question_answer 39)

                  The displacement of a string is given by \[y\,(x,\,t)=0.06\,\sin \,(2\pi x/3)\,\cos \,(120\,\pi t)\], where \[x\] and \[y\] are in \[m\] and \[t\] in \[s\]. The length of the string is 1.5 in and its mass is\[3.0\times {{10}^{-2}}\,k\,g\].                 (a) It represents a progressive wave of frequency 60 Hz.                 (b) It represents a stationary wave of frequency 60 Hz.                 (c) It is the result of superposition of two waves of wavelength 3 in, frequency 60 Hz each travelling with a speed of 180 m/s in opposite direction.                 (a) Amplitude of this wave is constant.

Answer:

                  (b, c) \[y\,(x,\,t)=0.06\,\sin \,\left( \frac{2\pi x}{3} \right)\cos (120\,\pi \,t)\]... (i)                 The given equation is the equation of a stationary wave, whose general form is                 \[y\,(x,\,t)=(2A\,\sin \,kx)\,\cos \,\omega t\]                ... (ii)                 Compare eqns. (i) and (ii)                 Amplitude \[=0.06\,\sin \,\left( \frac{2\pi x}{3} \right),\] which changes with \[x\].                 \[\omega =120\,\pi \] or \[v=60\,Hz\]                 \[k\,=\frac{2\pi }{3}\] or \[\frac{2\pi }{\lambda }\,=\,\frac{2\pi }{3}\] or \[\lambda \,=\,3m\]                 \[\upsilon \,=v\upsilon \lambda =180\,m{{s}^{-1}}\]