PhysicsLab

Waves · Longitudinal & Sound

Longitudinal & Sound Lab

s(x, t) = s₀ sin 2π(x/λ − t/T)
λ = v/f = 2.00 mT = 5.88 msdisplay ×6.0 visual gain
red ring = compression Cgreen ring = rarefaction Ramber column = watched particle · ticks = equilibrium

Graphs — same x-axis as the tube (0 → 4 m)

Δp–x = s–x shifted λ/4 = 0.500 m along +x (Δφ = π/2).

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Dashed guides drop from every C / R halo: on s–x they land exactly on zero crossings; on Δp–x they land on the extrema. Δp computed from Δp₀ = ρvωs₀ (ρ = 1.2 kg m⁻³) — huge only because s₀ is exaggerated; real speech is ≈ 0.02 Pa.

Try first: Pause, then Step T⁄8 — watch a C halo march right.

Controls

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340 m s⁻¹ = sound in air. s₀ is exaggerated (real audible sound ≈ µm) so bunching is visible; screen shifts get an auto-capped extra visual gain, all maths uses true SI values. At low f + high v, λ = v/f can exceed the 4 m tube — then less than one full wave fits.

Live measurements — read off the lattice, not the formula

λ set by sliders (v/f)2.00 m
λ measured C → next C—
v set by slider340 m s⁻¹
pattern speed, tracking one C—
max travel of any columns₀ = 3.0 cm
Δp₀ = ρvωs₀13.1 kPa
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C / R centres are found each frame by neighbour analysis on the 48 columns (sign change of displacement), then tracked between frames — a genuine measurement that keeps agreeing with v and v/f.

What this confirms — Wave Motion: Everything to Memorise
  • “Longitudinal wave” — every column oscillates ALONG the direction of energy travel (x); nothing moves sideways. “Displacement” here means shift along the tube.
  • “C and R live at ZERO displacement” — every red/green guide line lands exactly on a zero crossing of the s–x graph, never on a peak.
  • “Neighbour analysis” — the halos are computed from the columns themselves: left neighbour displaced +x and right neighbour −x → converging → compression; diverging → rarefaction.
  • “Pressure graphs — two flavours” — Δp–x is the s–x curve shifted λ/4 along +x (Δφ = π/2); its maxima sit under the C halos, minima under the R halos.
  • “Longitudinal bookkeeping” — the rose bracket (C → adjacent R) is λ/2, NOT λ; a full λ is C → next C (amber bracket).
  • “Sign conventions / graph shape ≠ wave type” — toggle the y–x plot: an identical sinusoid; only the axis meaning says longitudinal vs transverse.
  • “Progressive wave” — the C/R pattern travels at v (measured live ≈ set v) while the amber column only shuffles about its equilibrium tick: energy travels, the medium does not.