Distinguish between the phase and group velocity. Assuming that a 100 KHz plane wave is used to communicate a submerged submarine for sea water, σ = 4 s/m, ε = 80ε₀, µ = µ₀, compute the attenuation, phase velocity and skin depth of the wave.

• Phase velocity ($v_p$): This is the speed at which a point of constant phase on a single, monochromatic wave propagates through a medium. It describes how fast the crests or troughs of a wave move. For a wave with angular frequency $\omega$ and wave number $\beta$, the phase velocity is given by $v_p = \frac{\omega}{\beta}$.
• Group velocity ($v_g$): This is the speed at which the overall shape of a wave's amplitude envelope (the "group" of waves) propagates through a medium. It represents the speed at which energy or information is transmitted by the wave. For a wave in a dispersive medium, the group velocity is given by $v_g = \frac{d\omega}{d\beta}$. In non-dispersive media, phase and group velocities are equal.

Given:

• Frequency $f = 100 \text{ KHz} = 10^5 \text{ Hz}$
• Conductivity $\sigma = 4 \text{ S/m}$
• Relative permittivity $\epsilon_r = 80$
• Relative permeability $\mu_r = 1$ (since $\mu = \mu_0$)
• Permittivity of free space $\epsilon_0 = 8.85 \times 10^{-12} \text{ F/m}$
• Permeability of free space $\mu_0 = 4\pi \times 10^{-7} \text{ H/m}$