Transverse Waves Explained: Wavelength, Amplitude, Polarization

What Is a Transverse Wave?

A wave is a disturbance that transfers energy through a medium (or through empty space) without transferring matter. The medium itself doesn't travel — it oscillates locally while the wave pattern moves forward. Drop a stone into a pond and watch: the water doesn't rush outward, but the ripple pattern does. The energy travels; the water stays.

In a transverse wave, the oscillation of the medium is perpendicular to the direction of wave propagation. Picture a rope stretched horizontally and shaken up and down at one end: the rope moves vertically (up-down), but the wave travels horizontally along its length. The displacement and the travel direction are at right angles — that's the defining characteristic of every transverse wave.

This is distinct from a longitudinal wave, where the medium oscillates parallel to the direction of travel — compressions and rarefactions moving along the same axis. Sound is the most familiar longitudinal wave. Seismic P-waves (primary waves) are longitudinal; seismic S-waves (secondary, or shear waves) are transverse — and because S-waves cannot propagate through liquids (liquids can't sustain shear stress), seismologists use the absence of S-waves on the far side of the Earth to infer that the outer core is liquid.

Key Distinction

Transverse wave: displacement perpendicular to propagation. Examples: light, all electromagnetic radiation, guitar strings, water surface waves, seismic S-waves.
Longitudinal wave: displacement parallel to propagation. Examples: sound, seismic P-waves, ultrasound.

Electromagnetic waves — light, radio waves, X-rays, microwaves — are transverse waves, but with a key difference: they require no medium. They are oscillating electric and magnetic fields, perpendicular to each other and to the direction of travel, propagating through a vacuum at the speed of light c = 3 × 10⁸ m/s. For more on how Maxwell's equations predict this, see our Electromagnetism guide.

Wave Properties: Amplitude, Wavelength, Frequency, Period

Four quantities describe any periodic wave completely. Understanding what each one represents — and what it doesn't represent — is the foundation of every wave problem.

A Amplitude — the maximum displacement of the medium from its equilibrium position. Unit: metres. Determines the wave's energy: energy ∝ A².

λ (lambda) Wavelength — distance between two successive points in phase (crest to crest, or trough to trough). Unit: metres.

f Frequency — number of complete oscillations per second. Unit: hertz (Hz = s⁻¹). Determined by the source, not the medium.

T = 1/f Period — time for one complete oscillation. Unit: seconds. T and f are reciprocals.

These four quantities are connected through the fundamental wave speed equation:

v = fλ = λ/T

Wave speed (m/s) = frequency (Hz) × wavelength (m). The wave speed depends on the medium, not the frequency — this is why different notes on a guitar string have different wavelengths but travel at the same speed.

A subtle but important point: when a wave moves from one medium to another (say, light entering glass from air), its frequency stays the same — it's fixed by the source. What changes is the speed (and therefore wavelength). This is why refraction happens — wavelength changes, speed changes, direction bends. The frequency remains the fingerprint of the original source.

The Wave Equation — Derived

A sinusoidal transverse wave travelling in the positive x-direction can be written as:

y(x, t) = A sin(kx − ωt + φ)

y = displacement of medium at position x and time t. A = amplitude. k = wave number (rad/m). ω = angular frequency (rad/s). φ = initial phase.

Two new quantities appear here that need defining:

k = 2π/λ Wave number k — spatial frequency: how many radians of the wave fit in one metre. Analogous to ω for time.

ω = 2πf = 2π/T Angular frequency ω — temporal frequency in radians per second. ω = 2πf. Same as in simple harmonic motion.

The wave speed in terms of these quantities:

v = ω/k = fλ

Three equivalent expressions for wave speed. Choose whichever variables you have in a given problem.

The reason this function describes a wave is that any point of constant phase (where kx − ωt = const) moves at velocity dx/dt = ω/k = v in the positive x-direction. The entire pattern shifts forward at speed v. For a wave moving in the negative x-direction, the sign of ωt flips: y = A sin(kx + ωt).

The Partial Differential Equation

The wave equation in its full mathematical form is a partial differential equation (PDE). For a transverse wave on a string with tension T and linear mass density μ:

∂²y/∂t² = (T/μ) ∂²y/∂x² where v = √(T/μ)

The wave equation PDE. The wave speed on a string is √(tension/linear density). Heavier strings under the same tension produce slower, lower-frequency standing waves — this is why bass guitar strings are thicker.

Insight: Why Tight Strings Ring Higher

v = √(T/μ) explains guitar tuning directly. Tightening a string increases T → increases v → for fixed string length, higher harmonics → higher pitch. Thicker strings (larger μ) decrease v → lower pitch. Pressing a fret shortens the vibrating length, forcing higher frequencies for the same harmonics.

Energy Carried by a Transverse Wave

A wave carries energy from one place to another without net transport of matter. For a sinusoidal transverse wave on a string, the average power (energy per unit time) transmitted is:

P = ½μvω²A²

Average power carried by a sinusoidal wave. μ = linear mass density, v = wave speed, ω = angular frequency, A = amplitude.

Two results stand out here. First, power is proportional to A² — doubling the amplitude quadruples the power (energy). This is true for all wave types, including sound (loudness) and light (intensity). Second, power is proportional to ω² — doubling the frequency at the same amplitude quadruples the power. This is why high-frequency ultrasound is used for deep tissue imaging: it carries more energy per unit amplitude than audible sound.

For a wave spreading out in three dimensions (like light from a point source), the intensity I = P/Area decreases as the wavefront spreads. On a spherical surface of radius r, the area is 4πr², giving the inverse square law: I ∝ 1/r². Doubling your distance from a light source reduces its apparent brightness to one quarter.

Superposition and Interference

The principle of superposition states: when two or more waves overlap in the same medium, the resulting displacement at any point is the algebraic sum of the displacements of each individual wave. The waves pass through each other completely unchanged; the superposition is only local and temporary.

y_total(x,t) = y₁(x,t) + y₂(x,t)

Superposition principle. Applies to all linear waves (valid for small amplitudes in most media).

The result of superposition depends on the phase relationship between the waves:

Constructive interference: waves arrive in phase (crests meet crests). The combined amplitude equals the sum of individual amplitudes. Path difference = nλ where n = 0, 1, 2, …
Destructive interference: waves arrive exactly out of phase (crests meet troughs). They cancel. Path difference = (n + ½)λ.
Partial interference: all cases in between, producing a combined amplitude between the two extremes.

Interference is the phenomenon that proves the wave nature of light. Thomas Young's double-slit experiment (1801) produced alternating bright and dark fringes — impossible if light were purely a stream of particles. The interference pattern is a direct map of the wave's phase relationships. See our Optics guide for Young's experiment derived in full.

Standing Waves and Resonance

When a transverse wave reflects off a fixed boundary and overlaps with the incoming wave, the superposition creates a remarkable pattern: a standing wave. Unlike a travelling wave, a standing wave doesn't appear to move — certain points (nodes) are always stationary, and others (antinodes) oscillate with maximum amplitude.

Mathematically, adding two waves of equal amplitude and frequency travelling in opposite directions:

y₁ = A sin(kx − ωt) + y₂ = A sin(kx + ωt) = 2A sin(kx) cos(ωt)

Using the sum-to-product identity: sinα + sinβ = 2sin((α+β)/2)cos((α−β)/2). The result is a spatial pattern 2A sin(kx) that oscillates in time as cos(ωt). The wave pattern is stationary — no net energy transfer.

Nodes occur where sin(kx) = 0, i.e., at x = 0, λ/2, λ, 3λ/2, … (separated by λ/2). Antinodes occur where |sin(kx)| = 1, at x = λ/4, 3λ/4, 5λ/4, … (halfway between nodes).

Resonant Frequencies: Strings Fixed at Both Ends

A guitar string fixed at both ends can only sustain standing waves where nodes occur at both fixed endpoints. This means an integer number of half-wavelengths must fit in the string length L:

L = nλ/2 ⟹ λₙ = 2L/n ⟹ fₙ = nv/2L (n = 1, 2, 3, …)

Harmonics of a string fixed at both ends. f₁ = v/2L is the fundamental frequency. f₂ = 2f₁ is the first overtone (second harmonic). fₙ = nf₁.

This is why musical strings produce rich tonal quality: they vibrate simultaneously at f₁ (fundamental), f₂, f₃, and higher harmonics. The relative amplitudes of these harmonics — the timbre — is what makes a violin sound different from a guitar even when playing the same note at the same pitch.

Open vs Closed Pipes

Pipes in wind instruments follow analogous rules. An open pipe (open at both ends) has antinodes at both ends: fₙ = nv/2L, same harmonics as a string. A closed pipe (closed at one end) has a node at the closed end and antinode at the open end — only odd harmonics: fₙ = nv/4L for n = 1, 3, 5, … This is why clarinets (closed-end) sound differently from flutes (open-end) at similar lengths.

Polarization — A Property Only Transverse Waves Have

Polarization is the property that distinguishes transverse waves from longitudinal ones. A transverse wave on a rope can oscillate up-down, left-right, or at any angle — the direction of oscillation relative to the propagation direction is the polarization direction. Longitudinal waves (like sound) have no polarization, because there's only one axis of oscillation: along the direction of travel.

For light, the polarization direction is the direction of the oscillating electric field vector. Ordinary (unpolarized) light from a bulb or the sun has electric field vectors oscillating randomly in all transverse directions simultaneously — a superposition of all polarization angles. Polarized light oscillates in a single, fixed plane.

How Does Polarization Happen?

Three main mechanisms produce polarized light:

Polarizing filters (linear polarizers): materials (like Polaroid film) that contain molecules aligned in one direction. They absorb electric field components parallel to their absorption axis and transmit those parallel to the transmission axis. The result: linearly polarized light at the output, at half the original intensity.
Reflection: when light reflects off a flat non-metallic surface at a specific angle (Brewster's angle, θ_B = arctan(n₂/n₁)), the reflected beam is perfectly polarized parallel to the surface. Polarized sunglasses exploit this: they block the s-polarized reflected glare from roads and water.
Scattering: sky light is partially polarized because atmospheric molecules preferentially scatter polarized components at right angles to the sun. The sky is most polarized at 90° from the sun.

Malus's Law

When polarized light of intensity I₀ passes through a second polarizer (the "analyser") with its transmission axis at angle θ to the light's polarization direction, the transmitted intensity is given by Malus's Law:

I = I₀ cos²θ

Malus's Law (Étienne-Louis Malus, 1809). I₀ = initial intensity of polarized light, θ = angle between polarization direction and analyser axis, I = transmitted intensity.

The cos²θ factor comes directly from the vector projection of the electric field onto the analyser axis. The electric field component that passes through is E₀ cosθ; since intensity is proportional to E², the transmitted intensity is I₀ cos²θ.

Check the extremes: at θ = 0° (analyser aligned with polarization), cos²0° = 1 — full transmission. At θ = 90° (analyser perpendicular to polarization), cos²90° = 0 — no transmission. This is exactly what you observe when you hold two Polaroid lenses at right angles: complete blackout.

Practical Application

LCD screens work using polarization. Two perpendicular polarizers sandwich a liquid crystal layer. Without a voltage, the liquid crystal rotates the light's polarization direction by 90° — allowing light through the second polarizer. Apply a voltage: the crystal aligns, stops rotating the light, and the pixel goes dark. Every pixel in your screen is a voltage-controlled polarization switch.

Five Worked Examples

Example 1 — Basic

Find wavelength, wave number, and period from a wave equation

Giveny(x,t) = 0.04 sin(12x − 48t) metres | x in metres, t in seconds

Compare with y = A sin(kx − ωt): A = 0.04 m, k = 12 rad/m, ω = 48 rad/s

Wavelength: λ = 2π/k = 2π/12 = 0.524 m

Period: T = 2π/ω = 2π/48 = 0.131 s

Frequency: f = 1/T = 7.64 Hz

Wave speed: v = ω/k = 48/12 = 4.0 m/s (or v = fλ = 7.64 × 0.524 = 4.0 m/s ✓)

✓ A = 0.04 m | λ = 0.524 m | T = 0.131 s | f = 7.64 Hz | v = 4.0 m/s

Example 2 — Standard

Write the wave equation for a given wave

GivenTransverse wave: amplitude = 2 cm, frequency = 120 Hz, wave speed = 36 m/s, moving in the +x direction. At t = 0, x = 0, displacement = 0 and increasing.

ω = 2πf = 2π × 120 = 753.98 rad/s

k = ω/v = 753.98/36 = 20.94 rad/m (or k = 2π/λ, where λ = v/f = 0.3 m → k = 2π/0.3 = 20.94 ✓)

At t=0, x=0: y = A sin(0 − 0 + φ) = 0 and dy/dt > 0 → φ = 0 (sin starts at 0 and rises)

Full equation: y(x,t) = 0.02 sin(20.94x − 753.98t) metres

✓ y(x,t) = 0.02 sin(20.94x − 754t) m

Example 3 — Standing Waves

Guitar string harmonics

GivenGuitar string: length L = 0.65 m, linear density μ = 3.0 × 10⁻³ kg/m, tension T = 75 N. Find: wave speed, fundamental frequency, and first three harmonic frequencies.

Wave speed: v = √(T/μ) = √(75 / 3×10⁻³) = √(25,000) = 158.1 m/s

Fundamental (n=1): f₁ = v/2L = 158.1/(2×0.65) = 158.1/1.3 = 121.6 Hz

Second harmonic (n=2): f₂ = 2f₁ = 243.2 Hz

Third harmonic (n=3): f₃ = 3f₁ = 364.8 Hz

✓ v = 158.1 m/s | f₁ = 121.6 Hz | f₂ = 243.2 Hz | f₃ = 364.8 Hz

Example 4 — Malus's Law

Intensity through two polarizers

GivenUnpolarized light, intensity I₀ = 600 W/m² passes through two polarizers. The first polarizer is oriented vertically. The second is at 35° to the first. Find the final intensity.

After the first polarizer (unpolarized → polarized): intensity is halved. I₁ = I₀/2 = 300 W/m²

Apply Malus's Law for the second polarizer: I₂ = I₁ cos²(35°) = 300 × (0.8192)² = 300 × 0.671 = 201.3 W/m²

✓ Final intensity ≈ 201 W/m² — about 33.5% of the original.

Example 5 — Harder

Three polarizers — find angle for a target intensity

GivenUnpolarized light, I₀ = 800 W/m². Three polarizers in series: polarizer 1 at 0°, polarizer 2 at angle θ, polarizer 3 at 90°. Find θ that maximises the final transmitted intensity.

After P1 (unpolarized→polarized): I₁ = I₀/2 = 400 W/m². Polarization direction: 0°.

After P2 (angle θ from P1): I₂ = I₁ cos²θ = 400 cos²θ. Polarization direction: θ.

After P3 (at 90° from P1, so (90°−θ) from P2): I₃ = I₂ cos²(90°−θ) = 400 cos²θ · sin²θ

Using identity: cos²θ sin²θ = (1/4)sin²(2θ). So I₃ = 100 sin²(2θ). This is maximised when sin²(2θ) = 1, i.e., 2θ = 90° → θ = 45°.

Maximum I₃ = 100 × 1 = 100 W/m². Without the middle polarizer (P1 and P3 at 90°): I = 0. The middle polarizer at 45° allows 12.5% of original light through — from zero!

✓ Optimal angle: θ = 45° | Maximum transmitted intensity: 100 W/m² (12.5% of I₀)

Transverse vs Longitudinal: A Systematic Comparison

Property	Transverse Waves	Longitudinal Waves
Oscillation direction	Perpendicular to propagation	Parallel to propagation
Can be polarized?	Yes	No
Requires medium?	Not always (EM waves travel in vacuum)	Yes (must compress/expand something)
Speed in solids vs liquids	Slower in liquids (shear modulus → 0)	Similar or faster (bulk modulus)
Examples	Light, radio waves, seismic S-waves, guitar strings, surface water waves	Sound, seismic P-waves, ultrasound, pressure waves in pipes
Mathematical form	y(x,t) = A sin(kx−ωt) perpendicular to x̂	s(x,t) = A sin(kx−ωt) parallel to x̂

Real-World Transverse Waves

Electromagnetic spectrum: Every form of EM radiation — radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, gamma rays — is a transverse wave. They all travel at c = 3×10⁸ m/s in vacuum and differ only in frequency (and therefore wavelength and energy). The full picture is in our Waves guide.
Seismology: Seismic S-waves (shear waves) are transverse — the ground oscillates perpendicular to the direction the wave travels. They move at roughly 3.5 km/s in the Earth's crust and cannot pass through the liquid outer core. Seismologists use S-wave shadow zones to map the Earth's interior structure.
Musical strings: Every stringed instrument — guitar, violin, piano, harp — produces sound through transverse standing waves on strings. The fundamental frequency and harmonics determine pitch and timbre. Changing string tension (tuning), length (fretting), or density (different strings) changes the wave speed and hence the harmonic frequencies.
Polarized sunglasses: Glare from horizontal surfaces (roads, water) is partially polarized horizontally, because reflection at shallow angles preferentially polarizes in the horizontal plane. Polaroid lenses with a vertical transmission axis selectively block this glare — reducing reflected light without significantly dimming the overall scene.
Fibre optic communications: Light pulses in optical fibres are transverse EM waves. Polarization-maintaining fibres are used in precision sensing and quantum communication, where polarization state carries information.

Common Misconceptions

"The medium moves in the direction of the wave." It doesn't — the medium oscillates perpendicular to the direction of wave travel (for transverse waves). The pattern moves forward; the medium oscillates locally and returns to its rest position.
"Amplitude and wavelength are the same thing." Completely different properties. Amplitude is about how much the medium displaces (determines energy). Wavelength is about how long each cycle is in space (determines the wave's "size" and frequency at a given speed).
"Light slows down when it changes colour." In vacuum, all visible light travels at exactly c regardless of frequency. In a medium, different frequencies have slightly different speeds (dispersion) — which is why prisms separate white light into a spectrum. But in vacuum, the speed is always c.
"Standing waves are a different kind of wave." Standing waves are made of two travelling waves superposed. They're not a separate category — they're a consequence of reflection and superposition using the same physics as every other transverse wave.
"Polarizers block all the light." A single linear polarizer blocks half the intensity of unpolarized light (transmitting the component aligned with the transmission axis). Two polarizers at 90° block all light. But as Example 5 shows, a third polarizer at 45° in between restores partial transmission — a counterintuitive but real result.

Frequently Asked Questions

Polarization describes which direction in the plane perpendicular to propagation the medium oscillates. In a longitudinal wave, the oscillation is along the propagation axis — there's only one possible axis of oscillation, so there's no perpendicular plane to "choose from." Polarization requires the two degrees of freedom available in the transverse plane. Since longitudinal waves have only one degree of freedom in their displacement direction, the concept of polarization simply doesn't apply.

In a standing wave, a node is a point of permanent zero displacement — the medium at that position never moves. Nodes occur at fixed boundaries (like the ends of a guitar string) and at half-wavelength intervals between them. An antinode is a point of maximum displacement — the medium there oscillates with the greatest amplitude. Antinodes sit exactly halfway between nodes, displaced by λ/4 from each adjacent node.

It describes an idealised infinite sinusoidal plane wave moving in the +x direction — one that extends infinitely in both space and time with perfectly regular oscillations. Real waves are always finite wave packets, built from a superposition of many sinusoidal waves of slightly different frequencies (Fourier synthesis). The sinusoidal form is used because it's the solution to the wave equation PDE, and any real wave shape can be decomposed into sinusoidal components via Fourier analysis. For practical problems involving strings, light pulses, and sound bursts, the idealised form gives excellent approximations as long as the wave is reasonably monochromatic and extends over many wavelengths.

The wave speed v = √(T/μ) is the speed at which the wave pattern travels along the string — it depends on the physical properties of the string (tension and density) and is constant for a given string. The transverse particle velocity is how fast individual points on the string move up and down: vᵧ = ∂y/∂t = −Aω cos(kx−ωt). This varies with position and time, and its maximum value is Aω. The two speeds are completely different quantities — you could have a slow wave pattern (small T/μ) with fast-oscillating particles (large Aω), or vice versa.

When light strikes a dielectric interface (like air-glass or air-water) at Brewster's angle θ_B = arctan(n₂/n₁), the reflected beam contains only s-polarized light (oscillating parallel to the surface, perpendicular to the plane of incidence). The p-polarized component (oscillating in the plane of incidence) is completely refracted — none is reflected. This produces perfectly polarized reflected light without any special polarizing filter. For a glass surface (n≈1.5), θ_B ≈ 56°. For water (n≈1.33), θ_B ≈ 53°. Glare off roads and water occurs at roughly these angles, which is why polarized sunglasses are so effective at eliminating it.

Frequency is determined by the source — the rate at which it generates wave cycles. At the boundary between two media, for every cycle that arrives, one cycle must cross (otherwise crests would pile up or disappear at the interface, which can't happen physically). So the boundary forces the frequency on both sides to be equal. What changes is the wave speed (determined by the new medium), and therefore the wavelength (since v = fλ and f is fixed). This frequency-conservation at boundaries is the root cause of refraction.

No — energy is conserved even in destructive interference. Where two waves cancel completely (dark fringes in Young's experiment, nodes in standing waves), the energy is redistributed to the regions of constructive interference (bright fringes, antinodes). Globally, the total energy is the sum of each wave's energy — interference redistributes where the energy appears, but never creates or destroys it. In a standing wave, the nodes have zero displacement but the antinodes oscillate with double the amplitude, carrying four times the energy density (E ∝ A²), which on average balances out across the whole pattern.

The electromagnetic spectrum is the full range of EM radiation ordered by frequency (or wavelength): radio waves (longest λ, lowest f), microwaves, infrared, visible light, ultraviolet, X-rays, gamma rays (shortest λ, highest f). All are transverse waves because Maxwell's equations require the electric and magnetic fields to be perpendicular to the direction of propagation — this is a mathematical consequence of Gauss's law for magnetism (∇·B = 0) applied to a plane wave. There are no longitudinal EM waves in free space. The full derivation and spectrum breakdown is in our Electromagnetism guide.

Explore the Full Waves & Oscillations Guide

Transverse waves are one chapter in a much bigger story. Doppler effect, diffraction, interference, and the electromagnetic spectrum are all waiting in the complete topic guide.

Waves & Oscillations Guide → Optics: Reflection & Refraction

Sources & Further Reading

Halliday, D., Resnick, R., & Krane, K. S. (2002). Physics (5th ed., Vol. 1). Wiley. Chapter 17: Waves — I & Chapter 18: Waves — II.
French, A. P. (1971). Vibrations and Waves. MIT Introductory Physics Series. W. W. Norton. Chapters 7–9: Travelling and Standing Waves.
Hecht, E. (2017). Optics (5th ed.). Pearson. Chapter 8: Polarization of Light — including derivation of Malus's Law.
Crawford, F. S. (1968). Waves. Berkeley Physics Course Vol. 3. McGraw-Hill. Chapters 1–3: Wave Motion from first principles.
Griffiths, D. J. (2017). Introduction to Electrodynamics (4th ed.). Cambridge University Press. Chapter 9: Electromagnetic Waves — why EM waves are transverse.