Wave Attenuation

Consider a medium with nonzero conductivity σ(J = σE gives the current density) and no net charge (ρ = 0).

a) Write the set of Maxwell’s equations appropriate for this medium.

b) Derive the wave equation for E ( or B) in this medium,

c) Consider a monochromatic wave moving in the +x direction with E_­y (or E_z or B_y or B_z ) given by

Show that this wave has an amplitude which decreases exponentially; find the attenuation length (skin depth).

d) For sea water (σ ≈ 5mho/m, or 4.5 × 10¹⁰s^-1 in cgs units), and using radio waves of long wavelength ω = 5 × 10⁵s^-1, calculate the attenuation length. (Why is it hard to communicate with submerged submarines?) You can take for sea water ε = 1, μ = 1.

SOLUTION

a, b) We obtain the equation for the electric (magnetic) field in the same way as

(1)

c) Now, taking E_y in the form E_y = ѱ₀e^{i(kx – ωt)} and substituting into (1) yields

(2)

We have

(3)

To solve for the square root of the complex expression in (3), write

(4)

where a and are real, and

By squaring (S.3.47.4), we find

(5)

(6)

Taking a from (S.3.47.6) and substituting it into (S.3.47.5), we have for b

where we have chosen the branch of the root with the plus sign to satisfy σ = 0 (vacuum) b = k'' = 0 (no dissipation). So

(7)

Therefore, the attenuation length (1/e point) for the amplitude

(8)

whereas for the intensity, the attenuation length is δ/2.

d) For the frequency given in the problem,

and so we can disregard the 1’s in (8) and rewrite it as

At a depth of 10 m below the surface, the intensity attenuation at this frequency will be

which implies that transmission of signals to submerged submarines will require much lower frequencies, f ≤ 10 – 10² Hz.