Mathematical Physics Vol 1

7.8 Examples

423

Solution In this case we will look for the solution using the method of separation of variables

U = R ( r ) Z ( z ) .

By substitution in the initial equation we obtain two ordinary differential equations R ′′ R + 1 r R ′ R = − m 2 , Z ′′ Z = m 2 , (7.394) where m 2 is a constant. The second equation is a second order homogeneous differential equation with constant coefficients, whose solution is + mz . The first equation is a special form of Bessel equation of zero order, whose solutions are Bessel functions of the first and second type, of zero order. Bessel functions of the second type do not correspond to the nature of electricity potential of a point source, which tends to zero in infinity, and thus the solution of this equation will be of the form Z ( z )= C 1 e − mz + C 2 e

R ( r )= J 0 ( mr ) .

The final solution is

U ( r , z )= C 1 e −

+ mz J

mz + C

2 e

0 ( mr ) .

(7.395)

As the constant m cannot take all values, and given the fact that it was introduced in the expression (7.394) as m 2 , its change in the interval ( 0 , ∞ ) should be observed, so the most general form of the solution (7.395) will be obtained by integrating for parameter m U ( r , z )= ∞ Z 0 C 1 e − mz + C 2 e + mz J 0 ( mr ) d m . For determining the constants C i the boundary conditions are used.

Problem 276 Determine the motion of the points on a rectangular membrane.

Solution In this case we will represent the initial equation (7.13) in Cartesian coordinates, thus ∂ 2 u ∂ t 2 = c 2 ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 , (7.396)

with boundary conditions

u = 0 ,

( at the membrane borders for ∀ t ≥ 0 )

(7.397)