Table of Contents
The Harmonic Wave Equation
The harmonic wave equation y = A sin(kx - omega t + phi) is the mathematical description of a sinusoidal traveling wave. Each parameter has a physical meaning: A is the amplitude (maximum displacement), k is the wave number (spatial frequency), omega is the angular frequency (temporal frequency), and phi is the initial phase offset.
This equation describes waves on strings, sound waves in tubes, electromagnetic waves, and many other oscillatory phenomena. The minus sign between kx and omega t indicates a wave traveling in the positive x direction. A plus sign would indicate propagation in the negative x direction.
Wave Parameters
Parameter Relationships
| Parameter | Symbol | Unit | Formula |
|---|---|---|---|
| Wavelength | λ | m | 2π/k |
| Frequency | f | Hz | ω/2π |
| Period | T | s | 2π/ω |
| Phase velocity | v | m/s | ω/k |
Frequently Asked Questions
What is the phase constant?
The phase constant phi determines the initial displacement of the wave at x=0 and t=0. If phi=0, the wave starts at zero displacement. If phi=pi/2, it starts at maximum displacement (cosine wave). The phase constant shifts the entire wave pattern in space or time without changing its shape.
What is the difference between phase velocity and group velocity?
Phase velocity (v = omega/k) is the speed at which a single frequency component propagates. Group velocity is the speed at which the envelope of a wave packet (containing multiple frequencies) travels. In non-dispersive media they are equal, but in dispersive media they differ, which affects signal propagation.