- #1
crick
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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.
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).
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.
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).