Waves are mathematical marvels, described by elegant equations that capture their oscillating nature. These functions reveal how waves move through space and time, showing the interplay between amplitude, frequency, and wavelength.
Diving deeper, we see the difference between wave velocity and particle velocity. While waves propagate through a medium, particles oscillate in place. This distinction is crucial for understanding wave behavior in various physical phenomena.
Wave Mathematics
Mathematical function for constant velocity waves
- A sinusoidal function describes a wave moving at constant velocity
- General form: $y(x,t) = A \sin(kx - \omega t + \phi)$ or $y(x,t) = A \cos(kx - \omega t + \phi)$
- $A$ represents the wave amplitude
- $k$ represents the wave number, calculated as $2\pi/\lambda$, where $\lambda$ is the wavelength
- $\omega$ represents the angular frequency, calculated as $2\pi f$, where $f$ is the frequency
- $\phi$ represents the phase constant
- The phase is the argument of the sine or cosine function, expressed as $(kx - \omega t + \phi)$
- The wave function depends on both position $x$ and time $t$
- At a fixed position, the wave oscillates in time with a period of $T = 1/f$ (e.g., a buoy bobbing up and down in the ocean)
- At a fixed time, the wave oscillates in space with a wavelength of $\lambda$ (e.g., a snapshot of a wave on a string)
- The wave equation, a second-order partial differential equation, describes the propagation of waves in a medium
Particle velocity and acceleration in waves
- Particles in a wave-carrying medium oscillate about their equilibrium positions
- The wave function $y(x,t)$ gives the displacement of a particle from its equilibrium position
- The particle velocity is the time derivative of its displacement
- For a sine wave: $v(x,t) = \frac{\partial y}{\partial t} = -A\omega \cos(kx - \omega t + \phi)$
- For a cosine wave: $v(x,t) = -A\omega \sin(kx - \omega t + \phi)$
- The particle acceleration is the time derivative of its velocity or the second time derivative of its displacement
- For a sine wave: $a(x,t) = \frac{\partial v}{\partial t} = \frac{\partial^2 y}{\partial t^2} = -A\omega^2 \sin(kx - \omega t + \phi)$
- For a cosine wave: $a(x,t) = -A\omega^2 \cos(kx - \omega t + \phi)$
- Example: In a sound wave, air particles oscillate back and forth, with their velocity and acceleration determined by the wave function
Wave velocity vs particle velocity
- The wave velocity, or phase velocity, is the speed at which the wave propagates through the medium
- Calculated as $v_p = \frac{\omega}{k} = \frac{\lambda}{T} = \lambda f$
- Depends on the medium properties and wave type (e.g., sound waves in air vs. water)
- The particle velocity is the speed at which the particles oscillate about their equilibrium positions
- Given by the time derivative of the wave function, $v(x,t) = \frac{\partial y}{\partial t}$
- Varies with position and time, with a maximum value of $A\omega$
- The wave velocity and particle velocity are generally not the same
- In most cases, the wave velocity is much greater than the maximum particle velocity (e.g., sound waves in air)
- Particles do not move with the wave; they oscillate as the wave passes through the medium (e.g., water molecules in ocean waves)
- In dispersive media, the group velocity, which represents the velocity of a wave packet, may differ from the phase velocity
Advanced Wave Concepts
- The superposition principle states that when two or more waves overlap, the resulting displacement is the sum of the individual wave displacements
- Dispersion occurs when different frequency components of a wave travel at different velocities in a medium
- Fourier analysis allows complex waveforms to be decomposed into simpler sinusoidal components, enabling the study of wave spectra and harmonic content