-1
m
-1.570796
rad
3.141593
rad/m
31.415927
rad/s
10
m/s
0.2
s
-0.25
cycles
-1
-1
m
-1.570796
rad
3.141593
rad/m
31.415927
rad/s
10
m/s
0.2
s
-0.25
cycles
-1
The Harmonic Wave Equation Calculator evaluates the transverse displacement of a sinusoidal traveling wave at any point in space and time. Using the standard form y(x, t) = A sin(kx − ωt), it computes the wave number, angular frequency, phase, and instantaneous displacement from the amplitude, frequency, and wavelength you provide.
Harmonic waves are the building blocks of all wave phenomena. By Fourier's theorem every periodic disturbance can be decomposed into a sum of sinusoidal harmonics, making this single-frequency solution the cornerstone of acoustics, optics, seismology, and electromagnetic theory. The calculator also returns the wave speed v = fλ and the period T = 1/f for complete characterization of the wave.
A one-dimensional harmonic traveling wave propagating in the +x direction is described by:
$$y(x,t) = A\sin(kx - \omega t)$$
where the two derived quantities are:
The argument of the sine function, $$\phi = kx - \omega t$$, is the phase. Surfaces of constant phase travel at the phase velocity:
$$v = \frac{\omega}{k} = f\lambda$$
Key properties of this solution:
This calculator uses the convention of a wave traveling in the positive x-direction. For a wave moving in the negative x-direction the sign inside the sine changes to kx + ωt. Phase offsets can be added if the wave does not start at zero displacement at the origin at t = 0.
The harmonic wave equation applies to transverse waves on strings, longitudinal sound waves (where y represents pressure variation), electromagnetic waves, and surface water waves in the linear regime. It assumes a non-dispersive medium where wave speed is independent of frequency.
The displacement y(x,t) oscillates between −A and +A. Positive values indicate displacement above equilibrium; negative values below. If the phase is a multiple of π the displacement is zero (a node in a snapshot). The wave number k tells you how many radians of phase accumulate per meter; larger k means shorter wavelength and more tightly packed oscillations. The wave speed v = fλ is a property of the medium, not the wave itself — changing frequency at fixed speed changes the wavelength inversely.
Inputs
Results
A 440 Hz concert-A tone with 1 mm amplitude in air (v ≈ 343 m/s, λ ≈ 0.78 m). At x = 1.5 m and t = 2 ms the displacement is about −0.31 mm.
Inputs
Results
The E4 string vibrates at 330 Hz with a wavelength of 1.3 m (twice the string length for the fundamental). At the midpoint x = 0.65 m and t = 1 ms, the displacement is about 2.4 mm.
It describes a single-frequency sinusoidal traveling wave in one dimension. The equation y = A sin(kx − ωt) gives the displacement at any position x and time t for a wave of amplitude A, wave number k, and angular frequency ω propagating in the +x direction.
The wave number k = 2π/λ is the spatial analog of angular frequency — it counts radians per meter along the propagation axis. Angular frequency ω = 2πf counts radians per second in time. Together they define the phase velocity v = ω/k.
It works for any wave in a linear, non-dispersive medium — sound waves, waves on strings, electromagnetic waves in vacuum. In dispersive media the wave speed depends on frequency, and a single sinusoid still obeys this equation but a wave packet changes shape as it propagates.
Change the sign in the phase argument to y = A sin(kx + ωt). The plus sign means surfaces of constant phase move toward negative x values.
The time-averaged power transmitted by a harmonic wave on a string is $$P = \tfrac{1}{2}\mu\omega^2 A^2 v$$, where μ is the linear mass density. Energy is proportional to the square of the amplitude and the square of the frequency.
Both are equally valid — they differ only by a phase offset of π/2. The sine convention assumes zero displacement at the origin when t = 0. If your initial condition requires maximum displacement at the origin, use cosine or add a phase constant φ₀.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
How helpful was this calculator?
Be the first to rate!
