1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

5.7 Vibrations of a Circular Membrane 323
The rest of our problem can now be dispatched easily. Returning to Eq. (5),
we see that

Tn(t)=ancos(λnct)+bnsin(λnct),

and then for eachn= 1 , 2 ,...wehaveasolutionofEqs.(1),(2),and(8):

vn(r,t)=φn(r)Tn(t).

The most general linear combination of thevnwould be

v(r,t)=

∑∞

n= 1

J 0 (λnr)

[

ancos(λnct)+bnsin(λnct)

]

. (10)

The initial conditions Eqs. (3) and (4) are satisfied if

v(r, 0 )=

∑∞

n= 1

anJ 0 (λnr)=f(r), 0 <r<a,

∂v

∂t(r,^0 )=

∑∞

n= 1

bnλncJ 0 (λnr)=g(r), 0 <r<a.

As in the preceding section, the coefficients of these series are to be found
through the integral formulas

an=D^1

n

∫a

0

f(r)J 0 (λnr)rdr, bn=λ^1

ncDn

∫a

0

g(r)J 0 (λnr)rdr,

Dn=

∫a

0

[

J 0 (λnr)

] 2

rdr.

With the coefficients determined by these formulas, the function given in
Eq. (10) is the solution to the vibrating membrane problem that we started
with.

General Vibrations

Having seen the simplest case of the vibrations of a circular membrane, we
return to the more general case. The full problem was

1

r

∂

∂r

(

r∂u

∂r

)

+^1

r^2

∂^2 u

∂θ^2

=^1

c^2

∂^2 u

∂t^2

, 0 <r<a, 0 <t. (11)

u(a,θ,t)= 0 , 0 <t, (12)

∣∣

u( 0 ,θ,t)

∣∣

bounded, 0 <t, (13)

u(r,θ+ 2 π,t)=u(r,θ,t), 0 <r<a, 0 <t, (14)