Explanation:
Given, wave equation \(y=a \sin 2 \pi(b t-c x)\)
Compairing the above equation with the general equation of the progressive wave which is given as,
\(y=A_{0} \sin 2 \pi\left(f t-\dfrac{x}{\lambda}\right)\)
We get, frequency, \(f = b\), wavelength \(\lambda=\dfrac{1}{c}\) and amplitude of the wave \(A_{0}=a\)
As, we know that the maximum velocity of the particle,
\({v_{\max }} = {A_0}\omega = a \times 2\pi b\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( 1 \right)\)
Wave velocity, \(v_{\text {wave }}=f \lambda\)
\( \Rightarrow {v_{{\text{wave }}}} = \frac{b}{c}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( 2 \right)\)
It is given that,
\(v_{\max }=2 v_{\text {wave }}\)
So, by substituting the values from eq.(1) and eq.(2) in the above relation, we get
\(\begin{aligned}& \quad a 2 \pi b=2 \dfrac{b}{c} \\& \therefore c=\dfrac{1}{a \pi}\end{aligned}\)