Displacement Relation for a Progressive Wave

A progressive (or travelling) wave is one that carries its disturbance steadily onward through the medium, in contrast to the stationary wave met later in this chapter. For a simple harmonic progressive wave travelling in the positive x-direction, the displacement $y$ of the medium's particle located at position $x$, at time $t$, is written as $y(x,t) = A\sin(kx - \omega t)$ where $A$ is the amplitude (the maximum displacement any particle attains from its mean position), $k$ is the angular wave number (introduced in full in the next section), and $\omega$ is the angular frequency (also introduced below); the combined quantity $(kx-\omega t)$ is called the phase of the wave at that point and instant, and it is the phase, not $y$ alone, that fully describes how far along its oscillation cycle a given particle currently is. A wave travelling instead in the negative x-direction is written $y(x,t)=A\sin(kx+\omega t)$, the sign of the $\omega t$ term flipping to reflect the reversed direction of travel; a general initial phase constant $\phi_0$ may also be added inside the sine to fix the particle's displacement at $x=0,\,t=0$, giving the fully general form $y(x,t)=A\sin(kx-\omega t+\phi_0)$. Two complementary readings of this same equation are useful. Freezing time $t$ at some fixed instant and letting $x$ vary traces out the instantaneous shape (waveform) of the medium at that moment -- a snapshot, sinusoidal in $x$. Freezing position $x$ at some fixed particle and letting $t$ vary instead traces out that one particle's own displacement history over time -- and this history is itself simple harmonic motion, of amplitude $A$ and an …