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I Huygens principle from Kirchoff Integral (sign doubt)

  1. Sep 4, 2017 #1
    Consider the Kirkoff integral theorem and the Huygens -Fresnel principle/formula (both from Wikipedia):

    KIT

    The Kirchoff integral for monochromatic wave is:
    $$U({\mathbf {r}})={\frac {1}{4\pi }}\int _{S'}\left[U{\frac {\partial }{\partial {\hat {{\mathbf {n}}}}}}\left({\frac {e^{{iks}}}{s}}\right)-{\frac {e^{{iks}}}{s}}{\frac {\partial U}{\partial {\hat {{\mathbf {n}}}}}}\right]dS$$
    where the integration is performed over an arbitrary closed surface S' (enclosing r), s is the distance from the surface element to the point r, and ∂/∂n denotes differentiation along the surface normal (a normal derivative).

    HF fromula

    Consider the case of a point source located at a point P0, vibrating at a frequency f. The disturbance may be described by a complex variable U0 known as the complex amplitude. It produces a spherical wave with wavelength λ, wavenumber k = 2π/λ. The complex amplitude of the primary wave at the point Q located at a distance r0 from P0 is given by:

    $$U(r_0) = \frac {U_0 e^{ikr_0}}{r_0}$$
    The complex amplitude at P is then given by:

    $$ U(P) = -\frac{i}{\lambda} U(r_0) \int_{S} \frac {e^{iks}}{s} K(\chi)\,dS $$

    where S describes the surface of the sphere, and s is the distance between Q and P.

    300px-Huygens-Fresnel_BW.svg.png

    The surface ##S## in HF formula does not enclose the point ##P## (which whould be the ##r## in KIT). Is this correct?

    If so then is the following procedure correct to show that HF formula is an application of KIT theorem?

    1. Consider as ##S'## (surface for KIT) the closed surface ##S## used for HF formula plus a surface ##S_0## at infinity.
    2. ##S'=S+S_0## clearly contains the point ##P## (which is ##r##), while ##S## does not.
    3. The Kirchoff integral is zero on ##S_0## and on ##S## it is equal to the HF formula on ##S##.

    I don't want to prove passage 3. completely but I have a doubt on a sign: the normal vector of the surface ##S## (when considered as a part of ##S'=S_0+S##) should be ingoing in the volume ##V## enclosed by ##S## (which means outgoing from volume enclosed by ##S'##), while HF formula works with outgoing normal vector from ##S## I guess (##\chi## being measured with respect to ouitgoing normal).
     
  2. jcsd
  3. Sep 9, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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