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Homework Help: Resonant frequency of drums

  1. Jan 30, 2010 #1
    1. The problem statement, all variables and given/known data


    I'm new to this forum. I'm a musician and recording studio owner. I'm not good at physics although I read "The Physics Of Music" and I am interested when in physics when it concerns musical instruments in particular. What I would like to know is a formula to determine the resonant frequency of the individual drums of a drum set. The drums are cylinders made of wood except for the snare which is metal. There are heads on each end of the drums and a small vent hole about the circumference of a pencil. I'm not sure if these facts matter or not. The purpose is to get the best sound for each drum when tuning them.

    The size of the drums are:

    12" diameter by 8" deep hi tom
    14" diameter by 14" deep low tom
    20" diameter by 14" deep kick drum
    14" diameter by 4 1/2" deep snare drum

    Your help in this matter will be greatly appreciated. Thank You!



    2. Relevant equations

    I don't know any!

    3. The attempt at a solution

    I do it by ear now!
  2. jcsd
  3. Jan 30, 2010 #2
    I think you would need to solve the wave equation on a circular domain, and find the different vibrational modes. It is not easy to do. I think the resonance frequencies are given as zeros of some Bessel functions. If you are emathematically inclided, I think it is described here (allthough I'm not sure how well. Just search for the word "drum"):

    However, this is only based on an simplified approximation that ignores the influence of the rest of the drum and the air inside it/around it. The usual simplification in this case is that the skin of the drums is vibrating and fixed completely on its circular edge. Also, they assume that the vibrational amplitude is so small that the wave-equation is a valid approximation. Also that the skin of the drum has a certain tension that is exactly the same everywhere.

    That's probably the most accurate way to do it! :-)

    Maybe it would be possible to find a formula for the frequency of the first vibrational mode for a simplified drum of each size. But there is a big problem. In real life you don't know the tension of the skins on the drums, right? And you won't be able to calculate the frequency without knowing it...

  4. Jan 30, 2010 #3
    What it more interesting to me is your statement about finding the best sound in the drums. That would depend on the different overtones in the vibrations, and how fast the sound builds up and decays. However, you probably don't have a lot of adjustable parameters, apart from the two skin tensions, maybe the size of the vent hole and so on.

    If you think that the drums resonate too much, e.g. their sound doesn't die out quickly enough, then maybe you could try an uneven tension on the skin, and/or doing something asymmetrically to the skin that is on the underside, e.g. taping something to it on one side, etc.

    If you want a different type of resonance, maybe you could close the vent hole.

    The interesting things about drums is that the overtones are not integer multiples of each other. So is is not a completely "melodic" instrument, because the sound doesn't really define any particular note (although this probably depends a lot on the listener). It does of course to some degree because e.g. the funamental frequency is the strongest. Some drums change their "perceived note" as the sound dies out, since after a while, other overtones that are not an integer multiple of the fundamental can turn out to be louder.

  5. Jan 30, 2010 #4
    One can solve this problem with some effort with partial differential equation and is usually a typical school-book example. The vibrating circular membrane satisfies the following partial differential equation:

    [tex]\nabla^{2}U-\frac{1}{v^2}\frac{\partial^2 U}{\partial t^2}=0[/tex],

    where U is the function describing the surface at any moment and v is the speed of sound. Typically, when you solve this equation you will get a sum of contributions from many frequencies. The solution depends on the radius and involves a mathematical creature that goes by the name of Bessel function. The modes of oscillation can be found by looking at the zeros of these functions.

    [tex] J_{m}(ka) = 0[/tex]

    where [tex] J_m [/tex] is the m:th Bessel function and a is the radius of the drum. The variable k is the one giving the frequency.

    [tex] k=2\pi \frac{f}{v}[/tex].

    Thus if you know the zeros of [tex] J_m [/tex] it's easy to find the frequency. For this particular problem you might be interested only in the main mode of oscillation, i.e. the one with highest amplitude. This means you want to know the zeros of [tex] J_0[/tex]. The zero is 2.4048. You have to solve

    [tex] 2.4048 = 2\pi\frac{fa}{v}[/tex]

    for the varius radii of the drum. For instance the drum with 12" diameter (=6" radius =0.1524 meters), gives

    [tex] f=\frac{2.4048v}{2\pi a} =\frac{2.4048*343.3}{2*3.14*0.1524} = 863 Hz[/tex].

    If you find it easier you can work out the speed of sound (= 343.3 m/s) to inches per second instead and the you can use your values of the radii in inches.
  6. Jan 30, 2010 #5
    But you have used the speed of sound in air, where you should have used the speed at which waves on the skin of the drum travels? And that number depends on the skin tension.

  7. Jan 30, 2010 #6
    Yes, that's true. Didn't think of that.
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