Appendix B
Huygens’ Principle
One may conceive light to spread successively, by spherical
waves.
— Christiaan Huygens
In the analysis of how waves move in three dimensions, Christiaan Huygens, in
1690, poised a marvelous and useful idea: Each wavefront of a given wave may be
taken as a new source of spherical waves! (Fig. B.1). Consider, for example, a plane
wave traveling through a hole. If you are given the wavelength of the wave, the
interference and diffraction of the wave past the hole can be constructed as if new
little wavelets are being generated over small regions covering the hole, all initially
in phase with each other. The resulting wave produced past the hole generates a
diffracted wave. For light, the diffraction causes a diffraction pattern on a screen or
in a set of detectors.
G. Kirchhoff in 1882 indicated how the descriptions of wave propagation and
diffraction by Huygens, Young, and Fresnel, could come from the properties of
the solutions to the wave equation. Rayleigh and Sommerfeld later showed how
to express the theory of wave diffraction in a form amenable to a consistent
mathematical expansion for short wavelengths. If the wave with amplitude (r, t)
is taken to have one frequency f = ω/(2π), so that (r, t) = ψ(r)e −iωt , then the
wave equation requires that the space part satisfy:
∇
2 ψ(r) + k
2 ψ(r) = 0,
(B.1)
where k = ω/v, and v is the speed of the wave. The above is Helmholtz’ equation.
If the wave vanishes on a large surface area A except for a finite segment A , and
it propagates from that area into an effectively unbounded region, then an exact
solution to Helmholtz’ equation is given by
ψ(r) =
A
ψ(r
)∇G(r − r
) · dA
.
(B.2)
© Springer Nature Switzerland AG 2020
W. C. Parke, Biophysics, https://doi.org/10.1007/978-3-030-44146-3
633
Huygens’ Principle
One may conceive light to spread successively, by spherical
waves.
— Christiaan Huygens
In the analysis of how waves move in three dimensions, Christiaan Huygens, in
1690, poised a marvelous and useful idea: Each wavefront of a given wave may be
taken as a new source of spherical waves! (Fig. B.1). Consider, for example, a plane
wave traveling through a hole. If you are given the wavelength of the wave, the
interference and diffraction of the wave past the hole can be constructed as if new
little wavelets are being generated over small regions covering the hole, all initially
in phase with each other. The resulting wave produced past the hole generates a
diffracted wave. For light, the diffraction causes a diffraction pattern on a screen or
in a set of detectors.
G. Kirchhoff in 1882 indicated how the descriptions of wave propagation and
diffraction by Huygens, Young, and Fresnel, could come from the properties of
the solutions to the wave equation. Rayleigh and Sommerfeld later showed how
to express the theory of wave diffraction in a form amenable to a consistent
mathematical expansion for short wavelengths. If the wave with amplitude (r, t)
is taken to have one frequency f = ω/(2π), so that (r, t) = ψ(r)e −iωt , then the
wave equation requires that the space part satisfy:
∇
2 ψ(r) + k
2 ψ(r) = 0,
(B.1)
where k = ω/v, and v is the speed of the wave. The above is Helmholtz’ equation.
If the wave vanishes on a large surface area A except for a finite segment A , and
it propagates from that area into an effectively unbounded region, then an exact
solution to Helmholtz’ equation is given by
ψ(r) =
A
ψ(r
)∇G(r − r
) · dA
.
(B.2)
© Springer Nature Switzerland AG 2020
W. C. Parke, Biophysics, https://doi.org/10.1007/978-3-030-44146-3
633