PX154 - F4 - wave groups and group velocity

from PX154 - F3a - principle of superposition#^d12c0f: $$u_{total} = 2A\cos(kx-\omega t)\cos(\Delta kx -\Delta \omega t)$$
- where,
  $\cos (k x - ω t) = \frac{ω}{k} =$ wave speed,
  $\cos (Δ k x - Δ ω t) = \frac{Δ ω}{Δ k} =$ "group velocity"
- keeping $| Δ k | << | k |$ , and then, taking $\frac{Δ ω}{Δ k} \to \frac{d ω}{d k} \to g r o u p v e l o c i t y (v_{g})$
if any amplitude is considered maximum between two minima, this is the speed of this group of waves. $\frac{ω}{k}$ is the "phase velocity", $v_{p}$

dispersion

this is the relationship between $ω$ and $k$ for our wave: $ω (k)$
- waves of different wavelengths travelling at different speeds
eg: gravity waves on deep water (dimensional analysis}:

v_{p} \propto \sqrt{g λ}

- we have, $k = \frac{2 π}{λ}$ , $v_{p} = \frac{ω}{k}$
$$v_{p}= \frac{\omega}{k} \propto \sqrt{g .\frac{2\pi}{k}} \implies v_{p}= constant \times \sqrt{\frac{g}{k}}$$
- or, the "dispersion relation":
$$\omega(k) = constant \times \sqrt{gk}$$
- also, $v_{g} = \frac{d ω}{d k} = \frac{c o n s t a n t}{2} \sqrt{\frac{g}{k}}$
- so, $v_{g} = \frac{v_{p}}{2}$
- these waves are dispersive:
- wave speed( $v_{p}$ ) depends on $k$ , hence wavelength
- $v_{g} \neq v_{p}$

eg: capillary waves on water:

ω^{2} = \frac{σ}{ρ} k^{3}

where, $σ =$ surface tension, $ρ =$ density
$2 ω \frac{d ω}{d k} = 3 \frac{σ}{ρ} k^{2}$
$v_{g} = \frac{d ω}{d k} = \frac{3}{2} \sqrt{\frac{σ}{ρ} k}$
$v_{p} = \frac{ω}{k} = \sqrt{\frac{σ}{ρ} k}$
$∴ v_{g} = \frac{3}{2} v_{p}$
- waves are dispersive:
- $v_{p} \propto \sqrt{k}$
- $v_{g} \neq v_{p}$

eg: sound waves in air
- for sound:
$ω (k) = k \sqrt{\frac{B}{ρ}}$
where, $B =$ bulk radius, $ρ =$ density of air
$v_{p} = \frac{ω}{k} = \sqrt{\frac{B}{ρ}}$
$v_{g} = \frac{d ω}{d k} = \sqrt{\frac{B}{ρ}}$
- these are non-dispersive waves:
  - $v_{p}$ doesn't depend on $k$
  - $v_{g} = v_{p}$
also important for light:
- speed of light in vacuum is $c = 3 \times 10^{8} m s^{- 1}$
- now, snell's law:
$n_{1} \sin θ_{1} = n_{2} \sin θ_{2}$
where, $n = \frac{c}{v}$ is the refractive index, $θ$ is the angle with normal
- for glass: $n_{1} = 1, n_{2} = 1.5$ ; $v = \frac{c}{n}$
  $\sin θ_{2} = \frac{v_{2}}{c} \sin θ_{1}$
- light travelling through materials exhibits dispersion (eg: a prism)
eg: EM waves in the ionosphere:

(ω (k))^{2} = c^{2} k^{2} + ω_{p}^{2}

where, $w_{p} =$ cyclotron frequency (a constant)
- find $v_{p}$ and $v_{g}$
$v_{p} = \frac{ω}{k} = \sqrt{c^{2} + \frac{ω_{p}^{2}}{k^{2}}}$
$v_{g} = \frac{d ω}{d k} = \frac{2 c^{2} k}{\sqrt{c^{2} k^{2} + ω_{p}^{2}}}$
- the waves are dispersive:
$v_{p} \neq v_{g}$
$v_{p}$ depends on $k$
- any thing strange about $v_{p}$ ?
- faster than the speed of light