How does the head of a circular drum vibrate? Are the vibrations like water waves?
The wave equation
Unlike water waves, to a first order approximation the vibrations of a drum head can be considered as purely vertical perturbations (transverse to the plane of the drum head). As such, they can be shown to satisfy the wave equation for the height u of a vibration:
∂2u∂t2= c2 ∇2u
The term ∂2u∂t2 is the vertical acceleration of a point on the membrane; the Laplacian ∇2u is a second order derivative which measures the concavity of the drumhead at a given point, and c is the propagation speed of waves on the drum. The only boundary condition is that the height of the vibration at the edge of the drum is always 0.
Solution via separation of variables
Because the wave equation is linear, the sum of any number of solutions is also a solution. It is therefore possible to look for a basis for the solution set, and express any solution as a (possibly infinite) linear combination of elements of the basis. A convenient choice for the form of the basis elements is as a product of functions, each of which depends on just one of the coordinates. Choosing a cylindrical coordinate system for symmetry, we find solutions, called modes, of the form:
un, kr, θ,t = An, k⋅Jnλn, k ra⋅sinn⋅θ−ϑn,k⋅sinλn, k⋅c⋅t−τn,k,
where Jn is the n th order Bessel function (of first kind), λn,k is its k th root, a is the radius of the drum head, and An, k , ϑn, k , τn, k are arbitrary constants depending on n and k. In fact the general solution can be expressed as an infinite sum of such functions:
u = ∑n=0∞ ∑k=1∞un, kr, θ,t.
Modes of vibration
Each vibration mode un,k is characterized by:
n - the number of nodal diameters (diameters of the drum head having constant 0 vibration height) at θ=ϑn,k + j⋅πn, j=0.. n−1
k - the number of nodal circles (circles concentric with the drum head having constant 0 vibration height) at r =a⋅λn,jλn,k, j = 1.. k.
Adjust the numbers of nodal diameters and circles to see what the vibration modes look like. You can adjust the amplitude to switch between a more realistic, and a more exaggerated view. Press "Play" to start the animation, or adjust the time manually with the slider. The checkbox allows you to display or hide the nodal circles and diameters.
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