Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: SEPARATION OF VARIABLES AND OTHER METHODS

Laplace’s equation in cylindrical polars

Passing to three dimensions, we now consider the solution of Laplace’s equation

in cylindrical polar coordinates,

1

ρ

∂

∂ρ

(

ρ

∂u

∂ρ

)

+

1

ρ^2

∂^2 u

∂φ^2

+

∂^2 u

∂z^2

=0. (21.32)

We note here that, even when considering a cylindrical physical system, if there

is no dependence of the physical variables onz(i.e. along the length of the

cylinder) then the problem may be treated using two-dimensional plane polars,

as discussed above.

For the more general case, however, we proceed as previously by trying a

solution of the form

u(ρ, φ, z)=P(ρ)Φ(φ)Z(z),

which, on substitution into (21.32) and division through byu=PΦZ, gives

1

Pρ

d

dρ

(

ρ

dP

dρ

)

+

1

Φρ^2

d^2 Φ

dφ^2

+

1

Z

d^2 Z

dz^2

=0.

The last term depends only onz, and the first and second (taken together) depend

only onρandφ. Taking the separation constant to bek^2 , we find

1

Z

d^2 Z

dz^2

=k^2 ,

1

Pρ

d

dρ

(

ρ

dP

dρ

)

+

1

Φρ^2

d^2 Φ

dφ^2

+k^2 =0.

The first of these equations has the straightforward solution

Z(z)=Eexp(−kz)+Fexpkz.

Multiplying the second equation through byρ^2 ,weobtain

ρ

P

d

dρ

(

ρ

dP

dρ

)

+

1

Φ

d^2 Φ

dφ^2

+k^2 ρ^2 =0,

in which the second term depends only on Φ and the other terms depend only

onρ. Taking the second separation constant to bem^2 , we find

1

Φ

d^2 Φ

dφ^2

=−m^2 , (21.33)

ρ

d

dρ

(

ρ

dP

dρ

)

+(k^2 ρ^2 −m^2 )P=0. (21.34)

The equation in the azimuthal angleφhas the very familiar solution

Φ(φ)=Ccosmφ+Dsinmφ.