A transverse progressive wave travelling in the +x direction: $y = A\sin(kx - \omega t)$ or $y = A\sin(\omega t - kx)$ (differ by a sign/phase convention). The minus sign in $(kx - \omega t)$ indicates +x travel; a plus sign indicates -x travel. Wave speed $v = \omega/k = f\lambda$. Particle velocity $v_p = \partial y/\partial t = -A\omega\cos(kx - \omega t)$. Wave velocity $v_w = \omega/k$. The ratio $v_p/v_w = -\partial y/\partial x$ (slope of the wave at that point).