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Mathematics LibreTexts

5.3: Hyperbolic Equation


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As an example of a hyperbolic equation we study the wave equation. One of the systems it can describe is a transmission line for high frequency signals, 40m long.

\[\begin{aligned} \dfrac{\partial^2}{\partial x^2} V &=& \underbrace{LC}_{\text{imp}\times \text {capac}}\dfrac{\partial^2}{\partial t^2} V \nonumber\\ \dfrac{\partial}{\partial x} V (0,t) &=& \dfrac{\partial}{\partial x} V(40,t) = 0, \nonumber\\ V(x,0) &=& f(x),\nonumber\\ \dfrac{\partial}{\partial t} V(x,0) &=& 0,\end{aligned}\] Separate variables, \[V(x,t) = X(x) T(t).\] We find \[\frac{X''}{X} = LC \frac{T''}{T} = -\lambda .\]

Which in turn shows that

\[\begin{aligned} X'' &=&-\lambda X, \nonumber\\ T'' &=& -\frac{\lambda}{LC} T .\end{aligned}\]

We can also separate most of the initial and boundary conditions; we find \[X'(0) = X'(40)=0,\;\;T'(0)=0.\] Once again distinguish the three cases \(\lambda>0\), \(\lambda=0\), and \(\lambda<0\):
(almost identical to previous problem) \(\lambda_n = \alpha_n^2\), \(\alpha_n = \frac{n\pi}{40}\), \(X_n=\cos(\alpha_n x)\). We find that

\[T_n(t) = D_n\cos \left(\frac{n\pi t}{40\sqrt{LC}}\right) + E_n\sin\left(\frac{n\pi t}{40\sqrt{LC}}\right).\] \(T'(0)=0\) implies \(E_n=0\), and taking both together we find (for \(n \geq 1\)) \[V_n(x,t) = \cos\left(\frac{n\pi t}{40\sqrt{LC}}\right) \cos\left(\frac{n\pi x}{40}\right).\] \(X(x) = A + B x\). \(B=0\) due to the boundary conditions. We find that \(T(t) = D t + E\), and \(D\) is 0 due to initial condition. We conclude that \[V_0(x,t) = 1.\] No solution.

Taking everything together we find that

\[V(x,t) = \frac{a_0}{2} + \sum_{n=1}^\infty a_n \cos\left(\frac{n\pi t}{40\sqrt{LC}}\right) \cos\left(\frac{n\pi x}{40}\right).\] The one remaining initial condition gives

\[V(x,0) = f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty a_n \cos\left(\frac{n\pi x}{40}\right).\]

Use the Fourier cosine series (even continuation of \(f\)) to find \[\begin{aligned} a_0 & = & \frac{1}{20} \int_0^{40} f(x) dx, \nonumber\\ a_n & = & \frac{1}{20} \int_0^{40} f(x)\cos\left(\frac{n\pi x}{40}\right) dx.\end{aligned}\]