The wave equation calculator above evaluates the standard sinusoidal wave function y(x, t) = A sin(kx − ωt + φ). Give it an amplitude, any two of wavelength, frequency and speed, and a position and time, and it returns the displacement at that point along with the wave number, the angular frequency, and the velocity and acceleration of the medium at that instant.
Arb Digital builds free tools that compute the whole picture rather than one headline. The displacement is the obvious answer, but the particle velocity and acceleration are what most textbook problems actually ask for, and they are where the distinction between how fast the wave travels and how fast the medium moves becomes concrete instead of a slogan.
What This Wave Equation Calculator Does
A travelling harmonic wave is described by a function of two variables, position and time. Freeze time and you get a snapshot: a sine curve in space, repeating every wavelength. Freeze position and you get a history: a sine curve in time, repeating every period. The wave function encodes both at once, and the term kx − ωt is what makes the pattern move.
OpenStax University Physics Volume 1, section 16.2 on the mathematics of waves, develops the wave function in this form and shows how the position, velocity and acceleration of the particles of the medium follow from it. The wave number k = 2π/λ converts a distance into a phase angle, and the angular frequency ω = 2πf converts a time into one.
The particle velocity is the time derivative of the displacement, ∂y/∂t = −ωA cos(kx − ωt + φ), and the acceleration is the second derivative, −ω²y. That last identity is worth noticing: the acceleration of every point in the medium is proportional to its own displacement and directed back towards zero, which is exactly the condition for simple harmonic motion. A travelling wave is a field of simple harmonic oscillators, each running a fixed phase behind its neighbour.
How to Use It
- Choose which two of the trio you have. Wavelength, frequency and speed are locked together by v = fλ, so entering all three independently would over-specify the wave. The tool computes the third.
- Enter the amplitude, not the peak-to-peak swing. Halving that mistake is one of the most common errors in wave problems, and it propagates into every velocity and acceleration result.
- Set the position and the time you are interested in. Both may be negative; there is nothing special about x = 0 or t = 0 other than that the phase constant is defined there.
- Use the phase constant to match a known starting condition. A phase of 90 degrees turns the sine into a cosine, which is the form to use when the wave is at maximum displacement at the origin at t = 0.
- Choose the direction of travel. Positive x uses kx − ωt. Reversing it changes the sign of the time term, which reverses the particle velocity at every point.
The Formula and a Worked Example
k = 2π/λ, ω = 2πf, v = fλ = ω/k, and T = 1/f. The displacement is y = A sin(kx − ωt + φ), the particle velocity is −ωA cos(kx − ωt + φ) and the acceleration is −ω²y.
Take a clean case first. With A = 0.05 m, λ = 2 m and f = 5 Hz, the wave number is π = 3.1416 rad/m and the angular frequency is 10π = 31.416 rad/s, so the speed is 10 m/s and the period is 0.2 s. At x = 0.5 m and t = 0.1 s with no phase constant, the phase angle is 3.1416 × 0.5 − 31.416 × 0.1 = −1.5708 rad, which is exactly −90 degrees. The sine of that is −1, so y = −0.05 m: the point is at a trough, at maximum displacement, and therefore momentarily at rest.
The defaults loaded above use the same wave with a 30 degree phase constant, at x = 0.25 m and t = 0.05 s. The phase angle works out to −15 degrees, giving a displacement of 0.05 × sin(−15°) = −0.01294 m. The particle velocity is −31.416 × 0.05 × cos(−15°) = −1.5173 m/s, and the acceleration is −31.416² × (−0.01294) = +12.77 m/s². Displacement and acceleration have opposite signs, as they must.
Wave Speed and Particle Speed Are Not the Same Thing
This is the single most persistent confusion in the topic, and the numbers above make it concrete. The wave travels at 10 m/s. No particle of the medium travels anywhere at all; each one oscillates about a fixed point, with a maximum speed of ωA = 31.416 × 0.05 = 1.57 m/s. The two figures are unrelated in magnitude and can differ by orders of magnitude in either direction.
The reason is that a wave transports energy and phase, not material. A cork on a pond bobs up and down as a ripple passes; it does not travel with the ripple. The same is true of air molecules carrying sound and of the rope carrying a pulse.
Notice also what each depends on. The wave speed is set by the medium: tension and linear density for a string, bulk modulus and density for sound. It does not depend on how hard you shake the source. The maximum particle speed is ωA, which depends entirely on how hard and how fast you shake it, and not at all on the medium. Doubling the amplitude doubles the particle speed and leaves the wave speed untouched.
What the Phase Constant Is For
The phase constant φ is not a physical property of the wave; it is a bookkeeping choice about where to start the clock and the ruler. Changing it shifts the whole pattern along without altering its shape, its speed or its energy.
It earns its place when a problem tells you the state of the medium at a specific moment. If the wave is at maximum positive displacement at the origin when t = 0, a plain sine is wrong and you need φ = 90 degrees, which converts the expression into a cosine. If the medium at the origin starts at zero and moving downwards, φ = 180 degrees. Setting the constant to match the stated initial condition is usually the first step in solving a wave problem, and it is where sign errors get introduced.
It also matters when two waves are compared. The phase difference between two points on the same wave is kΔx, and between two waves of the same frequency it is the difference in their phase constants. That difference decides whether they reinforce or cancel where they meet, which is the whole of interference.
The Assumptions Behind This Model
The function evaluated here describes a single, unattenuated, purely sinusoidal wave travelling through a uniform non-dispersive medium in one dimension. Each of those words excludes something real.
Real waves lose amplitude with distance, through absorption in the medium and through spreading in two or three dimensions. Neither is in this expression, whose amplitude is the same at every x. The sound attenuation calculator covers that loss for acoustic waves.
Real media can be dispersive, meaning the wave speed depends on frequency. A single sinusoid is unaffected, but a pulse made of many frequencies spreads out as it travels, and this model cannot show that. Real disturbances are also rarely pure sinusoids; they are sums of them, which is why Fourier analysis exists. And large-amplitude waves become nonlinear, at which point the neat proportionalities here stop applying. OpenStax section 16.1 on travelling waves sets out the underlying relationship v = λ/T = λf that everything above rests on.
How This Differs From the Adjacent Arb Digital Tools
The boundary in one sentence: the wavelength calculator converts between wavelength, frequency and speed for a wave in a chosen medium, while this page takes those parameters and evaluates the wave function itself at a specific position and time. One gives you the wave's dimensions; the other tells you what the medium is doing at a point.
The frequency period calculator handles the reciprocal relationship between frequency and period, the speed of sound calculator supplies the wave speed for air at a given temperature, and the Doppler effect calculator covers what happens to the observed frequency when the source or observer moves. For the oscillation of a single body rather than a field of them, see the simple pendulum calculator, and for matter waves the de Broglie wavelength calculator.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Entering peak-to-peak as the amplitude — the amplitude is half the total swing, and doubling it doubles every velocity and acceleration result.
- Mixing degrees and radians — kx and ωt are in radians. Only the phase constant is entered in degrees here, and it is converted internally.
- Confusing wave speed with particle speed — the wave moves at v, the medium oscillates with a maximum speed of ωA, and the two are independent.
- Getting the sign of the time term wrong — kx − ωt travels in the positive x direction. Writing kx + ωt reverses it, and the displacement at a given point changes accordingly.
- Assuming a bigger amplitude means a faster wave — wave speed is a property of the medium, and shaking harder changes the energy carried, not the speed.
Related Free Tools From Arb Digital
For the relationship between wavelength, frequency and speed, use the wavelength calculator and the frequency period calculator. The speed of sound calculator gives the speed to feed in for acoustic problems, the Doppler effect calculator handles moving sources, and the sound attenuation calculator covers the amplitude loss this idealised model omits. The simple pendulum calculator treats the simple harmonic motion each particle here performs, the de Broglie wavelength calculator extends the idea to matter, and the full free online tools hub lists the rest.
Frequently Asked Questions
A is the amplitude in metres, k is the wave number in radians per metre and equals two pi over the wavelength, omega is the angular frequency in radians per second and equals two pi times the frequency, and phi is the phase constant that fixes the displacement at the origin at time zero.
Because it makes the pattern move in the positive x direction. As time increases, a given value of phase occurs at a larger x, so the whole waveform advances. Replacing the minus with a plus sends the wave the other way.
No, and they are usually very different numbers. The wave speed is how fast the pattern travels through the medium. The particle velocity is how fast one point of the medium is moving up and down at that instant, and its maximum value is the angular frequency times the amplitude.
Set the phase constant to 90 degrees. That is the correct choice when the wave is at maximum positive displacement at the origin at time zero, which is how many textbook problems are posed.
Because differentiating a sine twice returns the same sine with a factor of minus omega squared. Every point in the medium is therefore executing simple harmonic motion, accelerating back towards its rest position in proportion to how far it has been displaced.
Yes. Both axes have arbitrary origins, and negative values are perfectly meaningful. Only the phase constant depends on where you chose to put them.
The mathematics is identical for any sinusoidal travelling wave. For sound the displacement is of air molecules along the direction of travel rather than across it, and for light it is a field strength rather than a displacement, but the same function describes all of them.
Because this expression describes an idealised undamped wave in one dimension. Real waves lose amplitude to absorption and to spreading in two or three dimensions, and neither effect is included here.
