# Harmonic Wave Equation Calculator

URL: https://speedycalculator.com/calculators/physics/harmonic-wave-equation/
Last updated: 2026-10-30

> The harmonic wave equation calculator finds the displacement y(x, t) of a sinusoidal travelling wave from its amplitude, wavelength, speed, initial phase, position and time.

## Inputs

- Amplitude A (m, cm, mm, km, in, ft, yd, mi)
- Wavelength λ (m, cm, mm, km, in, ft, yd, mi)
- Velocity v (m/s, km/h, mph, ft/s, knots, km/s)
- Time t (s, ms, min, h, days)
- Initial phase φ (rad, °)
- Distance from the source x (m, cm, mm, km, in, ft, yd, mi)

## Results

- Wave number k
- Angular frequency ω
- Frequency
- Displacement y
- Transverse velocity

## Formula

```
y(x, t) = A·sin(ωt − kx + φ),  k = 2π/λ,  ω = 2πv/λ
```

## Explanation

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

FieldWhat to enterAmplitude AUnits: m, cm, mm, km, in, ft …Wavelength λUnits: m, cm, mm, km, in, ft …Velocity vUnits: m/s, km/h, mph, ft/s, knots, km/sTime tUnits: s, ms, min, h, daysInitial phase φUnits: rad, °Distance from the source xUnits: 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)0.025-0.01470.0375-0.0220.05-0.02940.0625-0.03670.075-0.04410.1-0.0588

## 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.

## FAQ

**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.


## Sources

- [OpenStax – University Physics Volume 1](https://openstax.org/details/books/university-physics-volume-1)
- [HyperPhysics (Georgia State University)](http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html)