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''Kirchhoff integral''
 
''Kirchhoff integral''
  
 
The formula
 
The formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k0554401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
u ( x , t )  =
 +
\frac{1}{4 \pi }
 +
 
 +
\int\limits _  \Omega
 +
 
 +
\frac{f ( y , t - r ) }{r }
 +
  d \Omega _ {y} +
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k0554402.png" /></td> </tr></table>
+
$$
 +
+
  
expressing the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k0554403.png" /> of the solution of the inhomogeneous [[Wave equation|wave equation]]
+
\frac{1}{4 \pi }
 +
\int\limits _  \sigma  \left [
 +
\frac{1}{r}
 +
 +
\frac{\partial  u }{\partial  n }
 +
- u
 +
\frac{\partial  ( 1 / r ) }{\partial  n }
 +
+
 +
\frac{1}{r}
 +
 +
\frac{\partial  u }{\partial  \tau }
 +
 +
\frac{\partial  r }{\partial  n }
 +
\right ] _ {\tau = t - r }  d \sigma _ {y} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k0554404.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
expressing the value  $  u ( x , t ) $
 +
of the solution of the inhomogeneous [[Wave equation|wave equation]]
  
at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k0554405.png" /> at the instant of time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k0554406.png" /> in terms of the retarded volume potential
+
$$ \tag{2 }
 +
u _ {tt} - u _ {x _ {1}  x _ {1} } -
 +
u _ {x _ {2}  x _ {2} } - u _ {x _ {3}  x _ {3} }
 +
= f ( x , t )
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k0554407.png" /></td> </tr></table>
+
at the point  $  x =( x _ {1} , x _ {2} , x _ {3} ) \in \Omega $
 +
at the instant of time  $  t $
 +
in terms of the retarded volume potential
  
with density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k0554408.png" />, and in terms of the values of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k0554409.png" /> and its first-order derivatives on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544010.png" /> of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544011.png" /> at the instant of time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544012.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544013.png" /> is a bounded domain in the three-dimensional Euclidean space with a piecewise-smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544015.png" /> is the outward normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544017.png" /> is the distance between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544019.png" />.
+
$$
 +
v _ {1} ( x , t )  = \
 +
 
 +
\frac{1}{4 \pi }
 +
 
 +
\int\limits _  \Omega
 +
 
 +
\frac{f ( y , t , r ) }{r }
 +
  d \Omega _ {y} ,\ \
 +
y = ( y _ {1} , y _ {2} , y _ {3} ) ,
 +
$$
 +
 
 +
with density $  f $,  
 +
and in terms of the values of the function $  u ( y , t ) $
 +
and its first-order derivatives on the boundary $  \sigma $
 +
of the domain $  \Omega $
 +
at the instant of time $  \tau = t - r $.  
 +
Here $  \Omega $
 +
is a bounded domain in the three-dimensional Euclidean space with a piecewise-smooth boundary $  \sigma $,  
 +
$  n $
 +
is the outward normal to $  \sigma $
 +
and $  r = | x - y | $
 +
is the distance between $  x $
 +
and $  y $.
  
 
Let
 
Let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544020.png" /></td> </tr></table>
+
$$
 +
v _ {1} ( x , t )  = \
 +
 
 +
\frac{1}{4 \pi }
 +
 
 +
\int\limits _  \sigma 
 +
\frac{1}{r}
 +
 
 +
\frac{\partial  \mu _ {1} ( y , t - r ) }{\partial  n }
 +
  d \sigma _ {y} ,
 +
$$
 +
 
 +
$$
 +
v _ {2} ( x , t )  =
 +
\frac{1}{4 \pi }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544021.png" /></td> </tr></table>
+
\int\limits _  \sigma 
 +
\frac{\partial  ^ {*} }{\partial  ^ {*} n }
 +
 +
\frac{
 +
\mu _ {2} ( y , t - r ) }{r }
 +
  d \sigma _ {y} ,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544022.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{\partial  ^ {*} }{\partial  ^ {*} n }
 +
 
 +
\frac{\mu _ {2} ( y , t - r ) }{r }
 +
  = \
 +
 
 +
\frac{1}{r}
 +
 
 +
\frac{\partial  r }{\partial  n }
 +
 
 +
\frac{\partial  \mu _ {2} ( y , t - r ) }{\partial  t }
 +
- \mu _ {2} ( y , t - r )
 +
 
 +
\frac{\partial  ( 1 / r ) }{\partial  n }
 +
.
 +
$$
 +
 
 +
The integrals  $  v _ {1} ( x , t ) $
 +
and  $  v _ {2} ( x , t ) $
 +
are called the retarded potentials of the single and the double layer.
 +
 
 +
The Kirchhoff formula (1) means that any twice continuously-differentiable solution  $  u ( x , t ) $
 +
of equation (2) can be expressed as the sum of the retarded potentials of a single layer, a double layer and a volume potential:
 +
 
 +
$$
 +
u ( x , t )  = v _ {1} ( x , t ) + v _ {2} ( x , t ) +
 +
v _ {3} ( x , t ) .
 +
$$
 +
 
 +
In the case when  $  u ( x , t ) = u ( x ) $
 +
and  $  f ( x , t ) = f ( x ) $
 +
do not depend on  $  t $,
 +
the Kirchhoff formula takes the form
 +
 
 +
$$
 +
u ( x)  =
 +
\frac{1}{4 \pi }
 +
 
 +
\int\limits _  \Omega
 +
 
 +
\frac{f ( y ) }{r }
 +
  d \Omega _ {y} +
 +
 
 +
\frac{1}{4 \pi }
  
The integrals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544024.png" /> are called the retarded potentials of the single and the double layer.
+
\int\limits _  \sigma
 +
\left [
  
The Kirchhoff formula (1) means that any twice continuously-differentiable solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544025.png" /> of equation (2) can be expressed as the sum of the retarded potentials of a single layer, a double layer and a volume potential:
+
\frac{1}{r}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544026.png" /></td> </tr></table>
+
\frac{\partial  u ( y) }{\partial  n }
  
In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544028.png" /> do not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544029.png" />, the Kirchhoff formula takes the form
+
- u ( y)
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544030.png" /></td> </tr></table>
+
\frac{\partial  ( 1 / r ) }{\partial  n }
  
and gives a solution of the [[Poisson equation|Poisson equation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544031.png" />.
+
\right ] d \sigma _ {y}  $$
  
The Kirchhoff formula is widely applied in the solution of a whole series of problems. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544032.png" /> is the ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544033.png" /> of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544034.png" /> and centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544035.png" />, then formula (1) is transformed into the relation
+
and gives a solution of the [[Poisson equation|Poisson equation]]  $  \Delta u = - f( x) $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544036.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
The Kirchhoff formula is widely applied in the solution of a whole series of problems. For example, if  $  \Omega $
 +
is the ball  $  | y - x | \leq  t $
 +
of radius  $  t $
 +
and centre  $  x $,
 +
then formula (1) is transformed into the relation
 +
 
 +
$$ \tag{3 }
 +
u ( x , t )  = \
 +
 
 +
\frac{1}{4 \pi }
 +
 
 +
\int\limits _ {r \leq  t }
 +
 
 +
\frac{f ( y , t - r ) }{r }
 +
\
 +
d y + t M _ {t} [ \psi ] +
 +
 
 +
\frac \partial {\partial  t }
 +
 
 +
t M _ {t} [ \phi ] ,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544037.png" /></td> </tr></table>
+
$$
 +
M _ {t} [ \phi ]  = \
 +
 
 +
\frac{1}{4 \pi }
 +
 
 +
\int\limits _ {| y | = 1 }
 +
\phi ( x + t y )  d s _ {y}  $$
  
is the average value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544038.png" /> over the surface of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544039.png" />,
+
is the average value of $  \phi ( x) $
 +
over the surface of the sphere $  | y - x | = t $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544040.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
\left . \phi ( x)  = u \right | _ {t = 0 }  ,\ \
 +
\left . \psi ( x )  = u _ {t} \right | _ {t = 0 .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544042.png" /> are given functions in the ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544043.png" />, with continuous partial derivatives of orders three and two, respectively, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544044.png" /> is a twice continuously-differentiable function for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544046.png" />, then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544047.png" /> defined by (3) is a regular solution of the [[Cauchy problem|Cauchy problem]] (4) for equation (2) when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544049.png" />.
+
If $  \phi ( x) $
 +
and $  \psi ( x) $
 +
are given functions in the ball $  | x | \leq  R $,  
 +
with continuous partial derivatives of orders three and two, respectively, and $  f ( x , t ) $
 +
is a twice continuously-differentiable function for $  | x | < R $,  
 +
0 \leq  t \leq  R - | x | $,  
 +
then the function $  u ( x , t ) $
 +
defined by (3) is a regular solution of the [[Cauchy problem|Cauchy problem]] (4) for equation (2) when $  | x | < R $
 +
and $  t < R - | x | $.
  
 
Formula (3) is also called Kirchhoff's formula.
 
Formula (3) is also called Kirchhoff's formula.
Line 57: Line 225:
 
The Kirchhoff formula in the form
 
The Kirchhoff formula in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544050.png" /></td> </tr></table>
+
$$
 +
u ( x , t )  = \
 +
t M _ {t} [ \psi ] +
 +
 
 +
\frac \partial {\partial  t }
 +
 
 +
t M _ {t} [ \phi ]
 +
$$
  
 
for the wave equation
 
for the wave equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544051.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$ \tag{5 }
 +
\Delta u  = u _ {tt}  $$
  
is remarkable in that the [[Huygens principle|Huygens principle]] does follow from it: The solution (wave) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544052.png" /> of (5) at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544053.png" /> of the space of independent variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544054.png" /> is completely determined by the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544057.png" /> on the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544058.png" /> with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544059.png" /> and radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544060.png" />.
+
is remarkable in that the [[Huygens principle|Huygens principle]] does follow from it: The solution (wave) $  u ( x , t ) $
 +
of (5) at the point $  ( x , t ) $
 +
of the space of independent variables $  x _ {1} , x _ {2} , x _ {3} , t $
 +
is completely determined by the values of $  \phi $,  
 +
$  \partial  \phi / \partial  n $
 +
and $  \psi $
 +
on the sphere $  | y - x | = t $
 +
with centre at $  x $
 +
and radius $  | t | $.
  
 
Consider the following equation of normal hyperbolic type:
 
Consider the following equation of normal hyperbolic type:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544061.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
+
$$ \tag{6 }
 +
\sum _ {i , j = 1 } ^ { {m }  + 1 }
 +
a  ^ {ij} ( x) u _ {x _ {i}  y _ {j} } +
 +
\sum _ { j= } 1 ^ { m+ }  1
 +
b  ^ {j} ( x) u _ {x _ {j}  } + c ( x) u  = \
 +
f ( x)
 +
$$
  
with sufficiently-smooth coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544064.png" />, and right-hand side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544065.png" /> in some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544066.png" />-dimensional domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544067.png" />, that is, a form
+
with sufficiently-smooth coefficients $  a  ^ {ij} ( x) $,  
 +
$  b  ^ {j} ( x) $,  
 +
$  c ( x) $,  
 +
and right-hand side $  f ( x) $
 +
in some $  ( m + 1 ) $-
 +
dimensional domain $  \Omega _ {m+} 1 $,  
 +
that is, a form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544068.png" /></td> </tr></table>
+
$$
 +
\sum _ {i , j = 1 } ^ { {m }  + 1 }
 +
a  ^ {ij} ( x) \xi _ {i} \xi _ {j}  $$
  
that at any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544069.png" /> can be reduced by means of a non-singular linear transformation to the form
+
that at any point $  x \in \Omega _ {m+} 1 $
 +
can be reduced by means of a non-singular linear transformation to the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544070.png" /></td> </tr></table>
+
$$
 +
y _ {0}  ^ {2} -
 +
\sum _ { i= } 1 ^ { m }
 +
y _ {i}  ^ {2} .
 +
$$
  
The Kirchhoff formula generalizes to equation (6) in the case when the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544071.png" /> of independent variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544072.png" /> is even [[#References|[4]]]. Here the essential point is the construction of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544073.png" /> that generalizes the [[Newton potential|Newton potential]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544074.png" /> to the case of equation (6). For the special case of equation (6),
+
The Kirchhoff formula generalizes to equation (6) in the case when the number $  m + 1 $
 +
of independent variables $  x _ {1} \dots x _ {m+} 1 $
 +
is even [[#References|[4]]]. Here the essential point is the construction of the function $  \phi  ^ {(} k) $
 +
that generalizes the [[Newton potential|Newton potential]] $  1/r $
 +
to the case of equation (6). For the special case of equation (6),
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544075.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>
+
$$ \tag{7 }
 +
u _ {tt} -
 +
\sum _ { i= } 1 ^ { m }
 +
u _ {x _ {i}  x _ {i} }  = 0 ,\ \
 +
m \equiv 1  ( \mathop{\rm mod}  2 ),
 +
$$
  
 
the generalized Kirchhoff formula is
 
the generalized Kirchhoff formula is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544076.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>
+
$$ \tag{8 }
 +
u ( x , t )  = \gamma
 +
\int\limits _  \sigma
 +
\sum _ { i= } 1 ^ { k }
 +
( - 1 )  ^ {k}
 +
\left \{
 +
 
 +
\frac{\partial  \phi  ^ {(} k- i+ 1) }{\partial  n }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544077.png" /></td> </tr></table>
+
\left [
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544078.png" /> is a positive number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544079.png" /> is the piecewise-smooth boundary of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544080.png" />-dimensional bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544081.png" /> containing the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544082.png" /> in its interior, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544083.png" /> is the outward normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544084.png" />. Further,
+
\frac{\partial  ^ {i-} 1 u }{\partial  t  ^ {i-} 1 }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544085.png" /></td> </tr></table>
+
\right ] \right . -
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544086.png" /></td> </tr></table>
+
$$
 +
- \left .
 +
\phi  ^ {(} k- i+ 1) \left [
 +
\frac{\partial  ^ {i} u }{\partial
 +
n \partial  t  ^ {i-} 1 }
 +
-
 +
\frac{\partial  r }{\partial  n }
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544087.png" /> denotes the retarded value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544088.png" />:
+
\left [
 +
\frac{\partial  ^ {i} u }{\partial  t  ^ {i}
 +
}
 +
\right ] \right ] \right \}  d \sigma _ {x} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055440/k05544089.png" /></td> </tr></table>
+
where  $  \gamma $
 +
is a positive number,  $  \sigma $
 +
is the piecewise-smooth boundary of an  $  m $-
 +
dimensional bounded domain  $  \Omega _ {m} $
 +
containing the point  $  y $
 +
in its interior, and  $  n $
 +
is the outward normal to  $  \sigma $.
 +
Further,
 +
 
 +
$$
 +
\phi  ^ {(} i)  = \gamma _ {i} r  ^ {-} k- i+ 1 ,\ \
 +
\phi  ^ {(} k)  = r  ^ {2-} m ,\ \
 +
r  =  | y - x | ;
 +
$$
 +
 
 +
$$
 +
\gamma _ {i}  = \textrm{ const } ,\  i  = 1 \dots k - 1 ; \  k  =  m-  
 +
\frac{1}{2}
 +
;
 +
$$
 +
 
 +
and  $  [ \psi ] $
 +
denotes the retarded value of  $  \psi ( x , t ) $:
 +
 
 +
$$
 +
[ \psi ( x , t ) ]  = \psi ( x , t - r ) .
 +
$$
  
 
Formula (8) for equation (6) is sometimes called the Kirchhoff–Sobolev formula.
 
Formula (8) for equation (6) is sometimes called the Kirchhoff–Sobolev formula.
Line 101: Line 357:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Bitsadze,  "Equations of mathematical physics" , MIR  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.S. Vladimirov,  "Equations of mathematical physics" , MIR  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Bateman,  "Partial differential equations of mathematical physics" , Dover  (1944)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M. Mathisson,  "Eine neue Lösungsmethode für Differentialgleichungen von normalem hyperbolischem Typus"  ''Math. Ann.'' , '''107'''  (1932)  pp. 400–419</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M. Mathisson,  "Le problème de M. Hadamard rélatifs à la diffusion des ondes"  ''Acta Math.'' , '''71''' :  3–4  (1939)  pp. 249–282</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S.G. Mikhlin,  "Linear partial differential equations" , Moscow  (1977)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  S.L. Sobolev,  "Sur une généralisation de la formule de Kirchhoff"  ''Dokl. Akad. Nauk SSSR'' , '''1''' :  6  (1933)  pp. 256–262</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  S.L. Sobolev,  "Applications of functional analysis in mathematical physics" , Amer. Math. Soc.  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  A.N. [A.N. Tikhonov] Tichonoff,  A.A. Samarskii,  "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Bitsadze,  "Equations of mathematical physics" , MIR  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.S. Vladimirov,  "Equations of mathematical physics" , MIR  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Bateman,  "Partial differential equations of mathematical physics" , Dover  (1944)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M. Mathisson,  "Eine neue Lösungsmethode für Differentialgleichungen von normalem hyperbolischem Typus"  ''Math. Ann.'' , '''107'''  (1932)  pp. 400–419</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M. Mathisson,  "Le problème de M. Hadamard rélatifs à la diffusion des ondes"  ''Acta Math.'' , '''71''' :  3–4  (1939)  pp. 249–282</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S.G. Mikhlin,  "Linear partial differential equations" , Moscow  (1977)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  S.L. Sobolev,  "Sur une généralisation de la formule de Kirchhoff"  ''Dokl. Akad. Nauk SSSR'' , '''1''' :  6  (1933)  pp. 256–262</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  S.L. Sobolev,  "Applications of functional analysis in mathematical physics" , Amer. Math. Soc.  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  A.N. [A.N. Tikhonov] Tichonoff,  A.A. Samarskii,  "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B.B. Baker,  E.T. Copson,  "The mathematical theory of Huygens's principle" , Clarendon Press  (1950)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Schwartz,  "Théorie des distributions" , '''2''' , Hermann  (1951)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.R. Kirchhoff,  "Vorlesungen über mathematischen Physik"  ''Ann. der Physik'' , '''18'''  (1883)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B.B. Baker,  E.T. Copson,  "The mathematical theory of Huygens's principle" , Clarendon Press  (1950)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Schwartz,  "Théorie des distributions" , '''2''' , Hermann  (1951)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.R. Kirchhoff,  "Vorlesungen über mathematischen Physik"  ''Ann. der Physik'' , '''18'''  (1883)</TD></TR></table>

Latest revision as of 22:14, 5 June 2020


Kirchhoff integral

The formula

$$ \tag{1 } u ( x , t ) = \frac{1}{4 \pi } \int\limits _ \Omega \frac{f ( y , t - r ) }{r } d \Omega _ {y} + $$

$$ + \frac{1}{4 \pi } \int\limits _ \sigma \left [ \frac{1}{r} \frac{\partial u }{\partial n } - u \frac{\partial ( 1 / r ) }{\partial n } + \frac{1}{r} \frac{\partial u }{\partial \tau } \frac{\partial r }{\partial n } \right ] _ {\tau = t - r } d \sigma _ {y} , $$

expressing the value $ u ( x , t ) $ of the solution of the inhomogeneous wave equation

$$ \tag{2 } u _ {tt} - u _ {x _ {1} x _ {1} } - u _ {x _ {2} x _ {2} } - u _ {x _ {3} x _ {3} } = f ( x , t ) $$

at the point $ x =( x _ {1} , x _ {2} , x _ {3} ) \in \Omega $ at the instant of time $ t $ in terms of the retarded volume potential

$$ v _ {1} ( x , t ) = \ \frac{1}{4 \pi } \int\limits _ \Omega \frac{f ( y , t , r ) }{r } d \Omega _ {y} ,\ \ y = ( y _ {1} , y _ {2} , y _ {3} ) , $$

with density $ f $, and in terms of the values of the function $ u ( y , t ) $ and its first-order derivatives on the boundary $ \sigma $ of the domain $ \Omega $ at the instant of time $ \tau = t - r $. Here $ \Omega $ is a bounded domain in the three-dimensional Euclidean space with a piecewise-smooth boundary $ \sigma $, $ n $ is the outward normal to $ \sigma $ and $ r = | x - y | $ is the distance between $ x $ and $ y $.

Let

$$ v _ {1} ( x , t ) = \ \frac{1}{4 \pi } \int\limits _ \sigma \frac{1}{r} \frac{\partial \mu _ {1} ( y , t - r ) }{\partial n } d \sigma _ {y} , $$

$$ v _ {2} ( x , t ) = \frac{1}{4 \pi } \int\limits _ \sigma \frac{\partial ^ {*} }{\partial ^ {*} n } \frac{ \mu _ {2} ( y , t - r ) }{r } d \sigma _ {y} , $$

where

$$ \frac{\partial ^ {*} }{\partial ^ {*} n } \frac{\mu _ {2} ( y , t - r ) }{r } = \ \frac{1}{r} \frac{\partial r }{\partial n } \frac{\partial \mu _ {2} ( y , t - r ) }{\partial t } - \mu _ {2} ( y , t - r ) \frac{\partial ( 1 / r ) }{\partial n } . $$

The integrals $ v _ {1} ( x , t ) $ and $ v _ {2} ( x , t ) $ are called the retarded potentials of the single and the double layer.

The Kirchhoff formula (1) means that any twice continuously-differentiable solution $ u ( x , t ) $ of equation (2) can be expressed as the sum of the retarded potentials of a single layer, a double layer and a volume potential:

$$ u ( x , t ) = v _ {1} ( x , t ) + v _ {2} ( x , t ) + v _ {3} ( x , t ) . $$

In the case when $ u ( x , t ) = u ( x ) $ and $ f ( x , t ) = f ( x ) $ do not depend on $ t $, the Kirchhoff formula takes the form

$$ u ( x) = \frac{1}{4 \pi } \int\limits _ \Omega \frac{f ( y ) }{r } d \Omega _ {y} + \frac{1}{4 \pi } \int\limits _ \sigma \left [ \frac{1}{r} \frac{\partial u ( y) }{\partial n } - u ( y) \frac{\partial ( 1 / r ) }{\partial n } \right ] d \sigma _ {y} $$

and gives a solution of the Poisson equation $ \Delta u = - f( x) $.

The Kirchhoff formula is widely applied in the solution of a whole series of problems. For example, if $ \Omega $ is the ball $ | y - x | \leq t $ of radius $ t $ and centre $ x $, then formula (1) is transformed into the relation

$$ \tag{3 } u ( x , t ) = \ \frac{1}{4 \pi } \int\limits _ {r \leq t } \frac{f ( y , t - r ) }{r } \ d y + t M _ {t} [ \psi ] + \frac \partial {\partial t } t M _ {t} [ \phi ] , $$

where

$$ M _ {t} [ \phi ] = \ \frac{1}{4 \pi } \int\limits _ {| y | = 1 } \phi ( x + t y ) d s _ {y} $$

is the average value of $ \phi ( x) $ over the surface of the sphere $ | y - x | = t $,

$$ \tag{4 } \left . \phi ( x) = u \right | _ {t = 0 } ,\ \ \left . \psi ( x ) = u _ {t} \right | _ {t = 0 } . $$

If $ \phi ( x) $ and $ \psi ( x) $ are given functions in the ball $ | x | \leq R $, with continuous partial derivatives of orders three and two, respectively, and $ f ( x , t ) $ is a twice continuously-differentiable function for $ | x | < R $, $ 0 \leq t \leq R - | x | $, then the function $ u ( x , t ) $ defined by (3) is a regular solution of the Cauchy problem (4) for equation (2) when $ | x | < R $ and $ t < R - | x | $.

Formula (3) is also called Kirchhoff's formula.

The Kirchhoff formula in the form

$$ u ( x , t ) = \ t M _ {t} [ \psi ] + \frac \partial {\partial t } t M _ {t} [ \phi ] $$

for the wave equation

$$ \tag{5 } \Delta u = u _ {tt} $$

is remarkable in that the Huygens principle does follow from it: The solution (wave) $ u ( x , t ) $ of (5) at the point $ ( x , t ) $ of the space of independent variables $ x _ {1} , x _ {2} , x _ {3} , t $ is completely determined by the values of $ \phi $, $ \partial \phi / \partial n $ and $ \psi $ on the sphere $ | y - x | = t $ with centre at $ x $ and radius $ | t | $.

Consider the following equation of normal hyperbolic type:

$$ \tag{6 } \sum _ {i , j = 1 } ^ { {m } + 1 } a ^ {ij} ( x) u _ {x _ {i} y _ {j} } + \sum _ { j= } 1 ^ { m+ } 1 b ^ {j} ( x) u _ {x _ {j} } + c ( x) u = \ f ( x) $$

with sufficiently-smooth coefficients $ a ^ {ij} ( x) $, $ b ^ {j} ( x) $, $ c ( x) $, and right-hand side $ f ( x) $ in some $ ( m + 1 ) $- dimensional domain $ \Omega _ {m+} 1 $, that is, a form

$$ \sum _ {i , j = 1 } ^ { {m } + 1 } a ^ {ij} ( x) \xi _ {i} \xi _ {j} $$

that at any point $ x \in \Omega _ {m+} 1 $ can be reduced by means of a non-singular linear transformation to the form

$$ y _ {0} ^ {2} - \sum _ { i= } 1 ^ { m } y _ {i} ^ {2} . $$

The Kirchhoff formula generalizes to equation (6) in the case when the number $ m + 1 $ of independent variables $ x _ {1} \dots x _ {m+} 1 $ is even [4]. Here the essential point is the construction of the function $ \phi ^ {(} k) $ that generalizes the Newton potential $ 1/r $ to the case of equation (6). For the special case of equation (6),

$$ \tag{7 } u _ {tt} - \sum _ { i= } 1 ^ { m } u _ {x _ {i} x _ {i} } = 0 ,\ \ m \equiv 1 ( \mathop{\rm mod} 2 ), $$

the generalized Kirchhoff formula is

$$ \tag{8 } u ( x , t ) = \gamma \int\limits _ \sigma \sum _ { i= } 1 ^ { k } ( - 1 ) ^ {k} \left \{ \frac{\partial \phi ^ {(} k- i+ 1) }{\partial n } \left [ \frac{\partial ^ {i-} 1 u }{\partial t ^ {i-} 1 } \right ] \right . - $$

$$ - \left . \phi ^ {(} k- i+ 1) \left [ \frac{\partial ^ {i} u }{\partial n \partial t ^ {i-} 1 } - \frac{\partial r }{\partial n } \left [ \frac{\partial ^ {i} u }{\partial t ^ {i} } \right ] \right ] \right \} d \sigma _ {x} , $$

where $ \gamma $ is a positive number, $ \sigma $ is the piecewise-smooth boundary of an $ m $- dimensional bounded domain $ \Omega _ {m} $ containing the point $ y $ in its interior, and $ n $ is the outward normal to $ \sigma $. Further,

$$ \phi ^ {(} i) = \gamma _ {i} r ^ {-} k- i+ 1 ,\ \ \phi ^ {(} k) = r ^ {2-} m ,\ \ r = | y - x | ; $$

$$ \gamma _ {i} = \textrm{ const } ,\ i = 1 \dots k - 1 ; \ k = m- \frac{1}{2} ; $$

and $ [ \psi ] $ denotes the retarded value of $ \psi ( x , t ) $:

$$ [ \psi ( x , t ) ] = \psi ( x , t - r ) . $$

Formula (8) for equation (6) is sometimes called the Kirchhoff–Sobolev formula.

References

[1] A.V. Bitsadze, "Equations of mathematical physics" , MIR (1980) (Translated from Russian)
[2] V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian)
[3] H. Bateman, "Partial differential equations of mathematical physics" , Dover (1944)
[4] M. Mathisson, "Eine neue Lösungsmethode für Differentialgleichungen von normalem hyperbolischem Typus" Math. Ann. , 107 (1932) pp. 400–419
[5] M. Mathisson, "Le problème de M. Hadamard rélatifs à la diffusion des ondes" Acta Math. , 71 : 3–4 (1939) pp. 249–282
[6] S.G. Mikhlin, "Linear partial differential equations" , Moscow (1977) (In Russian)
[7] S.L. Sobolev, "Sur une généralisation de la formule de Kirchhoff" Dokl. Akad. Nauk SSSR , 1 : 6 (1933) pp. 256–262
[8] S.L. Sobolev, "Applications of functional analysis in mathematical physics" , Amer. Math. Soc. (1963) (Translated from Russian)
[9] V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) (Translated from Russian)
[10] A.N. [A.N. Tikhonov] Tichonoff, A.A. Samarskii, "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)

Comments

References

[a1] B.B. Baker, E.T. Copson, "The mathematical theory of Huygens's principle" , Clarendon Press (1950)
[a2] L. Schwartz, "Théorie des distributions" , 2 , Hermann (1951)
[a3] G.R. Kirchhoff, "Vorlesungen über mathematischen Physik" Ann. der Physik , 18 (1883)
How to Cite This Entry:
Kirchhoff formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kirchhoff_formula&oldid=47500
This article was adapted from an original article by A.M. Nakhushev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article