PX154 - F2 - the wave equation

a fundamental equation, ubiquitous in physics
the wave function:

u (x, t) = A \cos (k x - ω t)

differentiating wrt $t$ :

\frac{\partial u}{\partial t} = A ω \sin (k x - ω t)

- note: $\frac{\partial u}{\partial t} \neq v$

for acceleration:

\begin{aligned} \frac{\partial^{2} u}{\partial t^{2}} & = - A ω^{2} \cos (k x - ω t) \\ \frac{\partial^{2} u}{\partial t^{2}} & = - ω^{2} u (x, t) . . . [1] \end{aligned}

differentiating wrt $x$ : $$\frac{\partial u}{\partial x} = -Ak\sin(kx-\omega t)$$
again:

\begin{array}{r} \frac{\partial^{2} u}{\partial x^{2}} = - A k^{2} \sin (k x - ω t) \\ \frac{\partial^{2} u}{\partial x^{2}} = - k^{2} u (x, t) . . . [2] \end{array}

from $[1]$ and $[2]$ :

\begin{aligned} \frac{1}{k^{2}} \frac{\partial^{2} u}{\partial x^{2}} & = \frac{1}{ω^{2}} \frac{\partial^{2} u}{\partial t^{2}} \\ \frac{\partial^{2} u}{\partial t^{2}} & = \frac{ω^{2}}{k^{2}} \frac{\partial^{2} u}{\partial x^{2}} \end{aligned}

the wave equation:

\frac{\partial^{2} u}{\partial t^{2}} = v^{2} \frac{\partial^{2} u}{\partial x^{2}} $ $ [Y F e q n 15.12] - * * * c o m m e n t s * * * : - a p p l i e s t o r i g h t + l e f t t r a v e l l i n g w a v e s, a n d a l s o t o s t a n d i n g w a v e s - t h i s i s i n 1 D, i n 3 D : $ $ \frac{\partial^{2} u}{\partial t^{2}} = v^{2} {\vec{\nabla}}^{2} u

where: ${\vec{\nabla}}^{2} u = (\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}) u$

electromagnetic waves:

\frac{\partial^{2} E}{\partial t^{2}} = \frac{1}{μ_{0} ϵ_{0}} \frac{\partial^{2} E}{\partial x^{2}}

where, $E =$ electric field, $μ_{0} =$ permeability of free space, $ϵ_{0} =$ permittivity of free space
- we can see here:

c = \frac{1}{\sqrt{μ_{0} ϵ_{0}}} = 3 \times 10^{8} m s^{- 1}