Mathematical Expression of a Wave

A progressive wave needs a single mathematical function that captures both HOW the wave's shape looks at any instant (a function of position x) and HOW each individual particle moves over time (a function of time t) -- so the function must depend on both x and t together. For a sinusoidal progressive transverse wave, with y(x,t) denoting the displacement of the particle located at position x at time t, this function is written

where a (amplitude), k (angular wave number), ω (angular frequency) and φ (an initial-phase constant) are all fixed constants for a given wave.

Why this form works. Freeze time at some instant $t=t_0$: then $y(x,t_0)=a\sin(kx+\text{constant})$, which as a function of x alone is exactly a sine curve -- so the wave's SNAPSHOT shape at any instant is sinusoidal, as required. Now instead freeze the position at some fixed point $x=x_0$: then $y(x_0,t)=a\sin(\text{constant}-\omega t)$, which as a function of t alone is again a sine function -- so every individual particle of the medium executes simple harmonic motion about its own rest position, exactly as double periodicity (section 8.2) demands. For the ARGUMENT $(kx-\omega t+\phi)$ to stay constant as t increases, x must increase along with it -- so this particular sign combination, minus between kx and ωt, describes a wave travelling along the POSITIVE x-direction. Flipping that one sign, $y(x,t)=a\sin(kx+\omega t+\phi)$, instead describes a wave travelling along the NEGATIVE x-direction. …