A progressive wave is a disturbance that travels through a medium, carrying energy from one place to another without transporting matter. Think of it like a ripple that moves across a pond when you drop a stone – the water particles move up and down, but the ripple itself moves outward.
Energy in a progressive wave is transferred by the oscillation of particles. The key point is that the energy moves with the wave, not the particles themselves. This is why a wave can travel across a long distance while the medium stays relatively unchanged.
Mathematically, the energy density (energy per unit volume) of a simple harmonic wave is given by:
\$\epsilon = \frac{1}{2} \rho \omega^2 A^2\$
where \$\rho\$ is the density of the medium, \$\omega\$ is the angular frequency, and \$A\$ is the amplitude.
The displacement of a particle in a progressive wave can be described by:
\$y(x,t) = A \sin(kx - \omega t)\$
where \$k = \frac{2\pi}{\lambda}\$ is the wave number and \$\lambda\$ is the wavelength.
The speed of the wave is:
\$v = f \lambda = \frac{\omega}{k}\$
⚡️ Tip: Remember that the product of frequency and wavelength gives you the wave speed.
Analogy: Imagine a line of people holding hands and passing a ball from one end to the other. Each person only moves a little, but the ball (energy) travels across the line.
| Concept | Key Formula |
|---|---|
| Wave speed | \$v = f \lambda\$ |
| Energy density | \$\epsilon = \frac{1}{2} \rho \omega^2 A^2\$ |
| Displacement | \$y(x,t) = A \sin(kx - \omega t)\$ |