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天河懸星銀蟲 (初入文壇)
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[求助]
如題，電報方程引入非線性微擾后該如何解答，緊急求助
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2. Consider a lossless ideal transmission line with inductance per unit length L, and capacitance per unit length C. The current and voltage distribution is V(x,t) and I(x,t) taken together, these two parameters determine the complete electromagnetic pattern in the system. Energy propagates in the system due to the telegrapher’s equations: ∂V = −L ∂I ∂x ∂t (2.1) ∂I = −C ∂V ∂x ∂t (2.2) At x=-‐∞, an RF source produces an excitation with amplitude 1 and angular frequency ω. We’ll consider this RF source to be an ideal, unidirectional source; that is, any incident radiation traveling in the negative x direction will not be reflected,but continue to ‐∞. Thus the system has the solution: V(x,t)=cos(kx-wt) (2.3) Here we have k = (LC)^0.5 ω (2.4) Now consider the possibility that the capacitance on the region (0,L) becomes slightly nonlinear. That is to say, the relationship between the charge per unit length q, and the voltage V, becomes: q = CV − δV 2 (2.5) Here δ is a small perturbation; we have δV< (2.2) and (2.3) would be the solution. Calculate the new solution for V as a function of x and t for the modified system to first order in δ. 我的想法是總有V的時間和空間二次偏微分方程，存在微擾后C'=(qV'-δV' 2)/V' =q- 2δV'，V’為單位長度兩端電勢差。取原電勢為 v ，V ' =v+U ，U 作為微擾項遠小于 v，有C'近似為q-2δv,對于v ，在固定單位區(qū)間上是含時的吧。。。。。不知道怎么處理。........還有我的理解有誤嗎 |
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