Question

Show that the particle speed can never be equal to the wave speed in a sine wave if the amplitude is less than wavelength divided by 2π.

Answer 1

Proved that Particle Speed ≠ Wave Speed

Explanation:

Equation of the wave is given by

$y=A\times sin(\omega t - kx)$

where

        A is the amplitude

          $\omega$ is the angular frequency

          k is the wave number

Velocity of wave, $v = \frac {\omega}{k}$

Velocity of particle,

Max. Velocity of particle, $v_p_m_a_x = A\omega$

As given, $A < \frac {\lambda }{2\pi}$

$v_p_m_a_x = \frac {\lambda\omega }{2\pi}$

$v_p_m_a_x < \frac {\omega}{k}$ [∵ $\frac {2\pi}{\lambda} = k$]