91Ó°ÊÓ

\(y\left( {x,t} \right) = \left( {2.75\,cm} \right)cos\left( {0.410\,{{rad} \mathord{\left/

{\vphantom {{rad} {cm}}} \right.

\kern-\nulldelimiterspace} {cm}}\,x + 6.20{{rad} \mathord{\left/

{\vphantom {{rad} s}} \right.

\\,t} \right)\)

Time period is defined as the time taken by the wave to complete one vibrational cycle.

Formula:

\(T = \frac{{2\pi }}{\omega }\)

Where,\(\omega \)is angular frequency in\({{rad} \mathord{\left/

{\vphantom {{rad} s}} \right.

\\nulldelimiterspace} s}\)

(a)

By comparison\(y\left( {x,t} \right) = A\cos \left( {kx + \omega t} \right)\)with given wave equation.

\(\begin{aligned}{l}A = 2.75\,cm\\k = 0.410\,{{rad} \mathord{\left/

{\vphantom {{rad} {cm}}} \right.

\\nulldelimiterspace} {cm}}\\\omega = 6.20\,{{rad} \mathord{\left/

{\vphantom {{rad} s}} \right.

\\nulldelimiterspace} s}\end{aligned}\)

\(\begin{aligned}{c}T = \frac{{2\pi }}{\omega }\\ = \frac{{2\pi \,rad}}{{6.20\,{{rad} \mathord{\left/

{\vphantom {{rad} s}} \right.

\\nulldelimiterspace} s}}}\\ = 1.013\,s\end{aligned}\)

Distance travels a wave crest in one cycle is

\(\begin{aligned}{c}\lambda = \frac{{2\pi \,}}{k}\\ = \frac{{2\pi \,rad}}{{0.410\,{{rad} \mathord{\left/

{\vphantom {{rad} {cm}}} \right.

\kern-\nulldelimiterspace} {cm}}}}\\ = 15.32\,cm = 0.153\,m\end{aligned}\)