question_answer 38)

                  A transverse harmonic wave on a string is described by \[y(x,\,t)=3.0\,\sin \,(36t+0.018x+\pi /4)\] where \[x\] and \[y\] are in cm and \[t\] is in \[s\]. The positive direction of \[x\] is from left to right.                 (a) The wave is travelling from right to left.                 (b) The speed of the wave is \[20\,\,m/s\]                 (c) Frequency of the wave is \[5.7\,Hz\]                 (d) The least distance between two successive crests in the wave is 2.5 cm.

Answer:

                  (a, b, c) \[y\,(x,\,t)=3.0\,\sin \,(36t+0.018\,x+\pi /4)\]                 Compare it with standard equation                 \[y\,(x,\,t)=A\sin \frac{2\pi }{\lambda }(\upsilon t-x+\phi )\]                                                                 [Along + ve \[x-\]axis]                 \[\therefore \] \[=\,\frac{2\pi \upsilon }{\lambda }\,=36\] or \[\frac{\upsilon }{\lambda }\,=\,\frac{36}{2\pi }\]                        ?? (i)                 and \[\frac{2\pi }{\lambda }=\,0.018\]                 or \[\lambda \,=\frac{2\pi }{0.018\,}=\,3.49\,m\]                 \[\therefore \] \[\upsilon =\,\frac{36}{2\pi }\,\times \,\frac{2\pi }{0.018}\,=\,2000\,cm\,{{s}^{-1}}\]                 \[v=\,\frac{\upsilon }{\lambda }\,=\,\frac{20}{3.49}\,=\,5.7\,Hz\]                 Distance between two successive crests \[=\lambda \]