Question

A stationary wave is represented by $y = 10 \sin \frac{\pi x}{4} \cos 20 \pi t$ where 'x' and expressed in cm and 't' in seconds. Distance between two consecutive nodes is

• A. 1 cm
• B. 2 cm
• C. 4 cm
• D. 8 cm

Answer 1

Answer: (C)

4 cm

Answer 2

The stationary wave is given by

$$y = 10 \sin\frac{\pi x}{4} \cos 20 \pi t.$$

Comparing with the standard form

$$y = A \sin(kx) \cos(\omega t),$$

we identify the wave number $k = \frac{\pi}{4}$.

Nodes occur where $\sin(kx) = 0$, i.e.,

$$kx = m\pi \quad \Rightarrow \quad x = \frac{m\pi}{k} = \frac{m\pi}{\pi/4} = 4m \quad (\text{cm}),$$

where $m$ is an integer. Hence, the distance between two consecutive nodes is

$$\Delta x = 4\, \text{cm}.$$