\(\vec{u}\) may be perpendicular or parallel to \(\vec{v}\) depending on whether it is a transverse or longitudinal wave.

\(y=A \sin k x-\omega t\)

\(\frac{d y}{d t}=A \omega \cos k x-\omega t\)

\(\frac{d y}{d t}=v\)

\(u=-A \omega \cos k x-\omega t\)

\(|v|=\frac{\omega}{k}\)

Slope of \(y=A \sin k x-\omega t\)

\(\frac{d y}{d x}=A k \cos k x-\omega t\)

\(\left|\frac{d y}{d x}\right|=|v|\left|\frac{d y}{d x}\right|\)

\(u=|v| \text { (Slope of waveform) }\)