HHDP: 16 – 03
Banjo Rim Height and Sound in the Pot
David Politzer ∗ (Dated: September 10, 2018)Rim and back geometry determine much of the behavior of sound inside the pot,whose effect on total, produced sound is subtle but discernible. The theory of soundinside a cylinder is reviewed and demonstrated. And previous work on the Helmholtzresonance and the interplay between the Helmholtz resonance and the lowest headmode is revisited using some improved techniques. ∗ a r X i v : . [ phy s i c s . pop - ph ] A ug Banjo Rim Height and Sound in the Pot
I. INTRODUCTION & OUTLINE
Some years ago, Joe Dickey offered a simple physics model of the banjo.[1] He consideredideal strings attached through a point mass to the center of an ideal drum head. With enoughapproximations and simplifications, such strings and drum head are soluble systems. Andthe model allows one to follow the action from an initial string pluck to the radiated sound.Another relevant system is the air motion inside the pot. With some simple approximations,it is also, by itself, a soluble system. However, there are two caveats. While its impact isrelevant to the concerns of builders and players, it is admittedly only a small piece of thetotal. And, perhaps more significantly, the coupling of the inside air to the head is strongbut not well understood. Internal air pressure variations make contact over the entire surfaceof the head, and that certainly effects how the head moves. But how that plays out has notbeen studied in any particular detail.This note is really just an addendum to an earlier work,
The Open Back of the Open-Back Banjo .[2] That was an investigation of the effects of rim height on air loading of thelowest frequency head motion and on the pot’s Helmholtz resonance. Here I compare thewell-understood calculation of sound resonances of cylindrical cavities to measurements onthose same three banjos, identical except for their rim heights (shown in FIG. 1). Again,admittedly, rim height relevance is somewhat indirect. The banjo’s sound comes overwhelm-ingly from the vibration of the head. The influence of the pot internal sound is throughits coupling to the head. That interaction is understood only qualitatively rather than indetail.Another caveat concerns the significant differences between the transient response due toa pluck and the steady-state response to continuous driving. Dickey’s strings and head andvirtually all discussions in the acoustics of musical instruments consider systems in terms oftheir steady-state response. Transients of coupled systems, even if they are linear, are morecomplex.[3] (For example, the modes of specific frequencies are, in general, not normal ororthogonal.) Nevertheless, for systems where the damping is weak, the steady-state resonantspectrum is a good starting point.
FIG. 1. shallow, standard, & deep
In the following, I give a verbal description of the sound resonances of cylindrical cavities.Rim height is identified as a crucial variable in determining the qualitative behavior of thespectrum. For the Helmholtz resonance, an essential feature is the air going in and out ofthe cavity. For the other cavity resonances, it is a huge simplification and a realistic ap-proximation to consider and measure a closed volume. Sound spectrum measurements withthree different rims are compared with each other and with the simple physics predictions.The previous Helmholtz resonance and head air-loading measurements[2] are repeated. Iattempt to explain the shortcomings of a ballistic picture of sound propagation.
II. CYLINDRICAL CAVITY AIR RESONANCES
For sound waves (unless we’re concerned with very narrow apertures), pressure is theonly relevant force. The equations of motion are standard undergraduate physics fare,and the cylinder solutions are among the simplest of three dimensional examples, after therectangular box. The relevant cylindrical coordinates are shown in FIG. 2. r i m z −r θ FIG. 2. defining directions and coordinates relative to the rim
The air pressure resonance solutions have specific frequencies and are products of a func-tion of r times a function of θ times a function of z . The z -dependence is the simplest andthe most relevant to the question of rim height effects. The z pressure function is sinusoidal,with maxima at the top and bottom of the cylinder. So this is a series of integer numbersof half waves that fit in the cylinder. Importantly, the series starts with zero. The lowestfrequency z contribution to the total pressure function is independent of z . So, for a squatcylinder, the several lowest resonances have oscillating pressures that are independent of z .The air motion at those resonant frequencies is purely in the r - θ plane. That also meansthat cylinders with the same diameter have the same resonant frequencies, independent oftheir z total dimension (rim height), at least until the first z -dependent resonance is reached.The lowest z -dependent resonance is independent of r and θ and is simply a single halfwave. So its wavelength is just twice the rim height, and its period is the time it takes forsound to bounce once back and forth from top to bottom to top. And the frequency is oneover the period.It is particularly noteworthy how the r , θ , and z motions combine to produce resonant fre-quencies above that lowest z -dependent resonance. In particular, for the combined motions,the frequencies of the separate factors “add in quadrature.,” i.e., we take the square rootof the sum of the squares — like finding the hypotenuse of a right triangle. In particular,there is a series of frequencies that involve motion in the r - θ plane that are independent ofrim height. Above the first z -dependent resonance, we multiply that z -dependence with theseries of r - θ resonance functions to get the total pressure dependence. The frequency of theproduct is the two separate frequencies added in quadrature. Again for the squat cylinder,there are many z -independent resonances before the first z mode appears. When that modeis “dressed” with the possible r and θ dependences, the resulting sequence is much closerspaced in frequency than the original z -independent series — at least in cases where thelowest z frequency is much higher than the r - θ frequencies in question. FIG. 3. Speaker and microphone mounted internally on a solid plywood head and a second plywooddisk to close the back; also the head-speaker combination used for FIG.s 6 and 9
Measurements were carried out on three Goodtime rims.[4] Their heights were 2 . (cid:48)(cid:48) ,2 . (cid:48)(cid:48) , and 5 . (cid:48)(cid:48) .[5] Acoustical investigations of guitars and violins have sometimes gone togreat lengths to decouple the soundboard, side, and back vibrations from the vibrations ofthe air inside. For example, the whole instrument might be buried in sand. For a banjo, it’smuch easier. I simply replaced the head with 3 / (cid:48)(cid:48) plywood, and attached another plywooddisk to the back. The sound was excited by a 3 / (cid:48)(cid:48) speaker mounted inside and recorded bya small microphone, also mounted inside. The speaker and microphone were diametricallyopposite, just under the head, as shown in FIG. 3. (Their locations and physical extent limitwhat resonances can be detected; for example, you cannot detect a mode if either speakeror mic lie on a node line.) I used a signal generator and audio amplifier to sweep linearly infrequency from 200 to 4000 Hz. And the signal was recorded and analyzed using Audacity R (cid:13) . !80$!70$!60$!50$!40$!30$!20$ 0$ 500$ 1000$ 1500$ 2000$ 2500$ 3000$ 3500$ 4000$ d B $ !! > $ Hz$!!>$ shallow$standard$deep$
FIG. 4. Measured spectra of the three rims; lines at the bottom are values computed from thediameters and heights; the black lines are the resonances, common to all, that have no variation inthe z direction. In FIG. 4, curves for the three rims are labeled shallow, standard, and deep for thethree rim heights, respectively. The straight lines at the bottom of the graph indicate thecalculated values of the resonant frequencies. The black lines are the z -independent r - θ modes for cylinders of internal diameter 9 . (cid:48)(cid:48) . (The three Goodtime inner diameters werewithin 0.6% of each other.) For each rim, there is a lowest frequency corresponding to ahalf-wave in z . Those frequencies are calculated to be 3322 Hz, 2421 Hz, and 1192 Hz,for shallow, standard, and deep, respectively. Those and the additional r - θ frequenciescombined in quadrature give the sequences of color-coded lines. Note that under 4000 Hz,the deep pot is the only one which has a second series starting at the z mode correspondingto two half wavelengths within the pot. That one starts at 2384 Hz.There are at least three noteworthy features, which I’ll discuss before addressing somecaveats and limitations below. First, the calculated values match the tall peaks in themeasurement. Impressive or not, that’s what physics is supposed to do. Second, for eachrim, at the lowest z resonance and above, the resonances are closer together than in theabsence of that z mode. And, of course, its location is a simple function of rim height.These resonances interact with the head and influence how well the head can convert stringvibrations into sound. So the closer internal cylinder resonances should give a more evenresponse. And third and somewhat more subtle, there is a step up in overall response (likelyrelated to the resonances being closer together in frequency) above the first z resonance. Infact, the deep pot exhibits a second step.The lowest calculated resonance is at 808 Hz, and, indeed, all three pots show a fine peakvery close to that value. The stuff below 808 Hz (albeit not vey loud) is a clear indicationthat there are other things vibrating besides the air in the cylinder. In fact, the observationsbetween 200 and 800 Hz varied from run to run, but I ran out of patience trying to trackdown every origin of the variability. The heavy plywood and firm bolting of the head werelikely not the culprits. But the double-sticky foam tape and masking tape mounting of thespeakers, microphones, and wires might not be as reproducible as some properly machinedapparatus. Similar small variability was also observed across the whole frequency range,likely of the same origin. However, the prominent, high peaks were always identifiable atnearly the same frequencies.The strength of a resonance in a system of multiple parts depends on two things. First,how well does the driving match the geometry of the resonance? To get a strong response,you have to push in the right place. Pushing in the wrong way might not get any response atall. And second, the effectiveness of a push of a fixed frequency depends on how close thatfrequency is to the resonant frequency. Both of these must be kept in mind when addressingthe real question of interest of does string vibration turn into sound. III. HELMHOLTZ AND HEAD RESONANCES
The theory here is crude but simple. For pots that differ only by their rim height, theHelmholtz frequency should be inversely proportional to the square root of the height. Thelowest head mode couples strongly to the Helmholtz mode. In the absence of that coupling,its frequency would be inversely proportional to the square root of an increasing, linearfunction of the height. (See ref. [2].)The following is an attempt to identify the lowest frequency head modes. All three headswere set to the same tension as determined by a DrumDial (at 89). The heads were tappedat their centers with a piano hammer. The sound of a long series of taps was recorded witha microphone at 12 (cid:48)(cid:48) in front of the center of the head. The (open) backs were left wideopen. Hence, there is substantially less air springiness than would be provided by a moreenclosed volume. Increased pot depth increases the “air loading.” That is usually thoughtof as the air that the head has to move if it moves. At a minimum, the effect is to increasethe effective inertia or mass of the head. Because heads are very thin, this is a much biggerdeal than it is for the soundboard of wood-topped instruments. An effect in the expecteddirection is evident in FIG. 5. However, interpreting these peak locations in the context ofthe head interaction with the Helmholtz resonance will prove problematic. !65$!60$!55$!50$!45$!40$!35$!30$!25$!20$!15$100$ 1000$ d B $ !! > $ Head%Taps%with%wide%open%back% shallow$standard$deep$
FIG. 5. The sound of head taps with wide open backs, recorded in front
The Helmholtz resonances can be decoupled from the head by using a plywood head.A 2 1 / (cid:48)(cid:48) speaker is mounted at the center in the head rather than on it. In particular,the diaphragm of the speaker forms part of the pot outer wall. Its motion compresses andexpands the air inside, which is precisely the Helmholtz resonance motion. In contrast, the3 / (cid:48)(cid:48) speakers mounted inside the sealed cylinders act as wigglers, producing both compres-sion and expansion (at slightly different places) inside the cylinder. (Both are pictured inFIG. 3.) !75$!65$!55$!45$!35$!25$!15$100$ 1000$ d B $$ !! > $ Speaker/Plywood.Head.00.angled.foam.or.wood.back. shallow$!$foam$standard$!$foam$deep$!$foam$shallow$!$wood$standard$!$wood$deep$!$wood$
FIG. 6. Helmholtz resonances excited by speaker in head with foam belly or wood back
For these measurements, the pots have backs that simulate open-back playing.[2] Inparticular, there are runs with the foam-cork-Hawaiian shirt synthetic belly and runs witha plywood back. Thin lines in FIG. 6 correspond to the foam back, and thick lines are forthe wood back. For both backs, the sound hole was defined by a 3 / (cid:48)(cid:48) spacer placed at onepoint between the back and the rim, with the back touching the rim diametrically opposite.This arrangement was the closest reproducible set-up I found to natural open-back playing,where the player’s body is the back.[2] The microphone was placed at 2 (cid:48)(cid:48) from the sound-holeopening in the back. The foam backs seem to smear out some of the detailed features thatare present with the wood backs. It is easy to imagine that the foam flexes a bit and absorbs,while the wood reflects better but might rattle.Finally, I examine the combination of a standard head and a normal (albeit synthetic)back. Again, a long series of center head taps with a piano hammer are recorded. In thiscase, the back is the foam-cork-Hawaiian shirt combination, spaced and angled as describedabove. For each of the three pots, FIG. 7 displays the spectra for two different microphonelocations. The thick lines are the result of mic placement at 2 (cid:48)(cid:48) from the sound hole in back.That emphasizes the sound of the Helmholtz resonance and head motions that require a netmovement of air in and out of the pot. The thin lines are for mic placement at 20 (cid:48)(cid:48) in frontof the head. That is closer to how the banjo is played and heard. The role of the in-and-out0 !75$!65$!55$!45$!35$!25$100$ 1000$ d B $$ !! > $ Head%Taps%with%Foam%Belly%Back% shallow$!$by$soundhole$standard$!$by$sound$hole$deep$!$by$soundhole$shallow$!$in$front$standard$!$in$front$deep$!$in$front$
FIG. 7. Head taps with the synthetic belly back and different mic positions air motion is still quite evident, but its magnitude in the signal is reduced relative to theother features of the sound.FIG. 7 offers an example of how the physics works in these situations. The idealizedversion of a drum tap on the head should be able to excite all resonances present — at leastto the extent that the tap is not near a node of that resonance. One example of this standsout. In addition to the low-lying resonances whose frequencies are rim-height dependent,there is clear evidence in FIG. 7 of a strong resonance which is nearly the same for allpots. That’s the one between 800 and 900 Hz. The obvious interpretation is that theseare the lowest “closed” cylinder resonances. As long as the taps were not all exactly at thecenter (which is on a pressure node line for those lowest cylinder resonances), they shouldbe excited to some extent. Not only are they rim-height independent as expected, they areclearly there when the pot is not sealed and when the head is allowed to vibrate. And thatis why the calculation and measurement of the sealed cylinders is not irrelevant.FIG. 7 suggests the presence of modes between 500 and 600 Hz as well as those around200. So I tried yet another hardware approach to explore this region. Classic studies ofguitar acoustics offer a very satisfactory picture of the Helmholtz resonance mixing with thelowest sound board resonance to produce two distinct combinations.[6] The spectrum hastwo resonances whose frequencies are functions of what would have been separate Helmholtzand sound board modes. In a nice bit of elementary physics, when the two interact, they1
FIG. 8. Set-up for one of the traces in FIG. 9: shallow pot; plywood/speaker back, angled with a3 / (cid:48)(cid:48) spacer; mic 2 (cid:48)(cid:48) from the center of the head combine to give two distinct combinations with two new frequencies whose sum of squaresis the same as the uncoupled case.In an attempt to get a clearer picture of that frequency region, I tried the following,shown in FIG. 8. I re-mounted the identical heads, tensioned again to 89 on a DrumDial,on the three rims. For backs, I used the 3 / (cid:48)(cid:48) plywood disks with 2 1 / (cid:48)(cid:48) speakers mountedin their centers. The sound hole was defined by the 3 / (cid:48)(cid:48) spacer at one point along the rimbottom, with the rim and back in contact at the diametrical opposite. And I recorded inthree different locations: 2 (cid:48)(cid:48) from the sound hole spacer; in front of the head at 2 (cid:48)(cid:48) fromthe center; and at 20 (cid:48)(cid:48) from the head center. (For the farther distance, the power to thespeaker was increased by a factor of 100 to get a recorded sound of comparable strength asthe others.)The results are plotted in FIG. 9. The clear similarity with FIG. 7 of the high amplitudefeatures is actually evidence in support of the head-tap method. From FIG. 9, in spite ofall the apparent wiggling, I conclude that there are, indeed, three relevant resonances foreach of the pots below 1000 Hz. The lowest ones are between 150 and 200 Hz; the next arebetween 425 and 575 Hz; and the highest are around 900 Hz. Theory suggests that these2 !65$!60$!55$!50$!45$!40$!35$!30$!25$!20$!15$100$ 1000$ d B $$ !! > $ mylar&head&&&plywood/speaker1drive&back& shallow$!$in$front$shallow$!$at$spacer$shallow$!$20$in$standard$!$in$front$standard$!$at$spacer$standard$!$20$in$deep$!$in$front$deep$!$at$spacer$deep$!$20$in$ FIG. 9. Response of the three rims to speaker in 3 / (cid:48)(cid:48) plywood back, with mic in three positions two lower resonances are both combinations of the Helmholtz and lowest head resonances.They decrease in frequency with increasing rim height. The highest is the same for all rimsand is the lowest “closed” cylinder mode. The subtle details give additional support to theseidentifications.The 900 Hz resonance appears with the microphone in front and not particularly whenthe mic is close “at the spacer,” i.e., at the sound hole. That makes sense because the lowestclosed-cylinder resonance involves air motion in the r - θ plane, moving from side to side. Thepressure is higher on one side than the other, alternating back and forth at ∼
900 Hz. Thosepressure variations push on the head and contribute to its up and down motion. However,there’s little reason for much air to venture out through the sound hole. In contrast, thelowest head mode pushes air in and out the sound hole. And similar air motion is a definingpart of the Helmholtz resonance. What happens typically[6] is that the lower frequencycombined motion occurs with both head and air volume pushing and pulling in the samedirection at the same time. In that case, there is air motion in and out of the sound hole andvibration of the enter of the head. The typical higher frequency combined motion has thetwo effects opposing each other. The head still moves and makes sound but the net motionat the sound hole is far smaller — because the head is pushing it one way and the internalair is pushing it the other way. In the measured sound, the lowest resonance is comparablyvisible in front and at the sound hole. The second higher resonance is much stronger in front3than at the sound hole.Recording in a modest size room definitely produces wiggles in measurements of this sort.The frequency of the speaker is swept very slowly through some range. There is always aroom resonance very near to the driving frequency. That sets up standing waves with nodalplanes. As the frequency slowly shifts, those planes move about. In particular, they passthrough the fixed location of the microphone. So the sound volume recorded at the micgoes up and down, even without any appreciable change in the whole-room average of thesound volume. This was particularly noticeable when I took the recordings, standing a fewfeet from the microphone. A display of the microphone voltage showed amplitude variationsthat were not particularly in sync with the variations I heard. Also, a slight motion of myown head could make the sound dramatically louder or softer. This would not happen in ananechoic chamber. One can also eliminate these effects my recording outdoors in a big fieldwith the microphone on the ground. Neither of those were practical, available alternatives.One further check of this mixing interpretation for the two lowest resonances is a com-parison of pots at different head tensions but the same rim height. (I chose the deep rimbecause the wood speaker back did not have to be removed to change the head tension.) Istarted with the carefully prepared 89 on the DrumDial by each hook, then did a run withall nuts tightened by 360 o , then one with all loosened by 180 o from the original 89 setting,and then loosened another 180 o . I did not take care to even out the tensions at these new !65$!60$!55$!50$!45$!40$!35$!30$!25$!20$!15$100$ 1000$ d B $$ !! > $ Mylar&Head,&Wood/Speaker&Back,&Mic&in&Front&
DrumDial$=$75$DrumDial$=$82$DrumDial$=$89$DrumDial$=$91$
FIG. 10. Different head tensions labeled by approx. DrumDial reading settings, but the approximate DrumDial readings were 91, the original 89, 82, and 75. (914is bright bluegrass tight, and any higher gets into the realm of “tighten ’till it breaks andthen back off a quarter turn.”) All tensions produced the same distinction between mic infront versus mic at the sound hole: The lowest frequency peaks were of similar strength atboth mic locations, while the second peaks, i.e., between 300 and 500Hz, were much weakerwith mic at the sound hole than ain from of the head. I only display the mic-in-front resultsin FIG. 10. Also, the FIG. 10 frequency resolution is double that of FIG. 9. That makesthe positions of the highest peaks clearer, but it also reports some of the jitters that areartifacts of room sound.The frequencies of the two lowest peaks increase with increased head tension. Thatmeans head motion is significant for both of them. On the other hand, there is a Helmholtzresonance in this region. Taken by itself, its frequency is independent of the head tension.So the simplest interpretation is that there is also only one head resonance in this region, butthe actual resonant modes are (orthogonal, linear) combinations of the head and Helmholtzmodes, with the resulting mic location dependence as discussed above.In viewing and interpreting FIG.s 9 and 10, it’s important to remember what’s goingon. These banjos are systems with a great many parts. The challenge is to understandimportant aspects of their performance by identifying a small number of crucial featuresand parts and developing a simple picture of how some idealization of those parts wouldbehave. Success is reckoned by how well a simple, understandable model represents andreproduces the observed features of the real systems. At least, that’s what interests me.
IV. WHY RESONANT MODES AND NOT BALLISTIC PROPAGATION — ORWAVES AND NOT PARTICLES
Under normal circumstances, we do not see sound waves. Nevertheless, people oftenimagine what’s going on. “The sound goes here, bounces off, and then goes there,” issometimes said, as if describing a stream of bullets or rubber balls. With somewhat greatersophistication, focusing, analogously to light and curved mirrors, is added to the description.Indeed, in the case of light, lenses and mirrors can often be well-described by ray optics, inwhich a bundle of bullet-like ray trajectories are analyzed and combined. There certainly arecircumstances where sound behaves quite analogously — but not in the case of the internalworkings of musical instruments themselves. For waves to act like particles and rays, their5wavelengths have to be small compared to the other lengths of interest. In optics, thosemuch longer sizes might be the dimensions and curvature of the mirrors. Musical soundshave wavelengths that start around 11 (cid:48) (around 100 Hz) and go to 1 3 / (cid:48)(cid:48) (around 10,000Hz). The physical features of musical instruments are not all much bigger, even for thehighest pitches.One basic aspect that distinguishes waves from particles is how they combine. With twosources of particles, the particles add. With two sources of waves, the waves combine sothat at some places and times they might completely cancel, while at other places and timestheir combined intensity (loudness) is greater than the sum of the two. There are ways tobuild up wave physics from trajectories, but those trajectories have to be combined with thisaddition/cancellation aspect strictly respected. There is certainly legitimacy to the notionthat sound bounces in and around an instrument. However, there are always many bouncesfor a musical sound, even if it dies off relatively quickly. In most optics situations, the lightbounces just once off a given mirror. (There are high tech and laboratory situations wheremultiple bounces are relevant. In such cases, considering the entire light field all at once isusually more effective than trying to combine the successive bounces.) There are acousticssituations where just one bounce is the dominant effect. But the wave aspects emerge asrelevant when combining waves, even with just a couple of overlapping waves. Musicalsounds almost invariably involve many reflections. Strictly speaking, the perspective offeredby resonances rests on an assumption of infinitely many reflections and assumes a steady-state situation. There are subtle issues for which that is misleading[3], but it is generally agood starting point. V. CONCLUSION
Everything works the way it should. To get to the produced sound, there are certainly alot of details and quantitative connections that are far beyond simple physics. But severalpieces of the puzzle have been examined here, and they behave as expected. Increasing thepot depth lowers the frequency of the lowest vibrations that the banjo can produce. Thatwas discussed in an earlier paper.[2] The additional perspective offered here is how the potdepth effects the whole spectrum of response. Missing is a detailed picture of how the airresonances talk back to the head. That’s really a good question.6 → PHYSICS → LAY LANGUAGE PAPER ON THEDYNAMICS OF THE BANJO:
The Banjo: the Model Instrument ; The structural dynamics ofthe American five-string banjo , J. Acoust. Soc. Am., (5) 11/2003, p 2958[2] D. Politzer,
The Open Back of the Open-Back Banjo ~ politzer . That paper gives a personal account of the whole adventureand, perhaps more importantly, includes sound files of the same tune being played on the threedifferent banjos.[3] D. Politzer, The plucked string: an example of non-normal dynamics , HDP: 14 – 04,
Am. J.Phys. ~ politzer; Zany strings andfinicky banjo bridges ~ politzer[4] I originally chose Deering Goodtime banjos for my acoustics investigations because 1) they areabout as identical as wood objects can be, being a combination of CNC fabrication and highquality hand finishing; 2) they are quality instruments; and 3) they are relatively inexpensive.When I fist approached Greg Deering, requesting some special items and perhaps a deal on theprice, he immediately offered to provide me with whatever I needed. He has been of great helpever since, including advice and fabrication.[5] 2 . (cid:48)(cid:48) is the standard Goodtime rim and typical of today’s banjos. No one plays a 2 (cid:48)(cid:48) rim. Theysound too thin. I know of only two 5 5 / (cid:48)(cid:48) rim banjos in existence, and neither gets playedregularly. :( The role of the deep pot in this study was to get a range of sound measurementsthat clearly tested the physics ideas.[6] The two coupled oscillator model of the guitar lowest modes and frequency sum rule are re-viewed in The Physics of Musical Instruments , N. H. Fletcher and T. D. Rossing, § The Science of String Instruments , T. D. Rossing ed., §§