Graduate Huygens Principle - how to explain this with classical language?

[filip97]

I was read this article(https://engineering.purdue.edu/wcchew/ece604f19/Lecture Notes/Lect31.pdf).

I was read this paper about Huygens' principle(https://engineering.purdue.edu/wcchew/ece604f19/Lecture Notes/Lect31.pdf)

Main idea of Huygens' principle is how wave function $ψ(r)$
$r∈S=∂V$(S is wave front in time t) affect on shape of of wave function $ψ(r′)$ and shape of wave front $S′=∂V′, r'∈S′$ in time$t′,t<t′$

. We have:

$(∇^2+k^2)ψ(r)=0$ (1)

$ψ(r′)=∮_{S}dS\hat{n}(G(r,r′)∇ψ(r)−ψ(r)∇G(r,r′)),r′∈V′$

,and

$ψ(r′)=0,r′∉V′$

where $G(r,r′)$
is Green's function of (1)

First term $\hat{n}G(r,r′)∇ψ(r)$
, can be interpeted how incident wave of point and unit source propagate in direction of surface normal of surface S′

Question is: How we can interpreted second term $−ψ(r)∇G(r,r′)$
,respectevely how identified this therm as new source of spherical waves in point r, respectively how explain this with classical language ?

P.S. If $−ψ(r)∇G(r,r′)$ is wave of unit and point source in $r'$ is clear that amplitude in this point depend of ampliture of wave $\psi(r)$

 

[tech99]

I think Huygens probably described it in classical language. Every point on a wavefront may be considered as a source of secondary wavelets. Whether it is proven, I don't know. I have never seen a finite wavefront which is not constrained in size by some object.

 

[vanhees71]

Huygen's principle works for the wave equation in odd-dimensional (configuration) space for $d \geq 3$. You get it by calculating the Green's function of the D'Alembert operator and using Green's theorem in $(1+d)$ dimensions. For a thorough treatment, see

S. Hassani, Mathematical Physics, Springer Verlag, Cham,
Heidelberg, New York, Dordrecht, London, 2 ed. (2013).
https://dx.doi.org/10.1007/978-3-319-01195-0