Harmonic Wave Equation Calculator
Turn a harmonic traveling wave into numbers: k, omega, speed and the displacement at any x and t, with a waveform plot.
🌊 What is the Harmonic Wave Equation?
The harmonic wave equation is the standard mathematical description of a sinusoidal traveling wave: y(x, t) = A sin(k x − ω t + φ). It gives the displacement y of the medium at every position x and every time t. The amplitude A is the largest displacement, the wave number k = 2π / λ counts how much phase the wave gains per metre, the angular frequency ω = 2π f counts the phase gained per second, and the phase constant φ fixes where the wave starts at x = 0 and t = 0. The same expression describes a wave on a string, a sound wave, a ripple on water and the electric field of light, which is why it is the first formula taught in any wave course. This calculator evaluates it for you, in two input styles, and plots the result.
Three results come out of the parameters automatically. The wavelength and the frequency fix the wave speed, v = ω / k = λ f. The sign in front of ω t fixes the direction: a minus sign gives a wave moving toward +x, because the pattern kx − ωt = constant is satisfied by larger x at later times, and a plus sign gives a wave moving toward −x. And the derivative with respect to time gives the velocity of a particle of the medium, −A ω cos(kx − ωt + φ) for the +x wave, which has a maximum of A ω. The distinction between the speed of the pattern and the speed of a particle is the most common source of confusion, and the results panel shows both side by side.
Typical uses include checking homework, designing a signal at a given wavelength, predicting where a crest will be at a later time, and comparing two waves with different phase constants. Because the equation is linear, two such waves can be added to model interference, and the phase constant is exactly the quantity that decides whether they add or cancel. The wave speed ties in with the speed of sound calculator and the speed of light calculator, and the oscillation of a single point is the same simple harmonic motion that governs a pendulum, covered in the physical pendulum calculator.
Keep a few cautions in mind. The equation describes an ideal wave with constant amplitude, so it ignores damping and dispersion. The phase is in radians inside the sine, which is why the phase constant you type in degrees is converted before use. And k and ω are not the wavelength and frequency, they are those quantities multiplied or divided by 2π.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — A Wave on a Rope
A = 0.5 m, wavelength 2 m, frequency 1.5 Hz, moving toward +x, at x = 0.5 m and t = 0.1 s
Example 2 — Starting From k and Omega
A = 0.05 m, k = 20 rad/m, ω = 600 rad/s, φ = 30°, toward +x, at x = 0.1 m and t = 0.002 s
Example 3 — A Wave Moving Toward Negative x
A = 2 m, wavelength 4 m, frequency 0.5 Hz, φ = 90°, toward −x, at x = 0.5 m and t = 1 s
Example 4 — An Ocean Swell
A = 1.2 m, wavelength 8 m, frequency 0.25 Hz, toward +x, at x = 3 m and t = 2 s
❓ Frequently Asked Questions
🔗 Related Calculators
What is the harmonic wave equation?
It is y(x, t) = A sin(k x - w t + phi), the displacement of a sinusoidal wave at position x and time t. A is the amplitude, k the wave number, w the angular frequency and phi the phase constant.
How do you find the wave number k?
The wave number is k = 2 pi / lambda, in radians per metre. A 2 m wavelength gives k = 3.1416 rad/m. It tells you how much phase the wave gains per metre of distance.
How do you find the angular frequency omega?
Use w = 2 pi f = 2 pi / T, in radians per second. A 1.5 Hz wave has w = 9.4248 rad/s. It is the phase gained per second at a fixed position.
How do you calculate wave speed from k and omega?
The speed of the wave pattern is v = w / k, which equals lambda times f. With k = 20 rad/m and w = 600 rad/s the speed is 30 m/s.
What does the minus sign in kx - wt mean?
It means the wave travels toward the positive x direction. A point of constant phase satisfies k x - w t = constant, so x grows as time grows. With a plus sign the pattern moves toward negative x.
What is the phase constant phi?
The phase constant sets the displacement at x = 0 and t = 0. For phi = 0 the wave starts at zero going one way, and for phi = 90 degrees it starts at its maximum A, because the sine becomes a cosine.
What is the difference between wave speed and particle speed?
The wave speed v = w / k is how fast the pattern moves along the medium. A particle only oscillates in place, with a velocity A w cos(phase) that peaks at A w. The two are generally different.
How do you find the displacement at a given x and t?
Compute the phase k x - w t + phi in radians and take A times its sine. For A = 0.5 m, lambda = 2 m, f = 1.5 Hz, x = 0.5 m and t = 0.1 s the phase is 0.6283 rad and the displacement is 0.2939 m.
Is a negative displacement possible?
Yes. The displacement swings between +A and -A, and a negative value just means the point is below the equilibrium position at that instant.
Does the wave equation work for sound and light?
Yes, for any sinusoidal wave. For sound y can stand for pressure or displacement, and for light for the electric field, with the speed v set by the medium.