(b) [15 marks] The Cable equation was used to model signal transmission in the first undersea telegraph cables. The volt
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- B 15 Marks The Cable Equation Was Used To Model Signal Transmission In The First Undersea Telegraph Cables The Volt 1 (161.78 KiB) Viewed 51 times
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(b) [15 marks] The Cable equation was used to model signal transmission in the first undersea telegraph cables. The voltage, V, in a long cable which consists of inner and outer conducting layers separated by insulating material is described by the following system V₁= DVxx - BV, 0<x<∞, -∞<t<∞, V(0, t) = A cos wt, V(x, t) →0 as x →∞. (i) Try a separation of variables-like solution of the form V(x, t) Av(x) cos(wt - kx), = for some constant k, and obtain an equation for v(x). (ii) Solve this equation for v(x): what does this tell you about the behaviour of signals in the cable? (iii) Show that w and k must satisfy the dispersion relation w = 2Dk√k² + B₁