question_answer 62)

                  In the given progression wave                 \[y=5\sin \,(100\,\pi t-0.4\,\pi x)\]                 where \[y\] and \[x\] are in \[m,\,t\] in \[s\]. What is the (a) Amplitude (b) wavelength (c) frequency (d) Wave velocity (e) Particle velocity amplitude?

Answer:

                  \[y=5\,\sin \,(100\pi t-0.4\pi x)\]                      ? (i)                 Compare is with standard equation                 \[y=a\sin (\omega t-kx)\]                                         ?. (ii) (a) Amplitude, \[a=\,5\,m\] (b) \[k=0.4\,\pi \] or \[\frac{2\pi }{\lambda }=\,0.4\pi \] or \[\lambda \,=\frac{2}{0.4}=\,5\,m\] \[\omega =100\,\pi \] or \[2\pi v=\,100\,\pi \] or\[v=50\,Hz\]. (c) \[\upsilon =v\lambda =50\,\times 5=250\,m\,{{s}^{-1}}\] (d) Differentiating eqn. (i) w.r.t. \[t,\] we get                 \[\frac{dy}{dt}\,=500\,\pi \,cos\,\,\pi t-\,0.4\,\pi x\]                 Where, \[\frac{dy}{dt}\,=\] particle velocity                 Particle velocity will be maximum (known as particle velocity amplitude) if \[\cos (100\,\pi t-0.4\,\pi x)=1\]                \            Particle velocity amplitude \[=500\,\,\pi \,m\,{{s}^{-1}}\]