Question

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

Answer 1

Answer: (D)

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 $\frac{2\pi x}{\lambda }=\frac{2\pi x}{3}$ $\Rightarrow$ $\lambda =3cm$ Hence, distance between two adjacent nodes is $\frac{\lambda }{2}$ . $=\frac{3}{2}=1.5cm$ 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.