PX275 - G4 - wave equation

caveat:
- PDEs only tend to give nice solutions in well posed problems
- so, only well posed problems with sufficient constraints are considered to get solutions
- considering examples that produce 'separable solutions'

considering the displacement of a vibrating spring, $u (x, t)$
- density waves in a medium, $ρ (x, t)$
forces act along the local gradient, $\frac{\partial u}{\partial x}$
given a string of unknown $τ$ , the force on a test element at a position, $x :$

F = τ [\frac{\partial u (x + Δ x)}{\partial x} - \frac{\partial u (x)}{\partial x}]

from newton's second law:

F = m a = m \frac{\partial^{2} u}{\partial t^{2}} = ρ Δ x \frac{\partial^{2} u}{\partial t^{2}}

where $ρ$ is the mass density (mass per unit length)

τ [\frac{\partial u (x + Δ x)}{\partial x} - \frac{\partial u (x)}{\partial x}] = ρ Δ x \frac{\partial^{2} u}{\partial t^{2}}

\frac{\partial^{2} u}{\partial t^{2}} = \frac{τ}{ρ} \frac{1}{Δ x} [\frac{\partial u (x + Δ x)}{\partial x} - \frac{\partial u (x)}{\partial x}] \to \frac{τ}{ρ} \frac{\partial^{2} u}{\partial x^{2}}

in the limit $Δ x \to 0$

so, the wave PDE:

\frac{\partial^{2} u}{\partial t^{2}} = \frac{τ}{ρ} \frac{\partial^{2} u}{\partial x^{2}}

the wave speed can be defined as: $c = \sqrt{τ / ρ}$

\frac{\partial^{2} u}{\partial t^{2}} = c^{2} \frac{\partial^{2} u}{\partial x^{2}}

if there is friction: friction $\propto \frac{\partial u}{\partial t}$ $⟹ \frac{\partial^{2} u}{\partial t^{2}} = - η \frac{\partial u}{\partial t} + c^{2} \frac{\partial^{2} u}{\partial x^{2}}$