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});

>