How the Particle Velocity Calculator Works
In acoustics, "particle velocity" is a specific, well-defined quantity: it is the speed at which the individual particles of a medium — air molecules, water molecules, or points inside a solid — oscillate back and forth around their resting position as a sound wave passes through. It is easy to confuse with the speed of sound, but the two describe completely different motions. The speed of sound, c, is how fast the pressure disturbance (the wave pattern) propagates through the medium. Particle velocity, u, is how fast each individual particle moves as it is pushed and pulled by that passing disturbance — typically a motion of millimeters per second, even when c is hundreds of meters per second. For a plane traveling wave in a lossless medium, sound pressure p and particle velocity u are tied together by the medium's specific acoustic impedance, z = ρc (density times speed of sound): p = z·u, so u = p / (ρc). This calculator uses that relationship to convert sound pressure into particle velocity, and also reports the specific acoustic impedance z, the sound intensity I = p² / (ρc), and the sound pressure level Lp = 20·log₁₀(p / 20 µPa).
The plane-wave impedance relationship
Specific acoustic impedance z = ρc depends only on the medium, not on the sound itself — it is roughly 420 rayl (Pa·s/m) for air at 20°C and sea level, and about 1.48 million rayl for fresh water at 20°C, because water is far denser and sound travels through it much faster. Because p = z·u, a given sound pressure produces a much smaller particle velocity in a high-impedance medium like water than in low-impedance air. This relationship is exact for a plane traveling wave far from any boundary or source; near a sound source (the "near field"), or for standing waves and reflections, pressure and particle velocity can be out of phase and this simple ratio no longer holds exactly.
Choosing between pressure in pascals and decibels
Sound pressure is often reported in decibels rather than pascals, since human hearing spans an enormous pressure range. Select "Sound Pressure Level (dB SPL)" if your source value is a decibel reading — the calculator converts it internally using p = 20 µPa × 10^(Lp / 20) before applying u = p / (ρc). If you already have a pressure reading in pascals (from a calibrated microphone or a textbook problem), select "Pascals (Pa)" and enter it directly. Make sure density and speed of sound match the same medium and, ideally, the same temperature — speed of sound in air changes by about 0.6 m/s per °C, and using a mismatched value will shift every result.