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# Lecture 26 ## The wave equation ### 1. Waves Waves can be described mathematically as functions of position and time: $\qquad u(x, t)=f(x-c t)$ where $c$ is the wave speed. Examples: * Wave on a string * Sound wave * Electromagnetic wave ### 2. Wave equation The wave equation is a second-o...

# Lecture 26 ## The wave equation ### 1. Waves Waves can be described mathematically as functions of position and time: $\qquad u(x, t)=f(x-c t)$ where $c$ is the wave speed. Examples: * Wave on a string * Sound wave * Electromagnetic wave ### 2. Wave equation The wave equation is a second-order partial differential equation that describes the propagation of waves: $\qquad \frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}}$ where $c$ is the wave speed. ### 3. Solution of the wave equation The general solution of the wave equation is: $\qquad u(x, t)=f(x-c t)+g(x+c t)$ where $f$ and $g$ are arbitrary functions. ### 4. Initial conditions To determine the particular solution of the wave equation, we need to specify initial conditions: * Initial displacement: $u(x,0) = f(x)$ * Initial velocity: $\frac{\partial u}{\partial t}(x,0) = g(x)$ ### 5. Boundary conditions To determine the particular solution of the wave equation on a finite domain, we need to specify boundary conditions: * Dirichlet boundary condition: $u(0,t) = a(t)$ and $u(L,t) = b(t)$ * Neumann boundary condition: $\frac{\partial u}{\partial x}(0,t) = a(t)$ and $\frac{\partial u}{\partial x}(L,t) = b(t)$ ### 6. Example Consider a string of length $L$ fixed at both ends. The string is initially displaced to form a triangle, and then released. Find the displacement of the string as a function of position and time. The wave equation is: $\qquad \frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}}$ The initial conditions are: $\qquad u(x,0) = \begin{cases} \frac{2h}{L}x, & 0 \le x \le \frac{L}{2} \\ \frac{2h}{L}(L-x), & \frac{L}{2} \le x \le L \end{cases}$ $\qquad \frac{\partial u}{\partial t}(x,0) = 0$ The boundary conditions are: $\qquad u(0,t) = 0$ and $u(L,t) = 0$ The solution is: $\qquad u(x, t)=\sum_{n=1}^{\infty} A_{n} \sin \left(\frac{n \pi x}{L}\right) \cos \left(\frac{n \pi c t}{L}\right)$ where $\qquad A_{n}=\frac{8 h}{n^{2} \pi^{2}} \sin \left(\frac{n \pi}{2}\right)$