634
B Huygens’ Principle
Fig. B.1 Huygens wavelets
construction: solid black lines
are wave fronts. Arrows show
direction of propagation. New
wave front to right is
constructed by wavelets from
prior wave front to left
In this relation, the so-called Green’s function, G(r − r ), is a solution to the wave
equation which vanishes when r locates points on a surface A which includes A
and the extension of A area to large distances:
G(r − r
)
r ∈A = 0 .
(B.3)
It can be expressed as
G(r − r
) =
1
|r − r |
+ F (r − r
)
(B.4)
where F (r − r ) is a solution of the Laplace equation
∇
2 F (r − r
) = 0
( B . 5 )
in the region into which the wave is propagating.
If the area segment A is reasonably flat (i.e. its curvature is large compared to
the wavelength of the wave) and is of extent much larger than the wavelength, then
Eq. (B.2) becomes
ψ(r) ≈
k
2πi
A
ψ(r
)
exp (ik
r − r
)
|r − r |
3
(r − r
) · dA
.
(B.6)
This shows that the propagated wave can be constructed from outgoing spherical
waves coming from the wave on the surface A , a statement of Huygens’ Principle.