Telegraph equation

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)