Question

If $\vec{u}$ is instantaneous velocity of particle and $\vec{v}$ is velocity of wave, then

• A. $\vec{u}$ is perpendicular to $\vec{v}$
• B. $\vec{u}$ is parallel to $\vec{v}$
• C. $|\vec{u}|$ is equal to $|\vec{v}|$
• D. $|\vec{u}|=$ (slope of wave form) $|\vec{v}|$.

Answer 1

Answer: (D)

$\vec{u}$ may be perpendicular or parallel to $\bar{v}$ depending on whether it is a transverse or longitudinal wave.
$y=A\sin kx-\omega t$
$\frac{dy}{dt}=A\omega \cos kx-\omega t$
$\frac{dy}{dt}=v$
$u=-A\omega \cos kx-\omega t$
$|v|=\frac{\omega }{k}$
Slope of $y=A\sin kx-\omega t$
$\frac{dy}{dx}=Ak\cos kx-\omega t$
$\left|\frac{dy}{dx}\right|=|v|\left|\frac{dy}{dx}\right|$
$u=|v|$ (Slope of waveform)