What Is a Longitudinal Wave? Definition, Diagram and Examples

Sound doesn't push air sideways. It squeezes and stretches it — alternating compressions and rarefactions travelling outward from the source. That's a longitudinal wave: the medium moves in the same direction the wave travels, not perpendicular to it. It's a fundamentally different kind of wave motion from light, and that difference explains why sound can't travel through a vacuum, why it bends around corners, and why you can't polarise it.

Longitudinal waves are everywhere: sound in air, seismic P-waves through rock, pressure waves in a doctor's ultrasound machine. Understanding how they work — really understanding, not just memorising the definition — gives you a physical picture that makes sense of all of these at once.

What Is a Longitudinal Wave?

The defining characteristic is the parallel relationship between oscillation direction and propagation direction. Push and pull one end of a horizontal Slinky: the coils compress and expand along the horizontal axis, and the wave pulse also travels horizontally. Both disturbance and propagation are parallel — that is a longitudinal wave.

This contrasts directly with a transverse wave, where oscillation is perpendicular to propagation. Shake the same Slinky up and down: the wave still travels horizontally, but the coils move vertically — transverse. The geometry of oscillation relative to propagation is the fundamental distinction.

Compressions and Rarefactions

Longitudinal waves are characterised by two alternating zones:

Compressions: regions where particles are pushed together — higher density and pressure than equilibrium. These are the "peaks" of the pressure variation.

Rarefactions: regions where particles are pulled apart — lower density and pressure. These are the "troughs" of the pressure variation.

The wavelength of a longitudinal wave is the distance between consecutive compression centres (or consecutive rarefaction centres) — exactly analogous to peak-to-peak distance in a transverse wave.

The Wave Equation

Longitudinal waves obey the universal wave equation:

where v is wave speed (m/s), f is frequency (Hz), and λ is wavelength (m). All wave properties — frequency, amplitude, period, wave speed, wavelength — apply identically to longitudinal and transverse waves. Only the direction of oscillation differs.

Worked Example: Wavelength of a Musical Note

A tuning fork vibrates at 440 Hz (concert pitch A) in air at 20°C, where the speed of sound is 343 m/s. Find the wavelength of the sound wave it produces.

Each compression-rarefaction cycle spans about 78 cm — roughly arm's length. Compare this to a seismic P-wave with a typical frequency of 1 Hz travelling at 6 km/s through the Earth's crust: λ = 6,000/1 = 6,000 m (6 km) — the same equation, but a wavelength thousands of times longer, since P-waves travel far faster and typically oscillate far slower than audible sound.

Examples of Longitudinal Waves

1. Sound waves

Sound is the most important longitudinal wave in everyday life. A vibrating speaker cone creates alternating compressions and rarefactions in air. These pressure variations travel outward at approximately 343 m/s in air at 20°C, 1,480 m/s in water, and 5,120 m/s in steel. Your ear detects the pressure variations as sound.

2. Seismic P-waves

During earthquakes, the Earth transmits primary waves (P-waves) — longitudinal waves where rock compresses and expands along the propagation direction. P-waves travel at 5–8 km/s through the crust. Crucially, they can pass through solids, liquids, and gases — unlike S-waves (transverse), which cannot pass through liquids. The detection of P-waves but not S-waves on the far side of the Earth from an earthquake epicentre was the evidence that clinched the liquid outer core model in 1936.

3. Ultrasound

Medical ultrasound uses longitudinal waves at frequencies above 20,000 Hz. A transducer sends compressions into the body; tissues reflect them at different strengths. Returning echoes construct images. The same principle works in sonar and industrial non-destructive testing of welds and castings.

4. Spring (Slinky) waves

The classic classroom demo: push and pull one end of a stretched Slinky. Compression zones travel from one end to the other with coils moving parallel to propagation. It is a textbook longitudinal wave — visible and slow enough to observe directly.

5. Infrasound

Frequencies below 20 Hz — inaudible to humans — are infrasound. Elephants, whales, and some birds use infrasound for long-distance communication. Volcanic eruptions and meteor strikes produce infrasound detectable thousands of kilometres away. All sound, including infrasound, is a longitudinal wave.

Speed of Longitudinal Waves Through Different Media

where B is the bulk modulus (resistance to compression) and ρ is density — this form applies directly to fluids like air and water, where B is the only relevant stiffness. Stiffer media transmit longitudinal waves faster; denser media transmit them more slowly.

For a solid rod or bar (like the steel figure in the table below), the more precise formula uses Young's modulus E instead of bulk modulus: v = √(E/ρ). For steel, E ≈ 200 GPa and ρ ≈ 7,850 kg/m³, giving v = √(200×10⁹/7,850) ≈ 5,050 m/s — consistent with the table. Using steel's bulk modulus instead (≈ 160 GPa, which resists volume change from all sides rather than stretching along one axis) would understate the speed, since a thin rod is free to expand sideways as it's compressed lengthwise in a way that bulk compression is not. Steel transmits sound faster than air despite being denser — its much greater stiffness outweighs the density increase.

Longitudinal vs Transverse Waves: Complete Comparison

Measuring the Speed of Sound

Historical methods: Marin Mersenne (1636) timed echoes from a wall at a known distance, getting ~450 m/s (too slow due to poor timing). Modern methods: measure the time for a sound pulse to travel a known distance using electronic timing (accurate to microseconds). Interference methods: two microphones at different distances from a speaker; the phase difference at each frequency determines the wavelength, and v = fλ gives the speed. Speed of sound in air is also used to measure temperature in meteorology — weather balloons carry sounders to probe atmospheric temperature profiles.

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