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 half the wave velocity, if \(\mathrm{c}=\)

General equation of a simple harmonic progressive wave,
\(y=A \sin 2 \pi\left[\frac{t}{T}-\frac{x}{\lambda}\right]\)
Given: \(y=a \sin 2 \pi(b t-c x)\)
\(\Rightarrow \mathrm{A}=\mathrm{a}, \frac{1}{\mathrm{~T}}=\mathrm{b} \text { and } \frac{1}{\lambda}=\mathrm{cx}\)
Also, \(\left(v_p\right)_{\max }=a \omega=a(2 \pi n)=\frac{a 2 \pi}{T}\)
\(=\frac{\mathrm{A} 2 \pi}{\mathrm{T}}=\mathrm{a} 2 \pi \mathrm{b}\)
From \(\mathrm{v}=\frac{\lambda}{\mathrm{T}}=\frac{1 / \mathrm{c}}{1 / \mathrm{b}}=\frac{\mathrm{b}}{\mathrm{c}}\)
Given: \(\left(\mathrm{v}_{\mathrm{p}}\right)_{\max }=\frac{1}{2} \mathrm{v}\)
\(\begin{aligned}
\Rightarrow & 2 \pi \mathrm{ab}=\frac{1}{2} \times \frac{\mathrm{b}}{\mathrm{c}} \\
\therefore \quad \mathrm{c} & =\frac{1}{4 \pi \mathrm{a}}
\end{aligned}\)