Wave Beats and Polarization

Beats are periodic variations in the loudness of a sound produced by the superposition of two waves of slightly different frequencies, with a beat frequency f_beat = |f₁ − f₂|, and polarization is the restriction of the oscillations of a transverse wave to a single plane; the intensity of polarized light passing through a polarizing filter follows Malus’s law, I = I₀ cos²θ. In brief: beats are used to tune instruments (the beats disappear when the frequencies match). Only transverse waves can be polarized, which shows that light is a transverse wave and sound in air is not. Unpolarized light passing through an ideal polarizer loses half its intensity. Polarizing sunglasses reduce glare because light reflected from horizontal surfaces is partly polarized.

Two guitar strings slightly out of tune produce a wobbling sound, and a pair of polarizing filters can turn from transparent to dark as one is rotated. Both are superposition effects. This guide explains the formation of beats, the polarization of light and Malus’s law, Brewster’s angle, compares the two phenomena in a table, works through numerical examples and corrects common mistakes.

What are beats?

When two sound waves of equal amplitude and close frequencies f₁ and f₂ are heard together, the resultant is y = 2A cos[π(f₁ − f₂)t] sin[π(f₁ + f₂)t]: a wave with the average frequency (f₁ + f₂) ÷ 2 whose amplitude varies slowly with the frequency |f₁ − f₂|/2. Since the loudness depends on the magnitude of the amplitude, which goes through a maximum twice per cycle of the envelope, the loudness rises and falls at the beat frequency f_beat = |f₁ − f₂|. The waves are in phase and reinforce each other at the loud points, and are in antiphase and cancel at the quiet points. Piano tuners and guitarists tune two strings to the same note by listening for beats and adjusting until they vanish. Beats are perceived clearly up to about 10 to 15 Hz; at higher differences, a rough or dissonant sound is heard. Beats occur with any waves: radio signals, light (beating of two lasers), and water waves.

What is polarization?

In a transverse wave, the oscillation is perpendicular to the direction of travel, but it can be in any direction perpendicular to it. A wave is plane polarized (linearly polarized) if the oscillations are confined to a single plane. Light from the Sun or a lamp is unpolarized: it consists of waves with oscillations of the electric field in all directions perpendicular to the ray. A rope passing through a vertical slit shows the idea: only vertical oscillations pass. A polarizing filter (Polaroid) has a transmission axis that allows only the component of the electric field along that axis to pass. Polarization is evidence that light is a transverse wave: longitudinal waves, such as sound in air, cannot be polarized.

What is Malus’s law?

If plane-polarized light of intensity I₀ falls on a polarizer (an “analyzer”) whose axis is at an angle θ to the plane of polarization, only the component E cos θ of the electric field passes. Since intensity is proportional to the square of the amplitude, I = I₀ cos²θ. At θ = 0° all light passes, at 90° (crossed polarizers) none does. Unpolarized light passing through an ideal polarizer emerges plane polarized with half the intensity: I = I₀/2 (the average of cos²θ over all angles is 1/2).

How can light be polarized?

• Selective absorption by a polarizing filter (long-chain molecules aligned in one direction).
• Reflection: light reflected from a nonmetallic surface is partly polarized, with the electric field mostly parallel to the surface. At Brewster’s angle θ_B, where tan θ_B = n₂ ÷ n₁, the reflected light is completely polarized (the reflected and refracted rays are perpendicular). For air to water, θ_B = 53°; for air to glass (n = 1.5), 56°. Polarizing sunglasses with vertical transmission axes block the glare from horizontal surfaces such as water and roads.
• Scattering: light from the blue sky is partly polarized (insects, such as bees, use it for navigation).
• Double refraction (birefringence) in crystals such as calcite splits light into two polarized rays.

Applications: LCD screens (liquid crystals rotate the plane of polarization between crossed polarizers), 3D cinema glasses, stress analysis of transparent plastics (photoelasticity), and camera filters to reduce reflections.

Beats and polarization compared

Worked example: beats, filters and Brewster’s angle

1. Beats. Two tuning forks have frequencies of 440 Hz and 444 Hz. f_beat = |444 − 440| = 4 Hz, so the sound gets loud four times per second (period 0.25 s). The average frequency heard is 442 Hz.
2. Tuning. A guitarist plays a string while a reference note of 220 Hz sounds, and hears 3 beats per second. The string frequency is either 217 Hz or 223 Hz. Tightening the string reduces the beat frequency to 1 Hz: the original frequency was 217 Hz (tightening raised it to 219 Hz). If the beat frequency had risen, it would have been 223 Hz.
3. Polarizers. Unpolarized light of intensity 100 W/m² passes through a polarizer: I = 50 W/m². A second polarizer with its axis at 60° to the first: I = 50 × cos²60° = 50 × 0.25 = 12.5 W/m².
4. Three polarizers. With crossed polarizers (0° and 90°), no light passes. Insert a third at 45° between them: after the first, 50; after the second, 50 × cos²45° = 25; after the third (45° to the second), 25 × cos²45° = 12.5 W/m². Adding a polarizer increases the transmitted light.
5. Brewster’s angle. For light on water (n = 1.33): θ_B = tan⁻¹(1.33) = 53.1° from the normal (36.9° above the surface). At this angle the reflected light is completely polarized horizontally.

How were they discovered?

Beats were described in the 17th century: Joseph Sauveur used them in 1701 to measure the frequency of organ pipes and introduced the term “beats”. Christiaan Huygens discovered polarization in calcite crystals in 1690; Étienne-Louis Malus observed polarization by reflection in 1808 and formulated the cosine-squared law in 1809. David Brewster found the relation for the angle of full polarization in 1815, and Fresnel and Young concluded in 1817 to 1821 that light waves are transverse. Edwin Land invented Polaroid sheet in 1928.

Which mistakes are common?

Mistake 1: Dividing the frequency difference by two for the beat frequency

The loudness varies at |f₁ − f₂|. The envelope amplitude itself has half of that frequency, but the loudness peaks twice per envelope cycle.

Mistake 2: Thinking that sound can be polarized

Sound waves in fluids are longitudinal and cannot be polarized.

Mistake 3: Expecting no light after a second filter with unpolarized light

The first polarizer polarizes the light. The second then follows Malus’s law.

Mistake 4: Applying Malus’s law to unpolarized light

For unpolarized light, the first ideal polarizer passes half of the intensity, independent of its angle.