Harmonic Wave Equation Calculator

Background.

Flick one end of a long rope up and down at a steady rhythm and a train of identical humps runs along it. Freeze the motion and the rope traces a sine curve in space; watch one spot on it and that spot bobs up and down as a sine curve in time. The harmonic wave function y(x, t) = A sin(kx − ωt + φ) captures both pictures in a single line, giving the sideways displacement of any point x at any instant t.

Each symbol controls one feature of the wave. The amplitude A is the height of a crest above the rest position. The wavenumber k sets how tightly the crests are packed, since the wavelength is 2π/k. The angular frequency ω sets how quickly each point cycles, since the period is 2π/ω. The phase constant φ slides the whole pattern so that the point at x = 0 can begin its cycle anywhere at t = 0, not just at zero.

The minus sign between kx and ωt carries the direction. As time increases, a crest keeps the same phase only if x increases too, so the pattern moves toward positive x at a speed of ω/k. Swap the minus for a plus and the identical wave runs the other way.

This is the simplest wave possible: one frequency, one dimension, no loss of energy and nothing for it to reflect from. Real waves on strings, in air or on water are sums of many terms like this one, which is why it is worth knowing thoroughly before tackling them.

What is harmonic wave equation calculator?

The harmonic wave function describes a sinusoidal wave travelling in one direction along a line. It gives the transverse displacement y, in metres, of the medium at position x and time t. It is one solution of the wave equation, the differential equation ∂²y/∂t² = v² ∂²y/∂x², with wave speed v = ω/k. Every point of the medium performs simple harmonic motion with the same amplitude and frequency, each one slightly behind its neighbour in phase.

How to use this calculator.

1. 01Enter the amplitude in metres and the wavenumber in radians per metre. If you know the wavelength λ instead, k = 2π/λ.
2. 02Enter the angular frequency in radians per second. From an ordinary frequency f in hertz, ω = 2πf, so a 2 Hz wave needs 12.566.
3. 03Enter the position x in metres, which may be negative, and the time t in seconds, which must be zero or more.
4. 04Enter the phase constant in radians; multiply degrees by π/180. A phase of π/2, about 1.5708, makes the point x = 0 start at a crest.
5. 05Read the displacement: positive above the rest line, negative below, and never larger in size than the amplitude.

The formula.

y = A sin(kx − ωt + φ). The bracket is the phase of the wave, in radians. The term kx turns distance into phase, adding k radians for every metre along the line; the term ωt turns elapsed time into phase, removing ω radians every second; φ is the starting offset. The sine of the phase lies between −1 and 1, and multiplying by A scales it to a displacement in metres. Following a point of constant phase, kx − ωt stays fixed, so x moves at ω/k, the wave speed. Wavelength λ = 2π/k, period T = 2π/ω and frequency f = ω/2π follow directly. Because the sine repeats every 2π, shifting x by one wavelength or t by one period leaves y unchanged. Amplitude, wavenumber and angular frequency must be positive.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, University Physics Volume 1, section 16.2, Mathematics of Waves.
2. [2]French, Vibrations and Waves (PRINT).
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.