Harmonic Wave Equation Calculator
Solve the harmonic wave equation y(x,t) = A sin(kx − ωt + φ). Enter amplitude, frequency, wavelength and phase to get wave speed, period, angular frequency, wave number and relative intensity.
m
Hz
m
°
v = f × λ — speed of the wave through the medium
- 1
Angular frequency (ω = 2π × f)
2π × 2 = 12.5664 - 2
Wave number (k = 2π ÷ λ)
2π ÷ 1.5 = 4.1888 - 3
Wave speed (v = f × λ)
2 × 1.5 = 3Speed at which the wave pattern moves through the medium.
How does this calculator work?
A harmonic wave y = A sin(kx − ωt + φ) is defined by amplitude A, frequency f, wavelength λ and phase φ. Wave speed v = f × λ; period T = 1/f; angular frequency ω = 2πf; wave number k = 2π/λ. Intensity scales as A². The wave profile is a sine snapshot that travels at speed v without changing shape (linear medium).
Formula
How this is calculated
A harmonic (sinusoidal) wave is fully described by four parameters: amplitude A (maximum displacement from equilibrium, in metres), frequency f (cycles per second, Hz), wavelength λ (metres per cycle), and phase offset φ (starting angle in degrees). This calculator derives the remaining wave quantities from these inputs.
The wave speed v = f × λ — the rate at which the wave pattern moves through the medium. The period T = 1/f is the time for one complete oscillation. Angular frequency ω = 2πf converts ordinary frequency into radians per second, while the wave number k = 2π/λ converts wavelength into radians per metre. The displacement at any point x and time t is y(x,t) = A sin(kx − ωt + φ).
The model assumes a linear, non-dispersive, isotropic medium in which wave shape is preserved as it travels. It applies to small-amplitude sound waves in air, light in vacuum, and vibrations on a taut string. For large amplitudes, nonlinear effects distort the shape. Intensity is proportional to A²: doubling the amplitude quadruples the power carried by the wave.
Frequently asked questions
Frequency f (in Hz) counts complete cycles per second. Angular frequency ω = 2πf (in rad/s) measures how many radians the wave phase advances per second. The factor 2π converts cycles to radians. Angular frequency is mathematically convenient because it appears naturally in the sine and cosine functions.
The phase offset shifts the wave left or right along the x axis at t = 0. φ = 0° gives a pure sine starting at zero; φ = 90° gives a cosine (starts at maximum); φ = 180° flips the sine. Phase differences between two waves determine whether they interfere constructively (in phase, φ = 0°) or destructively (out of phase, φ = 180°).
In a linear medium, no. The wave speed v = fλ is a property of the medium (its elasticity and density for mechanical waves, or permittivity and permeability for electromagnetic waves), not of the amplitude. Amplitude only affects intensity (power per area), which scales as A². In nonlinear media, amplitude can affect speed, causing wave distortion.
Also known as
TG we-Calculate Editorial Team. (2026). Harmonic Wave Equation Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/harmonic-wave-equation-calculator
TG we-Calculate Editorial Team. "Harmonic Wave Equation Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/harmonic-wave-equation-calculator.
TG we-Calculate Editorial Team, "Harmonic Wave Equation Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/harmonic-wave-equation-calculator
@misc{wecalculate_harmonic_wave_equation_calculator, title = {Harmonic Wave Equation Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/harmonic-wave-equation-calculator}}, year = {2026}, note = {TG we-Calculate} }
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