3.4 Important relations

  Motion of Waves

• Relation between wave velocity $v$, wavelength ${\left( \lambda \right)}$ and frequency $(f)$ $$v = \frac{\lambda }{T}$$ Also, $$v = f\lambda \quad \left( {As,\;f = \frac{1}{T}} \right)$$
• Relation between wave velocity $v$, angular frequency $\left( \omega \right)$ and wave number $k$ $$v = \frac{\omega }{k}$$
• Relation between wave velocity $v$, particle velocity $\left( {{v_p}} \right)$ and slope $m$ Suppose the equation of wave is, $$y(x,t) = A\sin (kx - \omega t)\quad ...(i)$$ Differentiate the above equation wrt time $t$, $$\begin{equation} \begin{aligned} \frac{{\partial y(x,t)}}{{dt}} = - Ak\cos \left( {kx - \omega t} \right) \\ {v_p} = - Ak\cos \left( {kx - \omega t} \right)\quad ...(ii) \\\end{aligned} \end{equation} $$ Also differentiate the above equation wrt displacement $x$, $$\begin{equation} \begin{aligned} \frac{{\partial y(x,t)}}{{dx}} = Ak\cos \left( {kx - \omega t} \right) \\ m = Ak\cos \left( {kx - \omega t} \right)\quad ...(iii) \\\end{aligned} \end{equation} $$ Dividing equation $(ii)$ & $(iii)$ we get,$$\begin{equation} \begin{aligned} \frac{{{v_p}}}{m} = \frac{{ - A\omega \cos \left( {kx - \omega t} \right)}}{{Ak\cos \left( {kx - \omega t} \right)}} \\ {v_p} = - \left( {\frac{\omega }{k}} \right)m \\ {v_p} = - vm\quad \left( {As,\;v = \frac{\omega }{k}} \right) \\\end{aligned} \end{equation} $$
• Relation between acceleration of particle $\left( {{a_p}} \right)$ and particle displacement $y$ Suppose the equation of wave is, $$y(x,t) = A\sin (kx - \omega t)\quad ...(i)$$ Double differentiate the above equation wrt time $t$, $$\begin{equation} \begin{aligned} \frac{{{\partial ^2}y(x,t)}}{{d{t^2}}} = - {\omega ^2}A\sin \left( {kx - \omega t} \right) \\ {a_p} = - {\omega ^2}A\sin \left( {kx - \omega t} \right)\quad ...(ii) \\\end{aligned} \end{equation} $$ From equation $(i)$ & $(ii)$ we get, $${a_p} = - {\omega ^2}y$$

The above figure shows the velocity $\left( {{v_p}} \right)$ and acceleration $\left( {{a_p}} \right)$ of a particle at two points on a wave propagating in positive $X$-direction.