Interesting Examples of Computational Physics
  • Welcome to the blog of Computational Physics at the IFM!
  • Dynamical Systems
    • The simple pendulum
    • The spherical pendulum
  • The wave equation
    • The simple discretization
    • Boundary conditions
    • Interfaces between mediums
    • Knots on ropes
    • 2D wave equation
  • Time dependent Schrödinger equation
    • The Crank-Nicolson method
    • Free wave packet
    • Quantum Tunneling
    • Harmonic Oscillator
    • Forced Harmonic Oscillator
    • 2D Harmonic Oscillator
    • Wave-particle duality
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The wave equation

PreviousThe spherical pendulumNextThe simple discretization

Last updated 9 months ago

In this chapter, we are going to solve numerically the wave equation. Unlike the previous equations we've been solving, this is a partial differential equation. For this new kind of equations, we are going to learn new methods, such as the method of lines.

Lets start by defining the problem for the 1 dimensional wave equation. We are trying to find a function f which depends of one spatial coordinate x and one temporal coordinate. All we know about f is that it satisfies the wave equation:

∂2f∂x2=1v2∂2f∂t2\frac{\partial^2f}{\partial x^2} = \frac{1}{v^2}\frac{\partial^2f}{\partial t^2}∂x2∂2f​=v21​∂t2∂2f​

and that it satisfies the following initial conditions:

f(x,t=0)=g(x)∂f∂t(x,t=0)=h(x)\begin{align}f(x,t=0) &= g(x)\nonumber\\ \frac{\partial f}{\partial t}(x,t=0) &= h(x) \nonumber \end{align}f(x,t=0)∂t∂f​(x,t=0)​=g(x)=h(x)​

In the following sections, we are going to show ways in which one can approximate the value of this function in a discrete domain.