question_answer

    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

A)  3 cm                                     

B)  4.5 cm

C)  6 cm                                     

D)  1.5 cm

Correct Answer: D

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.