Q. A string vibrates according to the equation $ y=5\sin \left( \frac{2\pi x}{3} \right)\cos 20\pi t $ where $ x $ and y are in cm and t in second. The distance between two adjacent nodes is
JamiaJamia 2006
Solution:
The nodes and antinodes are formed in a standing wave pattern as a result of the interference of two waves. Distance between two nodes is half of wavelength $ (\lambda ) $ standard equation of standing wave is
$ y=2a\sin \frac{2\pi x}{\lambda }\cos \frac{2\pi vt}{\lambda } $ ...(i) where a is amplitude, $ \lambda $ the wavelength, v the velocity and t the time. Given, equation is $ y=5\sin \frac{2\pi x}{3}\cos 20\pi t $ ...(ii) Comparing Eqs. (ii) with (i), we have $ \frac{2\pi x}{\lambda }=\frac{2\pi x}{3} $ $ \Rightarrow $ $ \lambda =3\,cm $ Hence, distance between two adjacent nodes is $ \frac{\lambda }{2} $ . $ =\frac{3}{2}=1.5\,cm $ Note: Standing wave is not actually a wave but rather a pattern which results from the interference of two or more waves. Since, standing wave is not technically a wave, then an antinode is not technically a point on a wave. The nodes and antinodes are merely points on the medium which up the wave pattern.
