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Submarine Cable Problems Solved by Contour Integration

Published online by Cambridge University Press:  03 November 2016

N. W. McLachlan
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Extract

The solution of the general case of a cable having constant resistance R, inductance L, capacitance C, and leakance G, all expressed per unit length, is usually obtained in the form of an expansion. Using his operational method, Heaviside was the first person to obtain a solution. F. W Carter points out that Heaviside’s result seems to be incorrect, and proceeds to find a solution in the form of a double summation. By aid of complex integration, the integrand being expanded in Bessel functions, H. Jeffreys gives another form of series solution. As this type of problem may arise in lectures on applied mathematics, it is proposed to show how a compact result can be got by contour integration, using a suitable transformation.

Type
Research Article
Information
The Mathematical Gazette , Volume 22 , Issue 248 , February 1938 , pp. 37 - 41
Copyright
Copyright © Mathematical Association 1938

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References

page no 37 note * Proc. Roy. Soc. A. 156, 1, 1936.

page no 37 note † Operational Methods in Mathematical Physics, 2nd edition (1931).

page no 37 note ‡ This avoids the necessity for assuming that the cable is of infinite length, in which case the steady state can never be attained!

page no 37 note § McLachlan, Bessel Functions for Engineers, pp. 107, 108.

page no 37 note ‖ The second solution is omitted, since by hypothesis there is no reflected wave.

page no 38 note * Proc. L.M.S. (2), 15, 401, 1916.