Sound Wave Propagation: Speed, Reflection and Absorption

Sound is a pressure wave, not a moving current of air

When a sound source vibrates, it doesn't push a stream of air molecules from source to listener — it launches a pressure disturbance that propagates through the medium while each individual air molecule only jiggles back and forth around a fixed average position. That disturbance obeys the acoustic wave equation, a direct consequence of combining Newton's second law (pressure gradients accelerate fluid) with conservation of mass (compressing a region of fluid raises its density, and pressure rises with density) for small perturbations around equilibrium.

∂²p/∂t² = c² · ∇²p
// p: pressure perturbation from equilibrium    c: speed of sound in the medium
// this is the same wave equation that governs light, water waves, and
// the string simulations elsewhere on this site — only c differs

Why speed depends on the medium, not the loudness

The speed of sound c is set entirely by the medium's mechanical properties — how stiff it is against compression (its bulk modulus, or elasticity) versus how much inertia it has (its density) — not by how loud the sound is or what frequency it carries. In an ideal gas this reduces to c = √(γRT/M), which is why sound in air (~343 m/s at 20°C) depends on temperature but, to good approximation, not on air pressure at a fixed temperature. The general pattern across media is that stiffer, less dense materials carry sound faster: sound travels around 343 m/s in air, roughly 1,480 m/s in water (about 4× faster, because water is far stiffer against compression despite being much denser too), and around 5,000 m/s in steel, where the extreme stiffness of the metal lattice dominates.

Reflection, transmission and the impedance mismatch

When a sound wave hits a boundary between two different media, what happens is governed by the acoustic impedance of each side, Z = ρc (density times sound speed). If the two media have similar impedance, most of the wave's energy transmits through with only a small reflection; if the impedance mismatch is large — air meeting a solid wall, for instance, where Z differs by roughly four orders of magnitude — almost all of the energy reflects back and only a tiny fraction transmits. This impedance-mismatch principle is why a wall stops nearly all airborne sound from getting through, why ultrasound gel is needed to couple an ultrasound probe to skin (removing the air gap that would otherwise reflect nearly everything), and why loudspeaker and microphone design spends so much effort on impedance matching between components.

R = ((Z2 - Z1) / (Z2 + Z1))²    // fraction of energy reflected at a boundary
T = 1 - R                        // fraction transmitted
// Z_air ≈ 415 Pa·s/m   Z_water ≈ 1.48×10⁶ Pa·s/m  → R ≈ 0.9989 at an air/water interface

Absorption: where the energy actually goes

Reflection redirects a wave's energy; absorption converts it to heat, through friction between vibrating molecules (viscous losses) and the finite time it takes molecular vibrations and rotations to exchange energy with translational motion (relaxation losses). Absorption in air rises steeply with frequency — high frequencies lose energy over much shorter distances than low frequencies, which is exactly why a passing car's bass line is audible long after its treble has faded into the background, and why distant thunder sounds like a low rumble even though a nearby lightning strike cracks sharply across the whole audible spectrum. Absorptive materials such as acoustic foam are engineered specifically to maximize this molecular friction at audible frequencies, converting incident sound energy to heat instead of letting it reflect back into a room.

Standing waves and resonance

In a bounded medium — a room, a pipe, the water tank in this simulation — reflections from the boundaries interfere with the outgoing wave, and at specific frequencies where the enclosure's dimension is a half-integer or quarter-integer multiple of the wavelength, that interference builds a stable standing wave: a pattern of fixed nodes (zero pressure variation) and antinodes (maximum pressure variation) that doesn't travel, just oscillates in place. These resonant frequencies are why a room has characteristic modes that colour its acoustics, why a organ pipe of a given length sounds a specific musical pitch, and why sealed cavities of very different sizes — from concert halls to Helmholtz resonators to the human vocal tract — can be tuned by adjusting their geometry alone.

What the simulation lets you explore

The demo propagates a pressure pulse or continuous wave through a 2D medium with adjustable frequency, wave speed and boundary absorption coefficient, so you can directly see the impedance-mismatch reflection at a boundary, watch a standing-wave pattern build up between two hard walls, and see how raising the absorption coefficient damps reflections into a cleanly absorbed pulse instead of a persistent echo — the same physics that separates a live, reverberant room from an acoustically treated one.

Frequently asked questions

Try it live

Everything above runs in your browser — open Sound Wave Propagation and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.