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I just learned in the context of a PDE course how the various modes of vibration arise as solutions to the 2D Wave Equation. I am curious as to how many of these describe the motion of actual drums.

Clearly the size and shape of the drum matter as far as the mathematical solutions are concerned. However, I would expect some qualitative invariance, for example, that the modes of one circular drum would be a scaled version of the modes of another circular drum of a different size. Can a single, circular drum be made to vibrate in any of the modes shown at the above link, depending on "performance" factors such as the location where the drum is struck or the ambient conditions, or is the vibration behavior restricted by external factors such as the type/material composition of the drum?

Also, I am assuming from the fact that there was a debate on whether or not the shape of a drum can be heard that the actual audible difference between the various modes is not immediately obvious. Is this assumption correct? Is the difference in sound extremely subtle, or do producers manipulate the various vibration mode behaviors when synthesizing drums to create unique sounds?

user82425
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  • The existing answers are good enough I can only think of one thing to clarify: yes real drum heads exhibit many vibrational modes, but it’s pretty much impossible for them to vibrate with only a single mode at a time. Also the resulting complexity of the simultaneous modes of vibration is important in the character of drum sounds and a large part of why most drums are I pitched or weakly pitched. – Todd Wilcox Oct 06 '21 at 23:06
  • @ToddWilcox Does this include digitally? So it is pretty much impossible to "hear the different modes"? – user82425 Oct 07 '21 at 16:33
  • This is probably a better fit on the Physics stack. – J... Oct 07 '21 at 19:15

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The math gets a bit beyond me, but I can address one question: Many instruments have not only a vibrating solid part (membrane, string, reed) but a body of air contained within a "vessel." In the case of a frame drum, this body is near non-existent, but for a djembe or tympani, the dimensions of the body matter a lot. Consider Hemlholtz's resonator vessels; simply by their proportions they isolate a frequency out of unpitched sound. All pitched instruments with "bodies" benefit by "tuning" their proportions to support their resonant needs. If you were to take a membrane of equal diameter and tension and put it on two cylindrical "tom" drums, one double the height of the other... then I'm not sure whether you'd get a pitch one octave lower, but you'd definitely get a difference.

Another question: Can you get different modes of vibration by striking in different places on the drumhead? Yes, absolutely. Compare the tones produced at this point in this frame-drum how-to, by striking near the frame:

... to the tone around the 2:00 mark. This is true of other vibrating bodies as well (a string struck or bowed near its fixed end will be very rich in upper partials).

Andy Bonner
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To my understanding the calculations presented in the Wikipedia page are done under assumption the membrane is infinitely thin and elastic, thus the vibration occur only because the membrane is stretched on the rim. Bending of a real membrane results in additional forces and dissipation of energy. This may affect the frequencies of the modes, and even more their relative amplitudes. This article: https://www.sweetwater.com/insync/choose-best-drumheads/ seems to confirm this: thicker or coated membranes sound less bright, as the energy of higher modes is absorbed.

Following, if a membrane made of the same material is stretched over a smaller rim, the radius of bending of the membrane will be smaller, thus the effect will be larger.

user1079505
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  • This is correct -- the standard physics joke of "first assume a spherical cow with a uniform distribution of milk" applies here. The closer you get to a massless membrane, and the farther you stay from high-amplitude oscillations (which run into elasticity limits), the closer you can get to the simple-model set of Zernicke polynomials – Carl Witthoft Oct 07 '21 at 17:08
  • I think the plastic drumheads used on most modern drumsets are to pretty good approximation bending-ideal though, and the reason coated heads sound different has more to do with the way they interact with the sticks than with their vibration characteristics per se. For natural hide heads it's a different story. – leftaroundabout Oct 07 '21 at 17:24
  • @leftaroundabout right, Fletcher and Rossing write the effect (frequency shift) is ~0.5% and thus negligible for timpani... but does it hold true for smaller drums? Also I'm thinking high frequency damping might be a more audible effect in drums than small frequency shifts which can be easily compensated by tuning anyway – user1079505 Oct 07 '21 at 20:00
  • @CarlWitthoft an ideal membrane isn't massless. It's stiffness what I'm discussing. – user1079505 Oct 07 '21 at 20:05
  • @user1079505 in first-order modeling, it's massless. "Stiffness" is actually the elasticity coefficient -- you don't want to split the membrane via shear force! – Carl Witthoft Oct 08 '21 at 14:14
  • @CarlWitthoft the equations in wikipedia specify density, thus mass. Concerning stiffness, I think you confuse two parameters: ability to stretch, and ability to bend – it's the second one I mention. – user1079505 Oct 08 '21 at 15:54
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I don't know if this qualifies as an answer, but I have some pointers to places that might help...

In the 1980's I briefly talked with Thomas Rossing, at NIU (Northern Illinois University), where he was studying the vibration modes of drums.

His books: "Science of Percussion Instruments", and "The Physics of Musical Instruments".

https://ccrma.stanford.edu/people/thomas-rossing

I think that some of the guys at Bell Labs also had a book on the physics of instruments, or something like that. Max Matthews, maybe?

Also, William Sethares has an interesting book "Tuning, Timbre, Spectrum, Scale". He does some analysis of drums (and rocks!) He ties together the spectrum of an instrument with the musical intervals which leads to the scale used in the music.

https://sethares.engr.wisc.edu

aMike
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