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Difference between revisions of "Telegraph equation"

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The partial differential equation
 
The partial differential 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/t/t092/t092340/t0923401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$ \tag{1 }
  
This equation is satisfied by the intensity of the current in a conductor, considered as a function of time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092340/t0923402.png" /> and distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092340/t0923403.png" /> from any fixed point of the conductor. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092340/t0923404.png" /> is the speed of light, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092340/t0923405.png" /> is a capacity coefficient and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092340/t0923406.png" /> is the induction coefficient.
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\frac{\partial  ^ {2} u }{\partial  t  ^ {2} }
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-
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c  ^ {2}
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\frac{\partial  ^ {2} u }{\partial  s  ^ {2} }
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+
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( \alpha + \beta )
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 +
\frac{\partial  u }{\partial  t }
 +
+
 +
\alpha \beta u  =  0.
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$$
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 +
This equation is satisfied by the intensity of the current in a conductor, considered as a function of time t $
 +
and distance $  s $
 +
from any fixed point of the conductor. Here, $  c $
 +
is the speed of light, $  \alpha $
 +
is a capacity coefficient and $  \beta $
 +
is the induction coefficient.
  
 
By the transformation
 
By the transformation
  
<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/t/t092/t092340/t0923407.png" /></td> </tr></table>
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$$
 +
e ^ {1/2 ( \alpha + \beta ) t }
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u ( s, t)  = \
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v ( x, y),\ \
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= s + ct,\ \
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= s - ct,
 +
$$
  
 
equation (1) is reduced to the form
 
equation (1) is reduced 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/t/t092/t092340/t0923408.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$ \tag{2 }
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v _ {xy} + \lambda v  = 0,\ \
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\lambda  = \left (
 +
 
 +
\frac{\alpha - \beta }{4c }
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 +
\right )  ^ {2} .
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$$
  
 
This equation belongs to the class of hyperbolic equations of the second order (cf. [[Hyperbolic partial differential equation|Hyperbolic partial differential equation]]),
 
This equation belongs to the class of hyperbolic equations of the second order (cf. [[Hyperbolic partial differential equation|Hyperbolic partial differential 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/t/t092/t092340/t0923409.png" /></td> </tr></table>
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$$
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v _ {xy} + av _ {x} + bv _ {y} + cv  = f,
 +
$$
  
in the theory of which an important part is played by the [[Riemann function|Riemann function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092340/t09234010.png" />. For equation (2) this function can be written in the explicit form
+
in the theory of which an important part is played by the [[Riemann function|Riemann function]] $  R ( x, y;  \xi , \eta ) $.  
 +
For equation (2) this function can be written in the explicit 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/t/t092/t092340/t09234011.png" /></td> </tr></table>
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$$
 +
R ( x, y; \xi , \eta )  = \
 +
J _ {0} ( \sqrt {4 \lambda ( x - \xi ) ( y - \eta ) } ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092340/t09234012.png" /> is the Bessel function (cf. [[Bessel functions|Bessel functions]]).
+
where $  J _ {0} ( x) $
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is the Bessel function (cf. [[Bessel functions|Bessel functions]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Courant,  D. Hilbert,  "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Courant,  D. Hilbert,  "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The special case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092340/t09234013.png" /> is treated in [[#References|[a1]]].
+
The special case $  \alpha = - \beta $
 +
is treated in [[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. John,  "Partial differential equations" , Springer  (1978)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. John,  "Partial differential equations" , Springer  (1978)</TD></TR></table>

Latest revision as of 08:25, 6 June 2020


The partial differential equation

$$ \tag{1 } \frac{\partial ^ {2} u }{\partial t ^ {2} } - c ^ {2} \frac{\partial ^ {2} u }{\partial s ^ {2} } + ( \alpha + \beta ) \frac{\partial u }{\partial t } + \alpha \beta u = 0. $$

This equation is satisfied by the intensity of the current in a conductor, considered as a function of time $ t $ and distance $ s $ from any fixed point of the conductor. Here, $ c $ is the speed of light, $ \alpha $ is a capacity coefficient and $ \beta $ is the induction coefficient.

By the transformation

$$ e ^ {1/2 ( \alpha + \beta ) t } u ( s, t) = \ v ( x, y),\ \ x = s + ct,\ \ y = s - ct, $$

equation (1) is reduced to the form

$$ \tag{2 } v _ {xy} + \lambda v = 0,\ \ \lambda = \left ( \frac{\alpha - \beta }{4c } \right ) ^ {2} . $$

This equation belongs to the class of hyperbolic equations of the second order (cf. Hyperbolic partial differential equation),

$$ v _ {xy} + av _ {x} + bv _ {y} + cv = f, $$

in the theory of which an important part is played by the Riemann function $ R ( x, y; \xi , \eta ) $. For equation (2) this function can be written in the explicit form

$$ R ( x, y; \xi , \eta ) = \ J _ {0} ( \sqrt {4 \lambda ( x - \xi ) ( y - \eta ) } ), $$

where $ J _ {0} ( x) $ is the Bessel function (cf. Bessel functions).

References

[1] R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German)

Comments

The special case $ \alpha = - \beta $ is treated in [a1].

References

[a1] F. John, "Partial differential equations" , Springer (1978)
How to Cite This Entry:
Telegraph equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Telegraph_equation&oldid=48953
This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article