How this instrument works
The equation y(x, t) = A·sin(kx − ωt) describes a single-frequency wave moving in the positive x direction. At any fixed instant t, plotting y against x gives a snapshot of the wave's shape — a sine curve of amplitude A and spatial period λ = 2π ⁄ k. At any fixed position x, plotting y against t gives the time history a stationary sensor would record — an oscillation of amplitude A and period T = 2π ⁄ ω. Wave number and angular frequency are the spatial and temporal versions of the same idea: radians of phase accumulated per metre, and per second.
The minus sign inside the sine is what makes the wave travel rather than merely oscillate in place. Keeping the argument kx − ωt constant as t increases requires x to increase too, at rate ω ⁄ k — that ratio is the phase velocity, the speed at which a crest physically moves through the medium. Flip the sign to kx + ωt and the identical shape travels the other way, toward negative x. The formula is the harmonic solution of the one-dimensional wave equation, the one that falls out of Newton's second law applied to a stretched string, a column of air, or an electromagnetic field, whenever that ratio ω ⁄ k stays fixed.
That fixed ratio is also the formula's real limit: it assumes one frequency in a non-dispersive medium, where every wavelength travels at the same speed. Pluck a real string or drop a real stone and the disturbance is a sum of many such harmonics — a Fourier series — and in a dispersive medium like deep water or an optical fibre, each component's ω ⁄ k differs slightly, so a pulse spreads out as it travels even though no single harmonic term does. This calculator evaluates one exact term of that sum: given A, k, ω, x, and t, it returns the displacement of that one component at that one point, with no damping and no boundary.
- Enter Amplitude (A) — the maximum displacement the wave reaches, in metres, measured from the resting line to a crest.
- Enter Wave number (k) in rad ⁄ m — the phase gained per metre of travel; k = 2π ⁄ λ, so a shorter wavelength means a larger k.
- Enter Angular frequency (omega) in rad ⁄ s — the phase gained per second at a fixed point; omega = 2π f.
- Set Position (x) and Time (t) to the exact point in space and instant you want to inspect — together they pick one spot on the wave's surface.
- Read Displacement (y) — the wave's instantaneous offset from rest at that (x, t) pair; the sign shows which side of the centerline it falls on.
Worked example — a rope wave at x = 1 m, t = 0.5 s
Take a demonstration rope wave with amplitude A = 0.05 m, wave number k = 2 rad ⁄ m, and angular frequency omega = 10 rad ⁄ s — a wavelength of λ = 2π ⁄ 2 ≈ 3.14 m and a frequency of f = 10 ⁄ 2π ≈ 1.59 Hz, travelling at phase velocity v = ω ⁄ k = 5 m ⁄ s. The question: where is the rope, in metres of displacement, at position x = 1 m and time t = 0.5 s? First the phase: kx − ωt = (2)(1) − (10)(0.5) = 2 − 5 = −3 rad.
Then the displacement: y = A·sin(−3 rad) = 0.05 × (−0.14112) = −0.007056 m, about −7.06 mm — the rope sits just over seven millimetres below its resting line at that instant. The phase −3 rad lands close to −π (−3.14159 rad), one of the wave's zero crossings, which is why the answer is small: −7.06 mm is only about 14 percent of the 5 cm amplitude, nowhere near a crest or trough. A crest that left x = 0 at t = 0 would have reached x = v·t = 5 × 0.5 = 2.5 m by this time, so the point at x = 1 m has already been passed by that crest and now sits on its falling edge.
Questions
What do wave number (k) and angular frequency (omega) actually mean?
k is the spatial rate of phase change — radians gained per metre, with k = 2π ⁄ λ where λ is the wavelength. Omega is the temporal rate — radians gained per second, with omega = 2π f where f is the frequency. Their ratio, v = ω ⁄ k, is the phase velocity: the speed at which a point of fixed phase, such as a crest, physically travels through the medium.
Why is there a minus sign in kx − ωt rather than a plus sign?
The minus sign is what makes the disturbance travel in the positive x direction. Holding the phase kx − ωt fixed as t grows forces x to grow too, at rate ω ⁄ k, so a crest moves toward positive x. Writing kx + ωt instead describes the mirror case: a wave travelling toward negative x with the same shape, amplitude, and speed.
How does this equation relate to the general wave equation?
It is the single-frequency, harmonic solution of the one-dimensional wave equation relating the second derivatives of y with respect to t and x, valid whenever the ratio v = ω ⁄ k stays constant. Substituting the sine form back into that equation is the standard textbook check that it works, and it is also why k and ω cannot be chosen completely independently once a medium fixes v.
What happens if the medium is dispersive, where v = ω ⁄ k is not constant?
Different frequencies then travel at different phase velocities, and a real signal — a sum of many harmonic terms, not one — spreads out or changes shape as it moves, a phenomenon called dispersion. This instrument evaluates a single exact A, k, omega term at one x, t point; it does not sum a spectrum, so it will not show dispersion on its own. You would need to add several evaluations at different omega values together.
Can this formula produce a standing wave instead of a travelling one?
Not by itself. A standing wave needs two travelling waves moving in opposite directions superposed: A sin(kx − ωt) plus A sin(kx + ωt) collapses to twice A sin(kx) cos(ωt), which separates into a fixed spatial envelope and a plain time oscillation. Run this calculator twice, once with omega negated, and add the two displacement results together to get a standing-wave value.
Why do I need to enter both Position (x) and Time (t)?
Because displacement is a function of both together, not either alone. Fixing t and varying x gives the wave's spatial profile at one instant, like a photograph; fixing x and varying t gives the motion at one location over time, like a single sensor's recording. The instrument needs one specific x, t pair to return one specific displacement.