6.1: The Wave Equation

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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.

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