Particle Velocity and Wave Velocity

Two distinct velocities arise in describing a progressive wave, and confusing them is one of the most common conceptual errors students make. The particle velocity, $v_P=\partial y/\partial t$, is the instantaneous rate of change of displacement of one single, specific particle of the medium at its own fixed location; for the standard sinusoidal wave $y(x,t)=A\sin(kx-\omega t)$, differentiating with respect to time at fixed $x$ gives $v_P=-\omega A\cos(kx-\omega t)$, which oscillates sinusoidally between $-\omega A$ and $+\omega A$ and differs from particle to particle across the wave at any given instant. The wave (or phase) velocity, $v=\omega/k$, by contrast, is the constant speed at which a single point of fixed phase -- such as a particular crest -- itself moves through the medium; it is derived by requiring that the total phase $kx-\omega t$ stay exactly constant as both $x$ and $t$ change together while tracking that one point, which on rearranging gives $dx/dt=\omega/k=v$. Physically these describe completely different motions: the wave velocity carries the overall pattern steadily forward through space at a fixed rate, exactly as computed from $v=f\lambda$, while any one individual particle of the medium never actually travels anywhere at all -- it only ever oscillates back and forth about its own fixed …