Progressive Wave

Picture the ripples that spread out on the surface of water when a stone is dropped into it. The water is displaced locally, right where the stone actually falls, and this disturbance slowly spreads outward: a distant particle of water is disturbed from its position of rest only once the ripple actually reaches it, then oscillates about that rest position for a short time as the disturbance passes, without ever being bodily carried away from where it started. This kind of localized, short-lived disturbance is called a wave pulse.

A wave in which the disturbance produced in the medium travels continuously in a given direction, without damping or obstruction, being passed on from one particle to the next indefinitely, is called a progressive wave or travelling wave. A sound wave is the standard example — a pressure wave made of a continuous train of compressions and rarefactions travelling along the direction of propagation.

Properties shared by every progressive wave:

1. Every particle of the medium executes the same type of vibration — each performs simple harmonic motion about its own mean position.
2. All the vibrating particles share the same amplitude, period and frequency.
3. The phase (the instantaneous state of vibration) differs from one particle to the next.
4. No particle stays permanently at rest — each one comes to rest only momentarily, at the extreme positions of its own vibration.
5. Each particle attains its maximum velocity as it passes through its own mean position.
6. As the wave propagates, energy is transferred along it, but there is no transfer of matter.
7. The wave propagates through the medium with a certain velocity that depends on the properties of that medium.
8. Progressive waves are of two types: transverse waves, in which particles vibrate perpendicular to the direction the wave travels, producing crests and troughs; and longitudinal waves, in which particles vibrate along the direction of travel, producing compressions and rarefactions.
9. Both transverse and longitudinal mechanical waves can propagate through solids, but only longitudinal waves can propagate through fluids (liquids and gases) — a fluid cannot sustain the shearing needed to carry a transverse disturbance.

The wave equation. When a mechanical wave travels through an elastic medium, the displacement of a particle at position $x$ at time $t$ can be written $y(x,t) = f(x - vt)$, where $v$ is the speed at which the disturbance travels (taking the wave to move along $+x$). The factor $(x-vt)$ appears because the disturbance produced at $x=0$ at some time reaches a point $x=x'$ only after a delay $x'/v$ — equivalently, whatever is seen at $x'$ at time $t$ actually originated at $x=0$ at the earlier time $(t-x'/v)$. This form represents a progressive wave of fixed shape travelling at constant speed $v$ along $+x$; the exact function $f$ depends on how the source of the disturbance itself moves.

If the source performs simple harmonic motion, the disturbance becomes a sine or cosine function of $(x-vt)$, conventionally written

$y(x,t) = A\sin(kx - \omega t)$ …