Read before you practise · 8 min

Read Amplitude, Wavelength and Speed Straight from y = A sin(ωt − kx)

Most wave questions hand you an equation or a statement about the type of wave. For the equation, compare it with the standard form to read off amplitude, angular frequency and wave number; for the statements, remember which waves need a medium with shear rigidity.

NSEJS · Wave Motion and Doppler's Effect · 13 linked questions · 100% coverage

Know this first

Motion and force questions become simpler when direction, starting condition, time and the acting forces are marked first.

Distance · Length of the path travelledDisplacement · Straight-line change in positionResultant force · Net force including direction

Study deeper: NCERT Science · Wave Motion and Doppler's Effect

Reading the equation

Compare with y = A sin(ωt - kx + φ): A is the amplitude, ω = 2π f, k = 2π/λ and the wave speed is v = ω/k = f λ

A term with +kx next to +ωt means travel in the -x direction, and -kx means travel in +x.

A constant added to y only shifts the mean position: for

y = 4 + 2 sin(6t - 3x) the amplitude is 2 and the speed is 6/3 = 2

The maximum particle velocity is A ω, so the ratio of maximum particle velocity to wave speed is A k: for y = 3 sin (π/2)(50t - x), ω = 25π, k = π/2, v = 50 and the ratio is 3π/2.

If 5 complete waves fit in 4 m, then

λ = 0.8 m, k = 7.85 rad/m and, at 128 m/s, ω = k v ≈ 1005 rad/s, giving y = 0.02 sin(7.85 x - 1005 t)

For Y = 4 sin{π(t/5 - x/9) + π/6} cm, k = π/9 so λ = 18 cm, and v = ω/k = 9/5 cm/s

Transverse, longitudinal and polarisation

Transverse elastic (mechanical) waves need shear rigidity, so they travel in solids (and along strings or on surfaces) but not in gases, where there is no restoring force against sideways shear. Electromagnetic waves are transverse and need no medium, so read whether a question says elastic or mechanical. Sound waves in a gas are longitudinal, so they cannot be polarised, since polarisation needs oscillations perpendicular to the direction of travel. Water surface waves are partly transverse and partly longitudinal, because the water particles move in circles or ellipses. In a longitudinal wave, energy and momentum propagate in the direction of travel, but mass does not: the particles only oscillate about their mean positions.

Phase, particle motion and the ω-k graph

For a profile moving to the right, y = f(x - v t), each particle's velocity is -v times the local slope of the profile, so particles on a rising slope move down and particles on a falling slope move up. The phase ωt - kx decreases along the direction of travel, so a point nearer the source has a larger phase than one further along.

For a non-dispersive medium

ω = v k, so the ω-k graph is a straight line through the origin with slope v

Reflection at a dense boundary

A wave reflected from a denser medium (a fixed end) undergoes a phase change of π, which shows up as a minus sign, and its direction reverses. Reflected energy is proportional to amplitude squared, so if 81% of the energy is reflected, the amplitude is 0.9 of the incident one.

For an incident wave

y = 10 sin(ax + bt), the reflected wave is y = -9 sin(bt - ax)

OlympIQ exam lens

✓y = A sin(ωt - kx): amplitude A, v = ω/k = fλ, k = 2π/λ; +kx means travel toward -x
✓A constant added to y shifts the mean level only; it does not change amplitude or speed
✓Maximum particle velocity = Aω; ratio to wave speed = A k
✓Elastic transverse waves need shear rigidity: solids yes, gases no; sound in gases is longitudinal and cannot be polarised
✓Reflection from a denser medium reverses the phase by π; reflected amplitude = sqrt(fraction of energy reflected) x incident amplitude

OlympIQ trap

Do not confuse the maximum particle velocity with the wave speed. The particle velocity is A ω and changes with time; the wave speed ω/k is fixed by the medium. Read k and ω separately from the equation before comparing them.

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