SearchOnMath

The wave equation, \(\frac{\partial^2u}{\partial t^2}= c^2\nabla^2 u\), relates how \(u\) varies with time and with the spatial coordinates as a wave propagates with speed \(c\).

The Taylor series for an infinitely differentiable \(f\) at point \(a\), given by \(f(x) = \sum_{k=0} ^ {\infty} \frac {f^{(k)}(a)}{k!} \, (x-a)^{k}\), allows \(f\) to be expressed in terms of its derivatives.

Maxwell's equations in a vacuum describe a wave propagating at speed \(c\) as the interaction of electric field \(E\) and magnetic field \(B\): \(\nabla \cdot E = 0\), \(\nabla \times E = -\frac{\partial B}{\partial t}\), \(\nabla \cdot B = 0\), \(\nabla \times B = \frac{1}{c^2} \frac{\partial E}{\partial t}\).