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Chapter 10: Problem 6
and hence one obtains (10.122).
Let \(u_{1}(x, t), u_{2}(x, t), u_{3}(x, t)\), respectively, be solutions of the
problems (for \(0
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Let \(m_{1}=m_{2}=\cdots=m_{N}=m, h_{\sigma}=0\) and \(F_{\sigma}(t) \equiv 0\) for \(\sigma=1, \ldots, N\) and \(u_{0}=\) \(u_{N+1}=0\) in (10.51), so that one has case (a), with equal masses. Show that the substitution \(u_{\sigma}=A(\sigma) \sin (\lambda t+\epsilon)\) leads to the difference equation with boundary conditions: $$ \begin{aligned} \Delta^{2} A(\sigma)+p^{2} A(\sigma) &=0, & & p^{2}=m \lambda^{2} / k^{2}, \\ A(0) &=0, & A(N+1)=0 . \end{aligned} $$ Use the result of Problem \(5(\mathrm{~d})\) to obtain the \(N\) normal modes a $$ \begin{aligned} u_{\sigma}(t) &=\sin \left(\frac{n \pi}{N+1} \sigma\right) \sin \left(\lambda_{n} t+\epsilon_{n}\right) \\ \lambda_{n} &=\frac{2 k}{\sqrt{m}} \sin \frac{n \pi}{2(N+1)}, \quad n=1 \ldots, N . \end{aligned} $$ Show that \(0<\lambda_{1}<\lambda_{2}<\cdots<\lambda_{N}\).
Prove that the function \(\Phi_{\mu}\) defined by \((10.147)\) attains its minimum value, among smooth functions \(u(x)\) satisfying the boundary conditions \(u(0)=u(L)=0\), when \(u\) is the solution of the equation \(-K^{2} u^{\prime \prime}(x)=F(x)\). [Hint: Take \(L=\pi\) for convenience. Then express the integral in terms of Fourier sine coefficients of \(u(x), u^{\prime}(x)\), and \(F(x)\), using Theorem 14 of Section 7.13. This gives a separate minimum problem for each \(n\), which is solved precisely when \(-K^{2} u^{\prime \prime}=F(x)\).]
Let \(u(x, t)\) be a solution of the partial differential equation
$$
\frac{\partial^{2} u}{\partial t^{2}}-\frac{\partial^{2} u}{\partial
x^{2}}=\sin x \sin \omega t, \quad 0
Show that separation of variables \(u(r, \theta)=R(r) \Theta(\theta)\) in the problem in polar coordinates for the domain \(r<1\) : $$ \nabla^{2} u+\lambda u \equiv \frac{1}{r^{2}}\left[r \frac{\partial}{\partial r}\left(r \frac{\partial u}{\partial r}\right)+\frac{\partial^{2} u}{\partial \theta^{2}}\right]+\lambda u=0, \quad u(1, \theta)=0 . $$ leads to the problems: $$ \begin{gathered} \left(r R^{\prime}\right)^{\prime}+\left(\lambda r-\frac{\mu}{r}\right) R=0, \quad R(1)=0, \\ \Theta^{\prime \prime}+\mu \Theta=0 . \end{gathered} $$ If we require that \(u(r, \theta)\) be continuous in the circle \(r \leq 1\), then \(\Theta(\theta)\) must be periodic in \(\theta\), with period \(2 \pi\). Show that this implies that \(\mu=m^{2}(m=0,1,2,3, \ldots)\) and that
Prove that under the conditions stated in the theorem of Section \(10.7\) the solution \(u(x, t)\) satisfying initial conditions (10.78) must have the form \(\left(10.75^{\prime}\right)\) and is hence uniquely determined. [Hint: Under the assumptions made, \(u(x, t)\) has a representation as a Fourier sine series in \(x\) : $$ u=\sum_{n=1}^{\infty} \phi_{n}(t) \sin n x, \quad \phi_{n}(t)=\frac{2}{\pi} \int_{0}^{\pi} u(x, t) \sin n x d x . $$ Differentiate the second equation twice with respect to \(t\), using Leibnitz's Rule (Section 4.9) and integration by parts to show that \(\phi_{n}^{\prime \prime}(t)+a^{2} n^{2} \phi_{n}(t)=0\). Hence \(\phi_{n}(t)=\alpha_{n} \sin (\) nat \()+\beta_{n} \cos (n a t)\).
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