General
Solution of 3D Wave Equation by Ronald H. Brady

Let U =
U(x,y,z,t) be a function of the spatial coordinates, of a rectangular
coordinate system, and also of the time t. Let U_{x }

and U_{xx
} denote the first and second order
partial derivatives of

U with
respect to x respectively. Also let the partial derivatives of U with respect
to y,z and t be denoted in a similar fashion.

Let us now
consider the equation

U_{xx}
+ U_{yy} + U_{zz}
– (1/c^{2})U_{tt}
= 0
(eq.1)

This
equation will be recognized immediately as the wave

equation of
mathematical physics. It describes wave motion

which has a
constant speed of propagation c.

It is
conventional to denote the constant speed of propa-

gation of
light or electromagnetic radiation in vacuum by

c. If the
wave motion involves a phenomenon other than light then the constant speed of
the wave motion is denoted by v.

The reader
will recall that the Laplacian partial dif-

ferential
operator ∆ (applied to a function U) is defined

by

∆U =
U_{xx} + U_{yy} + U_{zz
}(eq.2)

We may therefore re-write (eq.1) as

∆U - (1/c^{2})U_{tt} =
0

For a given function F the Laplacian may be applied

multiple times. For example

∆^{2}F = ∆(∆F)

∆^{3}F = ∆(∆^{2}F) = ∆(∆(∆F))

and more generally the symbol ∆^{n}F denotes the
appli-

cation of the Laplacian operator, to the operand F,

n successive times.

Often PDE with constant coefficients are solved by

the method of Separation of Variables. But we now will present
an alternative solution process that may be more straight-forward for many
applications.

Theorem

Let

F = F(x,y,z)
(eq.3a)

G = G(x,y,z)
(eq.3b)

be infinitely differentiable functions of x,y and z

in some domain D of R^{3}.
Also let

∆U -
(1/c^{2})U_{tt}
= 0
(eq.4)

then a solution U = U(x,y,z,t) of (eq.4) that

satisfies the auxiliary conditions

U(x,y,z,0) = G(x,y,z) (eq.5)

U_{t}(x,y,z,0)
= F(x,y,z)
(eq.6)

is given by

U = Ft + G +

∑(n=1 to ∞){c^{2n}[(1/(2n+1)!)(∆^{n}F)t^{2n+1}
+ (1/(2n)!)(∆^{n}G)t^{2n}]}

(eq.7)

provided
that the indicated infinite series on the

right side
of (eq.7) converges in the domain D. Note

that the
notation ∑(n=1 to ∞) indicates that the quantity

in the
curly braces (on the right side of (eq.7) is to

be summed
from n = 1 to n = ∞ .

The proof
of the Theorem is as follows:

By the
hypothesis that U is represented by a convergent power

series it
follows that term by term differentiation is valid.

Applying the Laplacian ∆ to both sides of (eq.7)

will give
us

∆U =
(∆F)t + ∆G +

∑(n=1 to ∞){c^{2n}[(1/(2n+1)!)(∆^{n+1}F)t^{2n+1}
+ (1/(2n)!)(∆^{n+1}G)t^{2n}]}

We obtain, after
writing the terms of the summation that

corresponds
to n = 1,2,3,4,..., the following

∆U =
(∆F)t + ∆G + c^{2}[(1/(3)!)(∆^{2}F)t^{3}+(1/2!)(∆^{2}G)t^{2}]
+

c^{4}[(1/5!)(∆^{3}F)t^{5} + (1/4!)(∆^{3}G)t^{4}]
+

c^{6}[(1/7!)(∆^{4}F)t^{7}
+ (1/6!)(∆^{4}G)t^{6}] +

c^{8}[(1/9!)(∆^{5}F)t^{9}
+ (1/8!)(∆^{5}G)t^{8}] +...
(eq.8)

Now for
each side of (eq.7) we will take the second

order
partial derivative with respect to t and then

multiply
the results by 1/c^{2 }. We will obtain

(1/c^{2})U_{tt}
= (1/c^{2})∑(n=1
to ∞){c^{2n}[((2n+1)2n)/(2n+1)!)(∆^{n}F)t^{2n-1}
+

(2n(2n-1)/(2n)!)(∆^{n}G)t^{2n-2}]}

We have,
after writing the terms of the summation that

corresponds
to n = 1,2,3,4,5,..., the following

(1/c^{2})U_{tt
}= (1/c^{2})[c^{2}[((3*2)/3!)(∆F)t
+ (2(1)/2!)(∆G)t^{0}] +

c^{4}[(5*4/5!)(∆^{2}F)t^{3} +
(4*3/4!)(∆^{2}G)t^{2}] +

c^{6}[(7*6/7!)(∆^{3}F)t^{5} + (6*5/6!)(∆^{3}G)t^{4}]
+

c^{8}[(9*8/9!)(∆^{4}F)t^{7} + (8*7/8!)(∆^{4}G)t^{6}]
+

c^{10}[(11*10/11!)(∆^{5}F)t^{9} +
(10*9/10!)(∆^{5}G)t^{8}] + ...]

which gives
us after simplification

(1/c^{2})U_{tt
}= [(∆F)t + (∆G) +

c^{2}[(1/3!)(∆^{2}F)t^{3} +
(1/2!)(∆^{2}G)t^{2}] +

c^{4}[(1/5!)(∆^{3}F)t^{5} + (1/4!)(∆^{3}G)t^{4}]
+

c^{6}[(1/7!)(∆^{4}F)t^{7} + (1/6!)(∆^{4}G)t^{6}]
+

c^{8}[(1/9!)(∆^{5}F)t^{9} + (1/8!)(∆^{5}G)t^{8}]
+ ...] (eq.9)

Now if we
substitute the expression ∆U from (eq.8)

and the
expression for (1/c^{2})U_{tt } from (eq.9) into

the left
side of (eq.4) and simplify we will

obtain 0 =
0, which indicates that U as defined by

(eq.7)
satisfies the wave equation (eq.4) in a

domain D
provided that the indicated infinite series

on the
right side of (eq.7) converges in the

Domain D.
Of course, the indicated infinite series will

always
converge if F and G are polynomials: because of the

repeated
application of the Laplacian ∆.

We will now
show that the auxiliary conditions are

also satisfied.
Set t = 0 on both sides of (eq.7) and

obtain

U(x,y,z,0)
= G

U(x,y,z,0)=
G(x,y,z)

which
agrees with the auxiliary conditions of (eq.5).

Now take
the partial derivative with respect to t of both sides of (eq.7). The resulting
equation is

U_{t }= F + ∑(n=1
to ∞){c^{2n}[(1/(2n)!)(∆^{n}F)t^{2n}
+

(1/(2n-1)!)(∆^{n}G)t^{2n-1}]}

now let t =
0 on both sides of the equation above. We will arrive at

U_{t}(x,y,z,0)
= F

U_{t}(x,y,z,0)
= F(x,y,z)

which
agrees with (eq.6) of the auxiliary conditions.

This completes
the proof.

Examples,
illustrations and actual applications will

be
available here