Question

The equation of stationary wave on a string clamped at both ends and vibrating in third harmonic is y = 0.5 sin (0.314x) cos (600$\pi$t) where x and y are in cm, t in seconds. The length of the vibrating string is

• A. 20 cm
• B. 30 cm
• C. 40 cm
• D. 10 cm

Answer 1

Answer: (B)

30 cm

Answer 2

For a string fixed at both ends and vibrating in the nth harmonic, the length $L$ is related to the wavenumber $k$ as:

$$kL = n\pi.$$

Given the wave is in the third harmonic ($n=3$) and the wave equation is:

$$y = 0.5 \, \sin(0.314x) \cos(600\pi t),$$

we have $k = 0.314$ rad/cm. Hence,

$$0.314\, L = 3\pi \quad \Rightarrow \quad L = \frac{3\pi}{0.314}.$$

Calculating,

$$L \approx \frac{9.4248}{0.314} \approx 30\, \text{cm}.$$