A harmonic wave is a sine wave moving through space – on a string, in water or in the air. Knowing its amplitude, wavelength and speed lets you predict the displacement at any point and any moment.
How to use the Harmonic Wave Equation Calculator
- Amplitude A
- Wavelength λ
- Velocity v
- Time t
- Initial phase φ
- Distance from the source x
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Enter the wave parameters, a distance from the source and a time. The phase φ shifts the wave; the wave number k and angular frequency ω are shown too.
Inputs explained
| Field | What to enter |
|---|---|
| Amplitude A | Units: m, cm, mm, km, in, ft … |
| Wavelength λ | Units: m, cm, mm, km, in, ft … |
| Velocity v | Units: m/s, km/h, mph, ft/s, knots, km/s |
| Time t | Units: s, ms, min, h, days |
| Initial phase φ | Units: rad, ° |
| Distance from the source x | Units: m, cm, mm, km, in, ft … |
What each result means
- Wave number k = 2×pi÷[Wavelength λ]
- Angular frequency ω = 2×pi×[Velocity v]÷[Wavelength λ]
- Frequency = [Velocity v]÷[Wavelength λ]
- Displacement y = [Amplitude A]×sin([Angular frequency ω]×[Time t] – [Wave number k]×[Distance from the source x] + [Initial phase φ])
- Transverse velocity = [Amplitude A]×[Angular frequency ω]×cos([Angular frequency ω]×[Time t] – [Wave number k]×[Distance from the source x] + [Initial phase φ])
Example values
How the main result changes when Amplitude A changes and everything else stays at the example values above:
| Amplitude A (m) | Displacement y (m) |
|---|---|
| 0.025 | -0.0147 |
| 0.0375 | -0.022 |
| 0.05 | -0.0294 |
| 0.0625 | -0.0367 |
| 0.075 | -0.0441 |
| 0.1 | -0.0588 |
Formula
y(x, t) = A·sin(ωt − kx + φ), k = 2π/λ, ω = 2πv/λ
Worked example
Inputs — Amplitude A: 0.05 m; Wavelength λ: 2 m; Velocity v: 10 m/s; Time t: 0.3 s; Initial phase φ: 0 rad; Distance from the source x: 1.2 m.
Results — Wave number k = 3.141593 rad/m; Angular frequency ω = 31.415927 rad/s; Frequency = 5 Hz; Displacement y = -0.029389 m; Transverse velocity = 1.270801 m/s.
Key terms
- Wave number
- k = 2π/λ.
Frequently asked questions
What does the minus sign in ωt − kx mean?
The wave travels in the positive x direction.
What is the wave number?
2π divided by the wavelength – the spatial frequency.
Is the wave speed the same as the particle speed?
No – particles oscillate in place; the pattern moves.
What is the phase?
The argument of the sine, setting where in the cycle the wave is.