Kirchhoff integral theorem

The Kirchhoff integral theorem (sometimes referred to as the Fresnel–Kirchhoff integral theorem) is a surface integral to obtain the value of the solution of the homogeneous scalar wave equation at an arbitrary point P in terms of the values of the solution and the solution's first-order derivative at all points on an arbitrary closed surface (on which the integration is performed) that encloses P. It is derived by using Green's second identity and the homogeneous scalar wave equation that makes the volume integration in Green's second identity zero. == Integral == === Monochromatic wave === The integral has the following form for a monochromatic wave: U ( r ) = 1 4 π ∫ S [ U ∂ ∂ n ^ ( e i k s s ) − e i k s s ∂ U ∂ n ^ ] d S , {\displaystyle 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 the observation point r {\displaystyle \mathbf {r} } , k {\displaystyle k} in e i k s {\displaystyle e^{iks}} is the wavenumber, s {\displaystyle s} in e i k s s {\displaystyle {\frac {e^{iks}}{s}}} is the distance from an (infinitesimally small) integral surface element to the point r {\displaystyle \mathbf {r} } , U {\displaystyle U} is the spatial part of the solution of the homogeneous scalar wave equation (i.e., V ( r , t ) = U ( r ) e − i ω t {\displaystyle V(\mathbf {r} ,t)=U(\mathbf {r} )e^{-i\omega t}} as the homogeneous scalar wave equation solution), n ^ {\displaystyle {\hat {\mathbf {n} }}} is the unit vector inward from and normal to the integral surface element, i.e., the inward surface normal unit vector, and ∂ ∂ n ^ {\displaystyle {\frac {\partial }{\partial {\hat {\mathbf {n} }}}}} denotes differentiation along the surface normal (i.e., a normal derivative) i.e., ∂ f ∂ n ^ = ∇ f ⋅ n ^ {\displaystyle {\frac {\partial f}{\partial {\hat {\mathbf {n} }}}}=\nabla f\cdot {\hat {\mathbf {n} }}} for a scalar field f {\displaystyle f} .

Source: Wikipedia — Kirchhoff integral theorem (CC BY-SA 4.0)

Kirchhoff integral theorem

The Kirchhoff integral theorem (sometimes referred to as the Fresnel–Kirchhoff integral theorem) is a surface integral to obtain the value of the solution of the homogeneous scalar wave equation at an arbitrary point P in terms of the values of the solution and the solution's first-order derivative at all points on an arbitrary closed surface (on which the integration is performed) that encloses P. It is derived by using Green's second identity and the homogeneous scalar wave equation that makes the volume integration in Green's second identity zero. == Integral == === Monochromatic wave === The integral has the following form for a monochromatic wave: U ( r ) = 1 4 π ∫ S [ U ∂ ∂ n ^ ( e i k s s ) − e i k s s ∂ U ∂ n ^ ] d S , {\displaystyle 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 the observation point r {\displaystyle \mathbf {r} } , k {\displaystyle k} in e i k s {\displaystyle e^{iks}} is the wavenumber, s {\displaystyle s} in e i k s s {\displaystyle {\frac {e^{iks}}{s}}} is the distance from an (infinitesimally small) integral surface element to the point r {\displaystyle \mathbf {r} } , U {\displaystyle U} is the spatial part of the solution of the homogeneous scalar wave equation (i.e., V ( r , t ) = U ( r ) e − i ω t {\displaystyle V(\mathbf {r} ,t)=U(\mathbf {r} )e^{-i\omega t}} as the homogeneous scalar wave equation solution), n ^ {\displaystyle {\hat {\mathbf {n} }}} is the unit vector inward from and normal to the integral surface element, i.e., the inward surface normal unit vector, and ∂ ∂ n ^ {\displaystyle {\frac {\partial }{\partial {\hat {\mathbf {n} }}}}} denotes differentiation along the surface normal (i.e., a normal derivative) i.e., ∂ f ∂ n ^ = ∇ f ⋅ n ^ {\displaystyle {\frac {\partial f}{\partial {\hat {\mathbf {n} }}}}=\nabla f\cdot {\hat {\mathbf {n} }}} for a scalar field f {\displaystyle f} .

Source: Wikipedia "Kirchhoff integral theorem" · CC BY-SA 4.0

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