Wave Equation - Mathematical Method

The $d$ -dimensional wave equation is given by the form:

u_{tt} = a^2 \nabla^2 u + f(\mathbf{x},t)

(1)

where $u(\mathbf{x,t}): \mathbb{R}^d\times \mathbb{R}\to \mathbb{R}$ .
The one dimensional wave equation $u_{tt}=a^2u_{xx}$ very naturally arise in modeling of a vibrating string.

The Vibrating String¶

Consider a horinzontally placed string with length $L$ . We make the following assumptions

Now to model the vibration mathematically, we place the string along the $x$ -axis, let $u(x,t)$ be the vertical displacement of the string at position $x$ from equilibrium position at time $t$ .

The Lagrangian Way¶

It’s clear that the kinetic energy of the string should be given by:

K=\frac{1}{2}\int_0^L mu_tdx

(2)