Wave Equation and its Solutions

Waves

• Described mathematically as functions of position and time: 𝑢(𝑥,𝑡)=𝑓(𝑥−𝑐𝑡)
• c denotes wave speed

Examples of waves

• Wave on a string
• Sound wave
• Electromagnetic wave

Wave Equation

• a second-order partial differential equation
• Describes the propagation of waves: $\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}}$
• c represents wave speed

Solution

• The general solution is: 𝑢(𝑥,𝑡)=𝑓(𝑥−𝑐𝑡)+𝑔(𝑥+𝑐𝑡)
• Where f and g are arbitrary functions

Initial conditions

• Conditions are needed to determine the particular solution
• Initial displacement: 𝑢(𝑥,0)=𝑓(𝑥)
• Initial velocity: $\frac{\partial u}{\partial t}(x,0) = g(x)$

Boundary conditions

• To determine the particular solution on a finite domain, boundary conditions are needed
• 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 $\frac{\partial u}{\partial x}(L,t) = b(t)$

Example: String of length L fixed at both ends

• The string is initially displaced to form a triangle, and then released.
• Task: Find the displacement of the string as a function of position and time.

Wave Equation

• Given by: $\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}}$

Initial Conditions

• Initial displacement: $u(x,0) = \begin{cases} \frac{2h}{L}x, & 0 \le x \le \frac{L}{2} \ \frac{2h}{L}(L-x), & \frac{L}{2} \le x \le L \end{cases}$
• Initial velocity: $\frac{\partial u}{\partial t}(x,0) = 0$

Boundary Conditions

• $u(0,t) = 0$ and $u(L,t) = 0$

Solution

• $u(x, t)=\sum_{n=1}^{\infty} A_{n} \sin \left(\frac{n \pi x}{L}\right) \cos \left(\frac{n \pi c t}{L}\right)$
• where $A_{n}=\frac{8 h}{n^{2} \pi^{2}} \sin \left(\frac{n \pi}{2}\right)$