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The Wave Equation in Cylindrical Coordinates
Overview and Motivation : While Cartesian coordinates are attractive because of
their simplicity, there are many problems whose symmetry makes it easier to use a
different system of coordinates. For example, there are times when a problem has
cylindrical symmetry (the fields produced by an infinitely long, straight wire, for
example). In this case it is easier to use cylindrical coordinates. So today we begin
our discussion of the wave equation in cylindrical coordinates.
Key Mathematics : Cylindrical coordinates and the chain rule for calculating
derivatives.
I. Transforming the Wave Equation
As previously mentioned the (spatial) coordinate independent wave equation
q t
q c
2 2
2 2
can take on different forms, depending upon the coordinate system in use. In
Cartesian coordinates the Laplacian ∇ 2 is expressed as
2
2 2
2 2
2 2 x y ∂ z
Our first goal is to re-express ∇ 2 in terms of cylindrical coordinates ( ρ , φ, z ), which are defined in terms of the Cartesian coordinates ( x , y , z )as
ρ = ( x^2 + y^2 )^1 2 , (2a)
x
φ arctan y , (2b)
z = z. (2c)
The following picture illustrates the relationships expressed by Eq. (2). For the point
given by the vector r = x x ˆ^ + y y ˆ+ z z ˆ, the coordinate ρ is the distance of that point
from the z axis, the coordinate φ is the angle of the projection of the vector onto the
x - y plane from the x axis toward the y axis, and z is the (signed) distance of the
point from the x - y plane. Note that ρ ≥ 0 and we can restrict 0 ≤ φ < 2 π.
In order to express ∇ 2 in terms of these new coordinates we start with a function
f ( ρ , φ, z )and consider it to be function of x , y , and z through the variables ρ , φ , and
z by writing
f = f [ ρ ( x , y , z ), φ( x , y , z ), z ( x , y , z )]. (3)
Then, for example, using the chain rule we can write
x
z z
f x
f x
f x
f ∂
Notice that this equation (as well as some later equations) have two types of terms.
The first type is a derivative of the function f , while the second type is a derivative of
a new coordinate with respect to an old coordinate. The goal here is to use the
relationship between the two coordinate systems [Eq. (2)] to write the second type of
term as a function of the new set of coordinates ρ , φ , and z. Then equations such
as Eq. (4) will be entirely expressed in terms of the new coordinate system.
For the particular case at hand the transformation is a bit simpler than the general
case because [see Eq. (2)] ρ = ρ( x , y ), φ = φ( x , y ), and z = z ( z ). Thus Eq. (3) simplifies
to
f = f [ ρ ( x , y ), φ( x , y ), z ( ) z ], (5)
x
y
z
r
φ
z
ρ
( ) (^) ( )φ
ρ =ρcos φ =cos
x
Similarly, using
[ ( )] x
u x u
u ∂
arctan 1 (13)
and Eq. (2b) we have
[ ( )] 1 ( ) 2 2 2 2
arctan 1 x y
y x
y x y x
y x x +
Using y =ρ sin( )φ and ρ 2 = x^2 + y^2 , we now re-express Eq. (14) in terms of the new
coordinates as
( ) ( )
φ ρsinφ sin
2
=− =^ −
x
In similar fashion one can express the second derivatives [Eq. (9c) and Eq. (9d)], in
terms of ρ and φ as
( )
2
(^2) sin
∂
x
and
( ) ( ) 2 2
(^2 2) cos sin
x
If we now insert Eqs. (12), (15), (16) and (17) into Eq. (9) we obtain
( ) ( )^ ( )^ ( )
( ) ( ) ( ) ( ) ( ) 2 2
2 2
2 2
2 2 2 2
2 2
2
sin cos sin 2 cos sin
cos sin cos sin
ρ
φ φ ρ φ
φ ρ φ
φ φ ρ φ
ρ
φ ρ ρ
φ φ φ ρ
φ ρ
f f f
f f f x
f
So we have now expressed the first term of the Laplacian (acting on a function f )
∂ 2 f ∂ x^2 in terms of cylindrical coordinates.
In a manner analogous to the procedure that we have just carried out one can also
derive the result 1
( ) ( )^ ( )^ ( )
( ) ( ) ( ) ( ) ( ) 2 2
2 2
2 2
2 2 2 2
2 2
2
sin cos cos 2 cos sin
sin sin cos cos
f f f
f f f y
f
And, of course, we also trivially have
2
2 2
2 z
f z
f ∂
Putting Eqs. (18) – (20) together then gives us the fairly simple result
2
2 2
2 2 2
2
2
2 2
2 2
2 2
1 1 z
f f f f
z
f y
f x
f f
Thus, in cylindrical coordinates the wave equation becomes
2
2 2
2 2 2
2 2
2 2
z
q q q q t
q c ∂
where now q = q ( ρ, φ, z , t ).
II. Separation of Variables
To look for separable solutions to the wave equation in cylindrical coordinates we
posit a product solution
q ( ρ , φ , z , t ) = R ( ρ) Φ( ) φ Z ( ) z T ( ) t. (23)
Substituting this into Eq. (22) produces
R ZT R ZT R ZT R ZT R Z T
c
(^1) Notice that Eq. (19) is the same as Eq. (18) with sin (φ ) → cos( )φ and cos( φ ) → −sin( )φ.
