MATH 373 -- Applied Mathematics (PDE)
April 24, 2001
Fundamental (or Normal) Modes of Vibration for the circular drumhead
We have seen that the fundamental modes of vibration for the circular
drumhead have the following forms, four for each pair , with m >= 0,
and n >= 1:
a)
b)
c)
d)
Here is the Bessel function of the first kind of order m, the
is the n th strictly positive zero of , and r = a is the
outer edge of the drumhead which is held at zero displacement for
all .
The basic shapes of all of these are similar, so we will only examine the
"cos-cos" form. We take c = 1 and a = 1 for convenience:
> FundMode:=(m,n,r,theta,t) -> [r*cos(theta),r*sin(theta),BesselJ(m,BesselJZeros(m,n)*r)*cos(m*theta)*cos(BesselJZeros(m,n)*t)];
> with(plots):
First examples: m = 1, n = 1, 2, 3 and m = 2, n = 2
For instance, here are animations of the m = 1, n = 1, 2, 3 fundamental modes through one complete
period:
> period:=(m,n)->2*Pi/BesselJZeros(m,n);
> animate3d(FundMode(1,1,r,theta,t),r=0..1,theta=0..2*Pi,t=0..period(1,1));
> animate3d(FundMode(1,2,r,theta,t),r=0..1,theta=0..2*Pi,t=0..period(1,2));
> animate3d(FundMode(1,3,r,theta,t),r=0..1,theta=0..2*Pi,t=0..period(1,3));
> animate3d(FundMode(2,2,r,theta,t),r=0..1,theta=0..2*Pi,t=0..period(2,2));
The general pattern
In general, we can describe the ( m, n ) fundamental mode in the following way: The circular
drumhead is divided into 2 m angular sectors by the zeroes of , and n "concentric rings"
by the zeroes of (so regions in all) which oscillate individually . The
divisions between the rings are the circles r = for j = 1, 2, ... , n-1, where the factor
has zeroes. These are known as nodal lines or curves of the fundamental mode --
the displacement stays zero for all time along a nodal line. The divisions between the angular
sectors are also nodal lines For instance, here is the (2,3) fundamental mode, with the nodal
lines marked.
> nm:=animate3d(FundMode(2,3,r,theta,t),r=0..1,theta=0..2*Pi,t=0..period(2,3)):
> nodallines:=plot3d({[BesselJZeros(2,1)/BesselJZeros(2,3)*cos(theta),BesselJZeros(2,1)/BesselJZeros(2,3)*sin(theta),0.01],[BesselJZeros(2,2)/BesselJZeros(2,3)*cos(theta),BesselJZeros(2,2)/BesselJZeros(2,3)*sin(theta),0.01],[r*cos(Pi/4),r*sin(Pi/4),0.01],[r*cos(3*Pi/4),r*sin(3*Pi/4),0.01],[r*cos(5*Pi/4),r*sin(5*Pi/4),0.01],[r*cos(7*Pi/4),r*sin(7*Pi/4),0.01]},r=0..1,theta=0..2*Pi,thickness=3,color=red):
> display3d({nodallines,nm});
>