Harmonic Wave Equation Calculator (y(x,t) = A sin(kx − ωt + φ))

Understanding the harmonic wave equation calculator — concept and principle

This calculator evaluates a sinusoidal travelling wave y(x, t) = A sin(kx ∓ ωt + φ) at a chosen position and time. It returns the displacement, the wave equation with numbers, the wave number, angular frequency, wave speed, period, and the transverse velocity and acceleration of the medium. It is a standard aid for physics courses on waves.

k and ω come from λ and f; putting x and t into the sine gives where that bit of string is at that instant.

How it is calculated

y(x, t) = A sin(kx ∓ ωt + φ); minus for a wave moving towards +x, plus towards −x

k = 2π ÷ λ; ω = 2π f; v = ω ÷ k = f λ; T = 1 ÷ f

Transverse velocity v_y = ∓A ω cos(kx ∓ ωt + φ)

Transverse acceleration a_y = −A ω² sin(kx ∓ ωt + φ)

Units follow the inputs: A in metres gives y in metres and v_y in m/s.

The principle

A travelling wave is a shape that moves without changing. Writing the shape as a sine of (kx − ωt) makes it move: if t increases, x must increase by ω/k to keep the same phase, so the pattern travels at v = ω/k. The wave number k sets how quickly the phase changes in space, and ω how quickly it changes in time. Each point of the medium only moves up and down in simple harmonic motion, so its velocity and acceleration come from differentiating with respect to time.

Worked example

Default inputs: A = 0.05 m, λ = 2 m, f = 3 Hz, φ = 0, towards +x, x = 0.5 m, t = 0.1 s.

k = 2π ÷ 2 = 3.142 rad/m; ω = 2π × 3 = 18.85 rad/s.
Equation: y = 0.05 sin(3.142x − 18.85t); v = 3 × 2 = 6 m/s; T = 0.3333 s.
Phase = 3.142 × 0.5 − 18.85 × 0.1 = −0.3142 rad, which is 5.969 rad in 0–2π.
y = 0.05 × sin(−0.3142) = −0.01545 m.
v_y = −0.05 × 18.85 × cos(−0.3142) = −0.8963 m/s; a_y = −18.85² × (−0.01545) = 5.49 m/s².

Each point is the calculator re-run with only “Position x (m)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Position x (m)	Displacement y(x, t)
0.125	-0.04985
0.25	-0.04455
0.375	-0.03247
0.5	-0.01545
0.625	0.003923
0.75	0.0227
1	0.04755

How to read the result

The displacement always lies between −A and +A. The transverse velocity is largest where y = 0 and zero at the crests, while the acceleration is largest at the crests and always points back towards y = 0. Do not confuse the speed of the medium, v_y, with the wave speed v.

Common mistakes

Using degrees instead of radians for the phase.
Mixing up the sign convention for direction of travel.
Confusing frequency f with angular frequency ω.

Keep in mind: The model describes an ideal, undamped sinusoidal wave in a uniform medium. Real waves can be damped, dispersive or non-linear.

Frequently asked questions

What does the wave number mean?

It is the number of radians of phase per metre, 2π divided by the wavelength.

How fast does the wave travel?

v = fλ, here 6 m/s.

Why is the transverse acceleration positive here?

The point is below equilibrium, and acceleration always points back towards it.

Sources: OpenStax, University Physics Volume 1 (2016), section 16.2 "Mathematics of Waves"

Checked 9 Nov 2026. OpenStax, University Physics Volume 1 (2016), §16.2 “Mathematics of Waves”: Eq. 16.2 (k = 2π/λ), Eq. 16.3 (v = ω/k), Eq. 16.4 (y = A sin(kx ∓ ωt + φ)) and the transverse velocity and acceleration of the medium.

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