Page 192 - Maxwell House

172 Chapter 4

As expected, all these solutions vanish as || → ∞. Evidently, the function 1 in (4.32) and
′
|− |
(4.33) describes the field created by a unit point-source located at position . Therefore,
′
′
(, ) = 1
′
|− | ⎫
⎪
1 ′ ′ ′
() = ∫ (, ) ( ) (4.34)
′
4 0
⎬
0 ′ ′ ′ ⎪
() = ∫ (, ) ( ) ⎭
4 ′
The function (, ) is called static free-space Green’s function and named after the British
′
physicist George Green who first introduced it in the 1830s.
4.1.8 Wave Equation. One-Dimensional Unbounded Space
Following the brief moment of amusement that we have succeeded and solved such complicated
equations as (4.24) let us transit to the more complex non-static case. To avoid too general
discussion, we will start from wave equation (4.22) in free-of-current sources space (i.e. =

0) assuming that the vector (, ) = [ (, ), 0, 0] is parallel to the -axis and depends on

space variables and time only. Then according to (4.22)
2
2
(,) − 1 (,) = 0 (4.35)
2 2 2
Note that (4.35) is invariant under the transformation → −. If so, we can expect as minimum
two identical solutions, one with t as the variable and the second with –t. The equation (4.35)
can be formally represented (factorized) as a product of two factors

1 1
� − � � + � (, ) = (, ) = 0 (4.36)

(−) (+)
Eventually, the general solution of (4.36) must satisfy two equations simultaneously
1 (,)
= 0
(−) � (4.37)
2 (,)
= 0
(+)
Moreover, it can be written as the superposition of two arbitrary continuous and twice
2
differentiable functions . Meanwhile, the first function depends on − ́ only and the
1
second one on + ́ only
2
(, ) = ( − ́) + ( + ́) (4.38)
2

1
Here ( − ́) and ( + ́) are the arbitrary functions in , ́ = − , is the
′
2
′
2
1
origin where the source of energy can be located and is the observer position. As it was
predicted, this solution is invariant under transformation → − and was published by French
scientist Jean le Rond d ’Alembert in 1747 and carries his name.
According to (4.38) the differential equation (4.35) yields an infinite set of functions. To get
the unique solution, we need to formulate some additional conditions in space or time domain.

2 Belong to class of function , in short notation.
2