Equations of Wave Motion

A wave function mathematically describes how particle displacement varies with both position and time in a medium. It represents the vertical position of particles oscillating in simple harmonic motion, where every particle moves up and down with the same frequency. The wave function is sinusoidal, translating in the positive x-direction as time progresses, allowing us to model and predict wave behavior.

Particles in a medium oscillate up and down in simple harmonic motion perpendicular to the wave's direction of travel. Each particle displaces from its equilibrium position by a certain height in both positive and negative directions within a period T. All particles oscillate with the same frequency, creating the characteristic sinusoidal wave pattern that translates through the medium.

Amplitude A determines the maximum displacement of particles from equilibrium, representing the peak height of the wave. In the wave function, amplitude multiplies the sine function to scale oscillations between +A and −A. This parameter directly controls the magnitude of particle motion and is essential for modeling the wave's vertical extent.

The wave number is the ratio 2π/λ, where λ is the wavelength. It modifies the position term in the wave function, determining how rapidly the wave oscillates spatially. The wave number quantifies the number of wave cycles per unit distance, connecting the spatial structure of the wave to its mathematical representation.

Angular frequency modifies the time term in the wave function, determining how rapidly particles oscillate temporally. It represents the rate of oscillation in radians per unit time and controls the wave's temporal behavior. Combined with the wave number, angular frequency fully characterizes how the wave evolves in both space and time.

The phase angle φ represents the initial phase of the wave, indicating the particle's initial position before wave motion begins. It shifts the entire wave function horizontally and determines various wave parameters at t=0. This angle is crucial for matching theoretical models to real wave conditions and predicting particle positions at any given time.

Wave velocity determines how far the wave travels in a given time interval. The wave moves a distance vt in time t, which modifies the wave function to account for spatial translation. This velocity term ensures the wave function accurately represents the wave's propagation through the medium and allows prediction of wave position at future times.