About the Harmonic Wave Equation Calculator
This harmonic wave equation calculator will help you find the displacement of any point along a harmonic wave traveling through space. Enter the amplitude, wavelength, position, wave velocity, time, and initial phase to instantly get the displacement, plus the wave number, frequency, period, and angular frequency.
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 λ), 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:
y = A · sin( (2π/λ) · (x − v·t) + φ )
where:
- y – Displacement of a given point along the wave;
- x – Position of that point (its distance from the source);
- t – Time point;
- v – Wave velocity;
- λ – Wavelength;
- A – Amplitude; and
- 𝜙 – Initial phase of the wave.
Supported unit systems
The calculator supports both measurement systems. Pick the one you work with — all the length inputs (amplitude, wavelength, and position) and the wave velocity automatically switch their units:
| Unit System | Length (A, λ, x) | Wave velocity (v) |
|---|---|---|
| Metric (SI) | meters (m) | meters per second (m/s) |
| American / Imperial | feet (ft) | feet per second (ft/s) |
The resulting displacement is shown in your chosen unit and also converted to meters, centimeters, millimeters, feet, and inches. Time is always in seconds and the initial phase in radians.
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 s.
- At x = 0 mm, the displacement was equal to y = −7 mm.
- At x = 10 mm, the displacement was also 7 mm, but in the opposite direction (y = 7 mm).
Let's work out the wave's wavelength!
- Determine the amplitude of the wave. In this case, we can assume it is equal to A = 14 mm, as that's how far we know the wave oscillates.
- Find out what the initial phase of the wave is. We can assume it is equal to 0 rad.
- 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 - 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 - Add the sides of the two equations together to find the velocity:
−0.083 λ + 0.083 λ = −v + 10 − v
0 = 10 − 2v
v = 5 mm/s - Calculate the wavelength:
−0.083 λ = −v
λ = 5 / 0.083
λ = 60.24 mm
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 the damping of an oscillating wave with the damping ratio calculator!
FAQs
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.
How do I calculate the displacement of a harmonic wave?
- Multiply the position x by the wave number k = 2π/λ.
- Subtract the product of wave velocity v and time t from the position before multiplying — that is, compute (x − v·t).
- Add the initial phase 𝜙 to get the total phase of the oscillation.
- Take the sine of that phase and multiply it by the amplitude A. The result is the displacement y.
What are the wave number, frequency, and period?
The wave number k = 2π/λ tells you how many radians of the wave fit into one unit of length. The frequency f = v/λ is the number of oscillations per second (in hertz), the period T = λ/v is the time for one oscillation, and the angular frequency ω = 2πf describes the rate of oscillation in radians per second.
Can the displacement be negative?
Yes. Because the displacement is the amplitude multiplied by a sine, it oscillates between +A and −A. A negative value simply means the point is displaced in the opposite direction to a positive value at that instant.