6.1: The Wave Equation
A wave can be described by a function \(f(x,t)\) , called a wavefunction , which specifies the value of a measurable physical quantity at each position \(x\) and time \(t\) . For simplicity, we will assume that space is one-dimensional, so \(x\) is a single real number. We will also assume that \(f(x,t)\) is a number, rather than a more complicated object such as a vector. For instance, a sound wave can be described by a wavefunction \(f(x,t)\) representing the air pressure at each point of space and time.
The evolution of the wavefunction is described by a partial differential equation (PDE) called the time-dependent wave equation : \[\frac{\partial^2 f}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 f}{\partial t^2}, \;\;\; v \in\mathbb{R}^+.\] The parameter \(v\) , which we take to be a positive real constant, is called the wave speed , for reasons that will shortly become clear.
Sometimes, we re-arrange the wave equation into the following form, consisting of a linear differential operator acting on \(f(x,t)\) : \[\left(\frac{\partial^2}{\partial x^2} - \frac{1}{v^2} \frac{\partial^2}{\partial t^2}\right) \; f(x,t) = 0.\] This way of writing the wave equation emphasizes that it is a linear PDE, meaning that any linear superposition of solutions is likewise a solution.