Harmonic Wave Equation Calculator

Created by Bogna Szyk
Last updated: Mar 10, 2022

This harmonic wave equation calculator will help you establish what is the displacement of any point along a harmonic wave traveling through space.

What is a harmonic wave?

A wave is a disturbance that propagates in space. When it moves through space, individual molecules oscillate back and forth. If the wave is harmonic, then it means that all particles are in simple harmonic motion.

Harmonic wave formula

If you know the fundamental properties of the wave, such as its wavelength, you will be able to determine the displacement of points along the wave. The displacement depends on two main variables; the time point t and position along the wave x.

To determine the displacement, you can apply the following harmonic wave formula:

y = A * sin[(2π / λ) * (x - vt) + Φ]

where:

  • y is the displacement of a given point along the wave,
  • x is the position of that point (its distance from the source),
  • t is the time point,
  • v is the wave velocity,
  • λ is the wavelength,
  • A is the amplitude, and
  • Φ is the initial phase of the wave.

Harmonic wave: an example of calculations

Let's assume that you want to find out what is the wavelength of a certain wave. You have measured its displacement at two points at time t = 1 s. At x = 0 mm, the displacement was equal to y = - 7 mm. At x = 10 mm, the displacement was also equal to 7 mm, but in the opposite direction (y = 7 mm).

  1. Determine the amplitude of the wave. In this case, we can assume it is equal to 14 mm.
  2. Find out what is the initial phase of the wave. We can assume it is equal to 0 rad.
  3. Plug the displacement of the first point into the harmonic wave equation and simplify it:

- 7 = 14 * sin[(2π / λ) * (0 - v * 1) + 0]

- 0.5 = sin[(2π / λ) * (- v)]

- 0.524 = (2π / λ) * (- v)

- 0.083 * λ = - v

  1. Plug the displacement of the second point into the harmonic wave equation and simplify it:

7 = 14 * sin[(2π / λ) * (10 - v * 1) + 0]

0.5 = sin[(2π / λ) * (10 - v)]

0.524 = (2π / λ) * (10 - v)

0.083 * λ = 10 - v

  1. Add the two equations together to find the velocity:

- 0.083 * λ + 0.083 * λ = - v + 10 - v

0 = 10 - 2 * v

v = 5 mm/s

  1. Calculate the wavelength:

- 0.083 * λ = - v

λ = 5 / 0.083

λ = 60.24 mm

  1. Plug these values of wave velocity and wavelength into the harmonic wave equation calculator to check whether the result is correct!

Now that you've learned how to find the displacement of any point along an oscillating wave, let's take a step further and learn about damping of an oscillating wave using these wonderful calculators - damping ratio and critical damping!

Bogna Szyk
Amplitude
in
Wavelength
in
Velocity
m/s
Time
sec
Initial phase
rad
Distance from the source
in
Displacement
in
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