Mathematical Physics Vol 1

Chapter 7. Partial differential equations

420

and b n

4 n 2 π 2

b n = c n −

q n .

The final solution is of the form

∞ ∑ n = 1

q n + c n −

q n e −

2 t sin

n π x 2

4 n 2 π 2

4 n 2 π 2

π 2 )

( n

u ( x , t )=

(7.370)

where q n and c n are determined by (7.365) and (7.368), respectively.

Problem 274 Find the equation for the motion of points on a circular membrane.

Solution In this case we will express the Delta operator in terms of polar coordinates, and thus the equation (7.13) becomes

∂ 2 u ∂ r 2

∂ 2 u ∂ϕ 2

∂ u ∂ r

1 r

1 r 2

∆ u =

= 0 .

(7.371)

+

+

The unknown function u is u = u ( r , ϕ , t ) . In the case of radial symmetry u does not dependon ϕ , and thus u = u ( r , t ) . For this case the initial equation (7.13) obtains the following form ∂ 2 u ∂ t 2 = c 2 ∂ 2 u ∂ r 2 + 1 r ∂ u ∂ r . (7.372) Assume that the membrane is fixed on its border (from where the boundary conditions follow), i.e.

u

r

0

r

Figure 7.6: Circular membrane.

u ( r 0 , t )= 0 , for t ≥ 0 . (7.373) Here r 0 represents the radius of the circular membrane. In addition, assume that – the initial value of the motion (initial condition) is

u ( r , 0 )= f ( r ) and

(7.374)