The equation of simple harmonic progressive wave is given by \(Y=a, \sin 2 \pi(b t-c x)\). The maximum…
The equation of simple harmonic progressive wave is given by \(Y=a, \sin 2 \pi(b t-c x)\). The maximum particle velocity will be twice the wave velocity if
\(c=\pi a\)
\(c=\frac{1}{2 \pi a}\)
\(c=\frac{1}{\pi a}\)
\(c=2 \pi a\)
Solution
Given, wave equation, \(y=a \sin 2 \pi(b t-c x)\)
Comparing the above equation with the general equation of the progressive wave which is given as,
\(y=A_0 \sin 2 \pi\left(f t-\frac{x}{\lambda}\right)\)
We get, frequency, \(f=b\), wavelength, \(\lambda=\frac{1}{c}\) and
amplitude of the wave, \(A_0=a\) As, we know, that the maximum velocity of the particle,
\(v_{\max }=A_0 \omega=a \times 2 \pi b \ldots(i)\)
Wave velocity, \(v_{\text {wave }}=f \lambda\)
\(\Rightarrow v_{\text {wave }}=\frac{b}{c} \ldots \text { (ii) }\)
It is given that,
\(v_{\mathrm{max}}=2 v_{\text {wave }}\)
So, by substituting the values from Eqs. (i) and (ii) in the above relation, we get
\(\begin{aligned}
& a 2 \pi b=2 \frac{b}{c} \\
& \therefore c=\frac{1}{a \pi}
\end{aligned}\)