question_answer

The equation of a stationary wave is,\[y=5\,\sin \,\frac{\pi x}{3\,}\,\cos \,40\,\pi t\] where\[x\] and \[y\] are in cm and t in second. The separation between two consecutive nodes is :

A)  1.5cm       

B)  6.0cm

C)  3cm         

D)  12cm

Correct Answer: C

Solution :

The standard equation of wave is \[y=2\,a\,\cos \frac{2\pi t}{T}\sin \frac{2\pi \,x}{\lambda }\] ... (i) where a is amplitude, T is periodic time, and  is wavelength. Given equation is \[y=5\sin \frac{\pi x}{3}\cos 40\,\pi t\] ... (ii) Comparing Eq. (i) with Eq. (ii), we get \[\frac{2\pi x}{\lambda }=\frac{\pi x}{3}\] \[\Rightarrow \] \[\lambda =6\,cm\]. Separation between consecutive nodes is given \[\frac{\lambda }{2}=\frac{6}{2}=3\,cm\]