Question

Two points x1 and x2 at x = 0 and x = 1 m are observed. The transverse motion of the two points are found to be as follows:
y1(x,t)=0.2 sin 3πt
and y2(x,t)=0.2 sin (3πt + π/8)
Calculate the frequency, wavelength and speed of the wave.

Answer 1

Two points $x1$ and $x2$ at $x = 0$ and $x = 1$ m are observed. The transverse motion of the two points are found to be as follows:

$$y1(x,t) = 0.2 \sin 3\pi t$$ $$\text{and } y2(x,t) = 0.2 \sin (3\pi t + \pi/8)$$

Calculate the frequency, wavelength and speed of the wave.

$$y_1(x,t) = 0.2 \sin 3\pi t = 0.2 \sin 2\pi \frac{t}{\frac{2}{3}}$$

where $T = \frac{2}{3}$ – period of motion

Frequency equals:

$$f = \frac{1}{T} = \frac{1}{\frac{2}{3}} = \frac{3}{2} \frac{1}{s}$$ $$y_2(x,t) = 0.2 \sin \left(3\pi t + \frac{\pi}{8}\right) = 0.2 \sin 3\pi \left(t + \frac{\pi}{24}\right)$$

where $\Delta t = \frac{\pi}{24}$ – delay time

Therefore, speed of the wave equals:

$$v = \frac{1m}{\Delta t} = \frac{24m}{\pi} \frac{m}{s}$$

The wavelength $\lambda$ of a sinusoidal waveform traveling at constant speed $v$ is given by:

$$\lambda = \frac{v}{f} = \frac{24}{\pi} \frac{2}{3} = \frac{16}{\pi} m$$

Answer: $f = \frac{3}{2} \frac{1}{s}, v = \frac{24}{\pi} \frac{m}{s}, \lambda = \frac{16}{\pi} m$