Wave quantities and representations

Amplitude A is the largest displacement from the equilibrium line, not the peak-to-trough height: a cord whose crests sit 12 mm above the rest line and whose troughs sit 12 mm below has A = 12 mm, not 24 mm. Beyond amplitude the wave repeats in two separate senses, and each needs its own instrument. Lay a ruler along the cord at one instant and the pattern repeats every wavelength λ, measured crest to crest, in metres. Stand at one point with a stopwatch and the displacement repeats every period T, in seconds, whose reciprocal f = 1/T is the frequency in hertz. Nothing so far ties λ to T. You could measure either one without ever discovering the other, which is exactly why they deserve separate symbols and separate axes.

Both graphs are sinusoids and both get called the graph of the wave, which is where the confusion starts. A snapshot plots y against position x with time frozen: it is a photograph of the whole cord at one instant, and the distance between two crests on it is λ. A history plots y against time t at one chosen x: it is what a single sensor records, and the gap between two crests on it is T. Read the horizontal axis label before anything else, because it carries the entire meaning. The slopes differ too. On a snapshot ∂y/∂x is a pure number, a gradient with no units. On a history ∂y/∂t is a speed in m s⁻¹, the transverse velocity of that one particle. Two curves that look identical on the page are answering different questions.

λ and T are what you measure; k and ω are what you calculate with. Feeding a length straight into a sine is meaningless, because sin takes a dimensionless argument in radians, so the wave function converts first: k = 2π/λ turns metres into radians of phase, and ω = 2π/T turns seconds into radians of phase. A wave of wavelength 2.5 m has k = 2π/2.5 = 2.51 rad m⁻¹, so two points 0.40 m apart sit 2.51 × 0.40 = 1.0 rad apart in phase. A 120 Hz oscillation has ω = 2π × 120 = 754 rad s⁻¹. Keep that factor of 2π in view: k is not 1/λ and ω is not 1/T, and dropping the 2π is the most common numerical slip in the whole unit.

Watch a single crest. In exactly one period the pattern reproduces itself shifted by exactly one wavelength, so the crest has travelled λ in time T and the pattern speed is v = λ/T = λf, or equivalently ω/k: phase per second divided by phase per metre. The division of labour matters. The medium fixes v, the source fixes f, so λ is the quantity forced to adjust. A 512 Hz tone in air at 343 m s⁻¹ has λ = 343/512 = 0.670 m; drive the same tone into water, where v = 1480 m s⁻¹, and f stays at 512 Hz because the water is pushed at the rate the air pushes it, while λ stretches to 1480/512 = 2.89 m, 4.3 times longer. A wavelength quoted with no medium named is an incomplete answer.

Write y = A sin(kx − ωt) and follow a chosen crest by holding its phase constant: kx − ωt = constant gives x = (ω/k)t + constant, so as t increases x must increase. The crest moves towards +x at speed ω/k. Flip the sign to kx + ωt and the same algebra gives x = −(ω/k)t + constant, a crest moving towards −x. One sign carries the whole direction, and it is the piece most often copied down wrong. A cross-check that needs no algebra: take the snapshot, redraw it shifted slightly in the proposed direction, and ask whether a chosen particle has moved up or down. If that agrees with the velocity arrow you were given, the sign is right. In symbols the same test reads u_y = −v (∂y/∂x) for a wave heading towards +x.

The medium itself does not travel. Differentiating y = A sin(kx − ωt) at fixed x gives u_y = −ωA cos(kx − ωt), so a particle oscillates transversely with peak speed ωA, reached as it crosses the equilibrium line, and is momentarily at rest at each crest and trough — exactly where the wave itself is still moving at v. The two speeds are unrelated in size, and their ratio is a pure geometric number: ωA/v = 2πA/λ. Take a rope carrying a wave with A = 3.0 mm, λ = 0.80 m and v = 24.0 m s⁻¹. Then f = v/λ = 30 Hz, ω = 188 rad s⁻¹, and the peak transverse speed is 0.57 m s⁻¹, about 2.4% of the wave speed. Small amplitude compared with wavelength always means slow particles compared with the wave.