What should be the minimum value of amplitude so that the particle speed can match or exceed the wave speed in sinusoidal progressive wave $y = A sin (ωt − \frac {2\pi}{\lambda}x)$?
(A) Particle speed can never exceed wave speed
(B) $A_{min} = \lambda$
(C) $A_{min} = \frac {\lambda}{2\pi}$
(D) Particles are not present as the wave travels in vacuum
Solution
Particle speed = $|\frac {\partial y}{\partial t}| = Aω |cos (ωt − \frac {2\pi}{\lambda}x)|$
So, particle speed is less than or equal to Aω.
Wave speed = $\nu.\lambda = \frac {ω}{2\pi}.\lambda$
If particle speed were to match or exceed wave speed,
$Aω |cos (ωt − \frac {2\pi}{\lambda} x)| \geq \frac {ω}{2\pi}.\lambda$
$\therefore A \geq \frac{{\frac{\lambda }{{2\pi }}}}{{\left| {cos\left( {\omega t - \frac{{2\pi }}{\lambda }x} \right)} \right|}}$
For A to be min. cosine should be maximum = 1
$A_{min} \geq \frac {\lambda}{2\pi}$
When $A_{min} = \frac {\lambda}{2\pi}$, particle speed can match wave speed periodically and when $A_{min} > \frac {\lambda}{2\pi}$, particle speed can exceed wave speed periodically.
Hence, (C).