Difference between revisions of "Dictionary:Kirchhoff’s equation"

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1. An integral form of the wave equation expressing the wave function ${\displaystyle \psi _{p}}$ at the point P as the sum of wave contributions from the surroundings. Wave contributions have to allow for the traveltime from the sources to P, that is, what the source does at an earlier time ${\displaystyle \tau =(t-r/V)}$ affects P at time t, where r is the distance from the source to P and V is the velocity. The earlier time ${\displaystyle \tau }$ is called retarded time. ${\displaystyle \psi _{p}}$ is expressed as an integral over the volume surrounding P (to accommodate sources within the volume) plus an integral over the surface surrounding the volume (to accommodate sources from outside). In source-free space in terms of the values of ${\displaystyle \psi }$ and its derivative on a surrounding surface S at the preceding time ${\displaystyle (t-r/V)}$:

${\displaystyle \psi _{p}={\frac {1}{4\pi }}\iint \left\{[\psi ]{\frac {\partial {\frac {1}{r}}}{\partial n}}-{\frac {1}{Vr}}{\frac {\partial r}{\partial n}}\left[{\frac {\partial \psi }{\partial t}}\right]-{\frac {1}{r}}\left[{\frac {\partial \psi }{\partial n}}\right]\right\}ds}$.

The terms in brackets are evaluated at the retarded time ${\displaystyle \tau =(t-r/V)}$, r is the distance from P to points on the surface S, and n is a unit vector normal to S. The Kirchhoff integral equation used in migration can be written

${\displaystyle \psi (x,z,t)={\frac {z}{\pi }}\int \left[1/r^{3}-(2/Vr^{2})\times \left({\frac {\partial }{\partial t}}\right)\psi (x',0,t+T)\right]dx'}$,

where x '; is position at z=0, ${\displaystyle \tau }$,is the two-way time 2r/V, and r is the distance from x ' to x. For r much longer than a wavelength this simplifies to the Rayleigh-Sommerfeld approximation,

${\displaystyle \psi _{p}(x,T,t)=-{\frac {2T}{\pi V^{2}}}\int {\frac {1}{T^{2}}}{\frac {\partial }{\partial t}}\psi (x',0,t+T)dx'}$,

where T=2z/V=vertical traveltime. This expresses migration by integration along a diffraction curve.

2. The radiation law that the ratio of emissivity to absorptance depends only on the wavelength and temperature, or that it is the same for all bodies as for an ideal blackbody for any wavelength at the given temperature.