Standing Waves in a Drum Membrane
Standing waves in a drum membrane are complicated and satisfactory analysis requires knowledge of Bessel functions. In contrast to standing waves in one dimensional media such as waves in a string, sound waves in a pipe, etc., the resonance frequencies of membrane standing waves are not integer multiples of the lowest order mode. Consider a circular membrane of radius a (m) and surface mass density
is subject to a surface tension of Ts (N/m). The velocity of transverse waves on the membrane is given by
The lowest resonance frequency is
where the numerical factor 2.40483 is the first root of the zero-th order Bessel function J0 (x). The entire membrane oscillates up and own in this lowest order standing wave as seen in the first animation. The second root of J0(x) = 0 is x2 = 5.52008 and the oscillation frequency is
The oscillation pattern is shown in the second animation. The second lowest resonance frequency occurs at
where 3.83171 is the first root of the first order Bessel function J1 (x) = 0. (The third animation.)

>

> with(plots):
animate3d([r,theta,BesselJ(0,2.40483*r)*sin(2*Pi*t)],r=0..1,theta=0..2*Pi,t=0..0.95,frames=20,coords=cylindrical);

> with(plots):
animate3d([r,theta,.5*BesselJ(0,5.52008*r)*sin(2*Pi*t)],r=0..1,theta=0..2*Pi,t=0..0.95,frames=20,coords=cylindrical);

> with(plots):
animate3d([r,theta,BesselJ(1,3.8317*r)*sin(theta)*sin(2*Pi*t)],r=0..1,theta=0..2*Pi,t=0..0.95,frames=20,coords=cylindrical);

> with(plots):
plot(BesselJ(0,x),x=0..10,color=red);

> plot(BesselJ(1,x),x=0..10,color=blue);

>