Follow longitudinal motion in a sound wave

Follow a parcel instead of a crest

In a small-amplitude sound wave traveling through a fluid, neighboring regions of fluid move back and forth along the propagation direction. This is longitudinal motion. A transverse string wave has local motion perpendicular to propagation; the fluid sound model here has motion parallel to it.

The sketch represents small fluid parcels by dots, not the random thermal paths of individual molecules. Open circles mark equilibrium positions. Filled circles show displaced positions. Sound propagates rightward while each parcel oscillates around its own reference position. The model contains no mean wind or streaming flow.

Read the exaggeration before measuring

The actual displacement amplitude is 1 micrometre, or 0.000001 m. That is too small to see against a four-metre spatial window at ordinary screen scale. The sketch therefore multiplies every displacement by 50,000, while keeping the equilibrium coordinate scale fixed. The drawn maximum offset corresponds to 0.050 m on that reference scale, not an actual five-centimetre air motion.

The numerical displacement graph and ledger use actual micrometre values. The pressure graph also uses actual model values; it is not multiplied by the sketch’s visual factor. Never estimate pressure or true density change from the exaggerated dot gaps. The sketch only helps identify the direction and relative pattern of compression and expansion.

Reconstruct one cycle manually

At 200 Hz, a period is 1/200 s = 0.005 s, or 5 ms. The time control selects a fraction of that period. A setting of 0.25 therefore means 1.25 ms here, not 0.25 s. When frequency changes, the physical time represented by the same cycle fraction changes too.

At the reference position x = 0, the model starts with displacement zero and leftward parcel velocity. At a quarter period the displacement is −1 micrometre and velocity is zero. At half a period it crosses zero moving right. At three quarters it reaches +1 micrometre, and after one full period it returns to its starting displacement and direction of motion.

The parcel has not moved one wavelength to the right. The pattern has advanced by one wavelength during that period. In the model, sound speed is 340 m/s, whereas the parcel’s peak speed at 200 Hz is about 1.256637 mm/s. These values describe different motions and have very different magnitudes.

A vertical curve does not make sound transverse

The displacement plot places a scalar value on its vertical axis so you can compare positions. Positive displacement places a parcel to the right of its reference position; negative displacement places it to the left. Neither sign alone determines its current velocity. The graph’s up/down direction is a plotting convention, not a physical air trajectory.

Likewise, the pressure graph’s vertical height represents pressure relative to ambient. It does not show the height of the air or the shape of a string. Read each axis and its units before interpreting a waveform.

Compare frequency and wavelength

At the fixed speed of 340 m/s, wavelength = speed/frequency. At 200 Hz, wavelength is 1.70 m. At 400 Hz, it is 0.85 m and the period is 2.5 ms. Doubling frequency halves both wavelength and period in this fixed-speed model. Displacement amplitude remains 1 micrometre; it is not a control in this investigation.

The assigned speed and equilibrium density are teaching parameters for a uniform idealized fluid. They are not measurements of the current room. Real sound speed depends on the medium and its conditions. The model omits reflections, attenuation, source startup and random molecular motion, and produces no audio. It therefore does not assess listening perception or loudness.

Transfer to an unfamiliar frequency

A wave in the same fixed-speed model has frequency 250 Hz. Predict wavelength, period and the elapsed time corresponding to a quarter period. Decide whether a representative parcel must move 1.36 m in that cycle.

Check the longitudinal transfer

Wavelength is 340/250 = 1.36 m. Period is 1/250 = 0.004 s = 4 ms, so a quarter period is 1 ms. The pattern advances 1.36 m per cycle, while a parcel oscillates locally and returns to its starting state. That wavelength is not its travel distance.

OpenStax sound waves describes sound using displacement and pressure. The parcel sketch and numerical comparisons here are original synthetic representations of the stated linear model.