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Wave Motion overview

Topic 3 of 6

Read longitudinal waves

In a longitudinal wave, particle displacement is parallel to propagation. Compression describes neighbouring particle spacing, not the height of a displacement graph.

Let x label a particle's equilibrium position along a right-moving wave. Let ξ denote its signed displacement: positive means to the right of equilibrium and negative means to the left. Plotting ξ on a vertical graph axis does not make the actual particle motion transverse.

Distinguish displacement from compression

A compression is a region where neighbouring particles are closer together than at equilibrium; a rarefaction has greater spacing. What matters is how displacements differ between neighbours.

If particles farther right have smaller displacements than their neighbours to the left, their spacing decreases. A negative spatial displacement gradient therefore corresponds to compression in this small-displacement model. The most negative gradient marks maximum compression; a positive gradient corresponds to increased spacing.

Displacement is not the same quantity as compression

A positive longitudinal displacement ξ means a particle is to the right of its equilibrium position. Vertical height on the graph encodes that signed horizontal displacement; the particles do not move up and down.

Spatial snapshot at t = 0

Compression is greatest where the displacement gradient is most negativeThe upper graph plots signed longitudinal displacement in millimetres against equilibrium position in metres. At positions zero, 0.15, 0.30, 0.45 and 0.60 metres the values are zero, positive two, zero, negative two and zero millimetres. Beneath it, one horizontal row shows 21 equally spaced equilibrium positions and a separate row shows the same ordered particles at that instant. Only their longitudinal displacements are magnified twenty times to make the spacing change visible; equilibrium spacing stays on the same position scale. The actual-particle row is closest-spaced near 0.30 metres, where displacement is zero with the most negative spatial gradient. It is widest-spaced near zero and 0.60 metres, where the gradient is positive. Brown rings identify the same equilibrium-coordinate 0.30-metre particle in both rows. No particle crosses another. The rows are a position comparison, not two successive times.0.000.150.300.450.60-20+2Displacement ξ / mmEquilibrium position x / mEquilibrium positionsPositions at this instantCompression near x = 0.30 mDisplacements in the dot row: ×20

Near x = 0.30 m, particles on the left are displaced right and particles on the right are displaced left, reducing their spacing. The displacement maxima at 0.15 and 0.45 m do not locate maximum compression. The widest spacing is near 0 and 0.60 m.

Time trace of the particle at equilibrium x = 0.30 m

The longitudinal particle starts at zero displacement and moves rightThe same particle with equilibrium coordinate 0.30 metres is followed through time. Signed longitudinal displacement in millimetres is plotted against time in milliseconds. At zero, 0.50, 1.00, 1.50 and 2.00 milliseconds the values are zero, positive two, zero, negative two and zero millimetres. The initial positive graph slope means motion to the right, not upward motion in space. Its two-millisecond period combines with wavelength 0.60 metres to give propagation speed 300 metres per second. This curve is displacement, not pressure or density.0.000.501.001.502.00-20+2Displacement ξ / mmElapsed time t / ms

This is the same fixed equilibrium position throughout. The rising trace initially means the particle moves right. A longitudinal displacement graph must not be relabelled as pressure or density.

The spatial graph plots signed displacement against equilibrium position, with a separate particle-spacing row. Displacements in that row are magnified 20 times. The time graph instead follows the fixed equilibrium position x = 0.30 m and initially rises.

The supplied model has displacement amplitude 0.0020 m = 2.0 mm, wavelength 0.60 m and period 0.0020 s = 2.0 ms. In the t = 0 snapshot:

  • Displacement is most positive at x = 0.15 m and most negative at x = 0.45 m.
  • Maximum compression is near x = 0.30 m, where displacement is zero but its spatial gradient is most negative.
  • Maximum rarefaction is near x = 0 and 0.60 m, where displacement is also zero but the gradient is positive.

A particle at a compression centre need not have a large displacement. Its neighbours determine the local spacing. A pressure or density trace would be a different plotted quantity; do not relabel this displacement curve as pressure.

Optional check In the longitudinal snapshot, signed displacement is largest positive at x = 0.15 m, zero with its most negative spatial gradient at x = 0.30 m, and largest negative at x = 0.45 m. Where is maximum compression?
In the longitudinal snapshot, signed displacement is largest positive at x = 0.15 m, zero with its most negative spatial gradient at x = 0.30 m, and largest negative at x = 0.45 m. Where is maximum compression?

Follow one particle through time

At equilibrium position x = 0.30 m, the displacements at t = 0, 0.50, 1.00, 1.50 and 2.00 ms are 0, +2.00, 0, -2.00 and 0 mm. At the first instant its displacement is zero and increasing, so the particle is moving right. One millisecond later it crosses equilibrium moving left.

The period is 2.0 ms, giving f = 1/0.0020 = 500 Hz. The wavelength is the spatial repeat distance, 0.60 m; it is not obtained from the horizontal axis of this time trace.

The particle moves back and forth along the same direction as propagation, but does not travel steadily with a compression. Matching a compression between frames tracks the wave; following the same labelled particle tracks local motion.

When interpreting a measured or simulated representation, check whether horizontal position means equilibrium position or instantaneous particle position. Also check any magnification of particle displacement. Use stated scales and readings rather than treating an enlarged particle-spacing sketch as a physical ruler.