" of the neck
Zrad the "inductance" of sound radiation at the opening
Zcavity, and Zradiation

Sound U (analogous to current i) flows through the resonator from one the opening, though the pipe to the cavity (finally out the tiny listening hole -- which we may neglect since it drains off a negligible amount of energy--it is not a "load"):

  .------> Zrad -----> Zpipe ------.
  |                                |
  |                                |
Source                             V
  ^                               Zcav
  |                                |
  |                                |
  `--------------------------------'

[ibid, p. 229]

By Ohm's Law:
V = Ztotal I
For resistances or impedences in series, Ztotal = Z1 + Z2 .. Zn. So:
p = (Zrad + Zpipe + Zcav) U [op. cit.]

The pipe is analogus to an inductor: it impedes sound. Instead of the electrical formula for the voltage V across an inductor L carrying current i:
V = L (di/dt) = jwLi [ibid. p. 227]
where:
L is inductance (in Henries)
i is current (in Amperes)
j is the imaginary unit (not using i, since it's traditional for current)
w (little omega) is angular frequency

Now the formula for the acoustic impedance of the neck pipe:
Zpipe = jw(pl/S) [ibid. p. 228]
where:
j is the imaginary unit
w (little omega) is angular frequency
p is acoustic pressure
l is length
S is size (area of cross-section)

Zrad is similar to electrical resistor: it causes a loss.
Unfortunately, there is no analogous formula to express an
elecrical resistor: it's a simple device. The calculation
for radiation from an unflaged pipe is "very difficult" [p. 200].
However, the result (which I will take on faith) is fairly
simple:
Zrad = 0.16 (pw²/c) + 0.6 (j w p / S) [op. cit.]
where:
p is acoustic pressure
w (small omega) is angular frequency
c is the speed of sound
j is the imaginary unit
S is the size (cross-sectional area) of the neck

Lastly, Zcav stores acoustical energy -- it is a reactor analogous to an electrical capacitor. The formula for impedance of a capacitor:
Zcap = 1 / (w C) * e^(j - (pi/2)) [source: https://en.wikipedia.org/wiki/Electrical_impedance]
where
j is the imaginary unit
w (little omega) is the angular frequency (in radians)
C is capacitance (in Farads)

The pressure inside the cavity is related to the current as follows:
p = yP / V ∫ U dt [Fletcher & Rossing 1998, p. 228]
where
p is acoustical pressure
P is atmosphereic pressure (constant)
y (small gamma) is the ratio of specific heats for air
V is the volume of the cavity
U is the acoustic current

Notice that ∫ U dt, flow of air into the cavity, is analogous to electrical change, so V/yP is analogous to electrical capacitance. A more convenient expression for "acoustic capacitance" is:
V / (p * c²)
which gives:
Zcav = - j p c² / (V w) [op. cit.]
where:
j is the imaginary unit
p is acoustic pressure
c is the speed of sound
V is volume of the cavity
w (small omega) is angular frequency

Since the three impedances are in series, they add together:
Ztotal = Zpipe + Zrad + Zcav
which gives the rather daunting equation
Ztotal = jw(pl/S) + ( 0.16 (pw²/c) + 0.6 (j w p / S) ) + (- j p c² / (V w) )
simplifying:
Ztotal = jwpl/S + 0.16pw²/c + 0.6jwp/S - jpc²/Vw
= (jwpl + 0.6jwp)/S + 0.16pw²/c - jpc²/Uw
= 0.16pw²/c + j(wpl/S + 0.6wp/S - pc²/Uw)
hence (using "approximately equal" sinc we did not include the effects of vicous and thermal losses to the pipe and cavity walls):
U ~= p / (0.16pw²/c + j(wpl/S + 0.6wp/S - pc²/Uw)
~= p / (0.16pw²/c + j(wp(l/S + 0.6/S) - pc²/Uw)

This is just a simple damped resonator equation, with the maximum acoustic flow and minimum acoustic pressure in the cavity occurring at the resonant frequency:

w* ~= c √ ( S / V (l + 0.6a)) [ibid., p. 229]

and since f = w / 2π

f* ~= c / 2π √ S/(V (l + 0.6a))

which is the familiar formula -- the one that wrongly gets used for ocarinas (a more complex network with flanged holes).

Edit: formatting

 

[EmilioL]

Here are the steps to extend the network analog to an ocarina:

Case 2. Piston-drive Helmholtz Resonator (simple conceptual model of fipple)
* Active: piston in base excites oscillation
* One neck
* No flange
* No listening hole

           .----.
          /      \
   ------'        `
                  ||
                  ||======   <-----> 
                  ||
   ------.        .
          \      /
           `----'

Case 3. Same, with flanged neck:

           .----.
   |      /      \
   `-----'        `
                  ||
                  ||======   <-----> 
                  ||
   .-----.        .
   |      \      /
           `----'

Same formula---different emprical constant for effective length.

Case 4. Simple Ocarina Model

* Active: piston in base excites oscillation
(replaced by fipple in actual ocarina)
* Two necks, possibly different lengths and diameters
-- fipple opening
-- finger hole
* Flanged
* No listening hole

           --. .--
             | |
           .-   -.
          /       \
   |     /         `
   `----`          ||
                   ||======   <-----> 
   .----.          ||
   |     \         .
          \       /
           `-----'

Network analog:

                 .-- Zpipe1 ---.
  .----> Zrad ---|             |---.
  |              `---Zpipe2 ---'   |
  |                                |
Source                             V
  ^                               Zcav
  |                                |
  |                                |
  `--------------------------------'

 

[EmilioL]

Any help solving the math of the above 2-pipe network would be greatly appreciated.