Compute wave displacement using the standard harmonic wave equation y(x,t) = A·sin(kx - ωt + φ). Visualize the waveform, extract wave speed, angular frequency, wavenumber, and analyze real-time propagation effects.
The harmonic wave equation (or sinusoidal wave) describes the propagation of a wave with constant frequency and amplitude. The fundamental formula is:
y(x,t) = A · sin(kx - ωt + φ)
where A = amplitude, k = wavenumber (2π/λ), ω = angular frequency (2πf), φ = phase constant.
This equation represents a wave traveling in the positive x-direction. The argument (kx - ωt + φ) is the phase. Points of constant phase move with velocity v = ω/k = fλ. This model is fundamental in acoustics, optics, quantum mechanics, and seismology.
The harmonic wave equation is a solution to the classical wave equation ∂²y/∂x² = (1/v²) ∂²y/∂t². By inserting y = A sin(kx - ωt + φ), we obtain the dispersion relation ω = v k. The wavenumber k = 2π/λ describes spatial oscillation frequency, while angular frequency ω = 2πf describes temporal oscillation. The phase velocity v = ω/k is constant for non-dispersive media, meaning all frequencies travel at the same speed — a key property in ideal strings and electromagnetic waves in vacuum.
Furthermore, using Euler's formula, sinusoidal waves can be represented as the imaginary part of complex exponentials: y = Im(A e^{i(kx - ωt + φ)}). This complex representation greatly simplifies analysis of superposition, interference, and Fourier transforms — the backbone of signal processing and modern physics.
Sound waves in air are longitudinal pressure variations, often modeled as harmonic waves. A tuning fork generates a nearly pure sinusoidal wave of frequency 440 Hz (Concert A). Using our calculator, set A = 0.02 Pa (pressure amplitude), λ = 0.78 m (at 343 m/s), f = 440 Hz, observe displacement amplitude and predict pressure nodes. Instrument makers rely on harmonic analysis to design resonators.
Light waves are transverse electromagnetic waves described by the same harmonic equation. For a laser beam with wavelength λ = 532 nm (green light), frequency f = c/λ ≈ 5.64×10¹⁴ Hz. Optical engineers use wave equations to compute interference patterns, grating diffraction, and thin-film coatings. This calculator helps conceptualize phase shifts and wave superposition.
Surface gravity waves in deep water are approximately sinusoidal for small amplitudes. Predicting wave displacement is crucial for offshore platform design and coastal erosion models. Input typical ocean swell: A = 1.5 m, λ = 80 m, f = 0.1 Hz → wave speed ≈ 8 m/s. Our tool visualizes the waveform and calculates critical parameters for engineering simulations.
By adjusting parameters and immediately seeing the waveform and displacement value, learners develop intuition for wave behavior: increasing frequency compresses the wave spatially; higher amplitude raises crests; adding a phase shift shifts the wave left/right. The tool also shows the instantaneous displacement at any user-defined position and time.
| Parameter change | Effect on waveform (snapshot at fixed t) | Effect on wave speed |
|---|---|---|
| Increase λ (wavelength) | Wave stretches horizontally, fewer cycles in same range | Increases v if f constant |
| Increase f (frequency) | More cycles per unit distance (k increases) | Increases v if λ constant |
| Increase A (amplitude) | Taller peaks and deeper troughs | No change |
| Add phase φ > 0 | Wave shifts to the left (negative x direction shift) | No change |