Harmonic Wave Equation Calculator

Created by Bogna Szyk
Reviewed by Rijk de Wet
Last updated: Dec 21, 2022

This harmonic wave equation calculator will help you find 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 (see simple harmonic motion calculator).

Harmonic wave formula

If you know the fundamental properties of the wave (such as its wavelength λ\lambda – see our wavelength calculator), you can determine the displacement of points along the wave. The displacement depends on two main variables; the time point tt and the position along the wave xx.

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

y=Asin(2πλ(xvt)+ϕ),\footnotesize y = A\sin\left(\tfrac{2π}{λ} \cdot (x - vt) + \phi\right),

where:

  • yy – Displacement of a given point along the wave;
  • xx – Position of that point (its distance from the source);
  • tt – Time point;
  • vv – Wave velocity;
  • λλ – Wavelength;
  • AA – Amplitude; and
  • ϕ\phi – Initial phase of the wave.

Harmonic wave: an example of calculations

Let's assume that you want to find out what the wavelength of a certain wave is. You have measured its displacement at two points, both at time t=1 st = 1 \text{ s}.

  • At x=0 mmx = 0 \text{ mm}, the displacement was equal to y=7 mmy = - 7 \text{ mm}.
  • At x=10 mmx = 10 \text{ mm}, the displacement was also 7 mm7 \text{ mm}, but in the opposite direction (y=7 mmy = 7 \text{ mm}).

Let's work out the wave's wavelength!

  1. Determine the amplitude of the wave. In this case, we can assume it is equal to A=14 mmA = 14\text{ mm} as that's how far we know the wave oscillates.

  2. Find out what the initial phase of the wave is. 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=14sin(2πλ(0v1)+0)0.5=sin(2πλ(v))0.524=2πλ(v)0.083λ=v\footnotesize \begin{aligned} - 7 &= 14 \sin\left(\tfrac{2π}{\lambda}\!\cdot\!(0 - v\!\cdot\!1) + 0\right) \\ -0.5 &= \sin\left(\tfrac{2\pi}{\lambda}\cdot(-v)\right) \\ -0.524 &= \tfrac{2\pi}{\lambda}\cdot(-v) \\ -0.083\,\lambda &= -v \end{aligned}
  1. Plug the displacement of the second point into the harmonic wave equation and simplify it:
7=14sin(2πλ(10v1)+0)0.5=sin((2π/λ)(10v)]0.524=2πλ(10v)0.083λ=10v\footnotesize \begin{aligned} 7 &= 14 \cdot \sin\left(\tfrac{2π}{\lambda}\!\cdot\!(10 - v\!\cdot\! 1) + 0\right) \\ 0.5 &= \sin\left((2π / \lambda) \cdot (10 - v)\right] \\ 0.524 &= \tfrac{2π}{\lambda} \cdot (10 - v) \\ 0.083\, \lambda &= 10 - v \end{aligned}
  1. Add the sides of the two equations together to find the velocity:
0.083λ+0.083λ=v+10v0=102vv=5 mm/s\footnotesize \begin{aligned} - 0.083λ + 0.083\,\lambda &= - v + 10 - v \\ 0 &= 10 - 2 v \\ v &= 5 \text{ mm/s} \\ \end{aligned}
  1. Calculate the wavelength:
0.083λ=vλ=5/0.083λ=60.24 mm\quad\footnotesize \begin{aligned} - 0.083\, \lambda &= - v \\ \lambda &= 5 / 0.083 \\ \lambda &= 60.24 \text{ mm} \\ \end{aligned}
  1. Plug the values of wave velocity and wavelength into the harmonic wave equation calculator and check whether the result is correct! (Spoiler alert: it is.)

✅ 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!

FAQ

What is a harmonic wave function?

A harmonic wave function is a periodic function expressed by a sine or cosine. The harmonic waves have the form of y = A sin(2π/λ × (x - vt) + 𝜙), and their final form depends on the amplitude A, the wavelength λ, the position of point x, wave velocity v, and the phase 𝜙.

What is the equation for simple harmonic motion?

The generalized equation for simple harmonic motion can be written as x(t) = A cos(ωt + 𝜙), where t is time in seconds, A is the amplitude, ω = 2π/T = 2πf is the angular frequency, and 𝜙 is the phase shift.

Note that depending on the phase shift, harmonic motion can be modeled either by sine or cosine function.

What are the examples of harmonics waves?

Harmonic waves are widespread in many fields, such as electronic power transmission, music, radio, and all technologies that use waves in other forms.

In physics, you can use them to describe the properties of alternating current (AC), an electromagnetic field, or a sound wave.

How can I find the wavelength of y = 0.07 sin(2π/0.4 ×(x - 320t) + π/6)?

The wavelength is 0.4 m. Compare the wave equation with the basic formula:

y = A sin(2π/λ × (x - vt) + 𝜙)

y = 0.07 sin(2π/0.4 × (x - 320t) + π/6)

Note that the wavelength is 0.4 m, the amplitude is equal to 0.07 m, the velocity is 320 m/s, and the phase shift is π/6.

How can I write equation of harmonic wave?

To find the equation of a 15 m wavelength wave with an amplitude of 0.5 m at a velocity of 14 m/s:

  1. Write down the harmonic wave equation:

    y = A sin(2π/λ × (x - vt) + 𝜙)

  2. Substitute the known data:

    y = 0.5 sin(2π/15 × (x - 14t) + 𝜙)

  3. Calculate the phase shift if the wave is shifted horizontally.

  4. Check if the wave propagates along the +X-axis direction. Otherwise, change (x - 14t) to (x + 14t).

Bogna Szyk
Amplitude (A)
in
Wavelength (λ)
in
Velocity (v)
m/s
Time (t)
sec
Initial phase (𝜙)
rad
Distance from the source (x)
in
Displacement (y)
in
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