Transverse Waves & Polarization

Every travelling wave has a propagation direction, set by the wave vector k. It also has an oscillation direction — whatever is actually displaced or changing. A longitudinal wave puts them on the same axis: the air in a sound wave moves back and forth along the direction the sound travels, so once k is given nothing is left to specify. A transverse wave puts the oscillation in the plane perpendicular to k, and a plane contains a whole circle of directions. That leftover choice is the polarization. For light the transverse quantities are E and B, perpendicular to each other and to k; by convention the polarization is the direction of E, because it is E that pushes the charges in a detector or an eye, and B follows from it.

Take a wave along z and resolve its field into components Eₓ cos(kz − ωt) and E_y cos(kz − ωt + δ). Only three numbers matter: the two amplitudes and the phase difference δ. If δ = 0 the components rise and fall together, the tip of E runs back and forth along a fixed line, and the wave is linearly polarized at arctan(E_y/Eₓ) from the x axis. If δ = π the y component is simply inverted, so the state is linear again, along the mirrored direction −arctan(E_y/Eₓ). If δ = ±π/2 with Eₓ = E_y, one component peaks while the other is zero, so the tip sweeps a circle of constant radius at angular frequency ω — circular polarization, right or left handed according to the sign of δ. Every δ other than 0 and π traces an ellipse, with the circle as the special case whose axes are equal. Notice what stays fixed: a circularly polarized beam has |E| constant, so its irradiance is steady even though its direction never is.

A filament or the Sun is an enormous number of atoms radiating independently. Each emits a wave train of order 10⁻⁸ s with its own polarization direction, and the next train points somewhere else. A detector averaging over even a microsecond sees every direction equally represented, which is what unpolarized means — not that the light has no polarization, but that it has no persistent one. Send it through an ideal polarizer and each train contributes cos²θ of its own intensity, with θ spread uniformly round the circle. The average is (1/2π)∫₀^(2π) cos²θ dθ = ½. So the first sheet passes exactly half whatever angle it is set to, and rotating that sheet while watching the transmitted power is precisely how you test whether a beam was unpolarized to begin with.

Model an ideal sheet as a device that transmits the component of E along its transmission axis untouched and absorbs the perpendicular component completely. If the light arrives linearly polarized at angle θ to that axis, the surviving amplitude is E₀ cos θ. Irradiance goes as the square of amplitude, so I = I₀ cos²θ, which is Malus's law. The squaring is the whole content. At θ = 30° the amplitude keeps 0.866 of its value while the irradiance keeps 0.750; at 45° the amplitude keeps 0.707 and the irradiance exactly half; at 60° the amplitude is halved and only 0.250 of the irradiance survives. And whatever angle the light came in at, what emerges is linearly polarized along the new axis.

Cross two ideal polarizers at 90° and nothing emerges: whatever leaves the first is perpendicular to the second, and cos 90° = 0. Now slide a third sheet between them, its axis at θ to the first. The cascade on unpolarized input I₀ runs I₀ → ½I₀ → ½I₀ cos²θ → ½I₀ cos²θ cos²(90° − θ), and with cos(90° − θ) = sin θ and sin θ cos θ = ½ sin 2θ that last expression tidies to (I₀/8) sin²2θ. It is zero at θ = 0° and 90° but rises to I₀/8 at 45°. The point is not the algebra but the re-preparation: the middle sheet does not select from a fixed population, it hands the last sheet a beam polarized at θ, which the last sheet no longer sees edge on.

None of this works for sound. A longitudinal wave has no perpendicular plane in which to choose a direction, so no orientation of any filter can change what gets through, and no experiment on sound has ever found one that does. Light behaves the other way. Malus found in 1808 that light reflected from a window and viewed through a calcite crystal brightened and dimmed as he rotated the crystal, and within a decade Young and Fresnel had concluded that only a transverse wave can behave like that. That killed the longitudinal pressure wave in an elastic ether, and it was one of the strongest hints before Maxwell that light's oscillating quantity is a vector perpendicular to travel. Seismology repeats the argument: S waves in rock can be polarized, P waves cannot.