# Harmonic Wave Equation Calculator

Created by Bogna Szyk
Reviewed by Rijk de Wet
Last updated: Jul 22, 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 simple harmonic motion.

## Harmonic wave formula

If you know the fundamental properties of the wave (such as its wavelength $\lambda$), you can determine the displacement of points along the wave. The displacement depends on two main variables; the time point $t$ and the position along the wave $x$.

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

$\footnotesize y = A\sin\left(\tfrac{2π}{λ} \cdot (x - vt) + \phi\right)$

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
• $\phi$ is the 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 $t = 1 \text{ s}$.

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

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 $A = 14\text{ mm}$ as that's how far we know the wave to oscillate.
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:
\footnotesize \begin{aligned} - 7 &= 14 \sin\left(\tfrac{2π}{λ}\!\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:
\footnotesize \begin{aligned} 7 &= 14 \cdot \sin\left(\tfrac{2π}{λ}\!\cdot\!(10 - v\!\cdot\! 1) + 0\right) \\ 0.5 &= \sin\left((2π / λ) \cdot (10 - v)\right] \\ 0.524 &= \tfrac{2π}{λ} \cdot (10 - v) \\ 0.083 λ &= 10 - v \end{aligned}
1. Add the sides of the two equations together to find the velocity:
\footnotesize \begin{aligned} - 0.083λ + 0.083λ &= - v + 10 - v \\ 0 &= 10 - 2 v \\ v &= 5 \text{ mm/s} \\ \end{aligned}
1. Calculate the wavelength:
\quad\footnotesize \begin{aligned} - 0.083 λ = - v \\ λ = 5 / 0.083 \\ λ = 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!

Bogna Szyk
Amplitude (A)
in
Wavelength (λ)
in
Velocity (v)
m/s
Time (t)
sec
Initial phase (ϕ0)
Distance from the source (x)
in
Displacement (y)
in
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