A wave in a string has an amplitude of $2 ext{ cm}$. The wave travels in the +ve direction of x-axis with a

Question

A wave in a string has an amplitude of $2 	ext{ cm}$. The wave travels in the +ve direction of x-axis with a speed of $128 	ext{ ms}^{-1}$ and it is noted that $5$ complete waves fit in $4 	ext{ m}$ length of the string. The equation describing the wave is:

Accepted Answer

1. **Determine the Amplitude ($A$):** The amplitude is given as $2 	ext{ cm} = 0.02 	ext{ m}$.
2. **Calculate the Wavelength ($\lambda$):** It is given that $5$ complete waves fit in a length of $4 	ext{ m}$. Therefore, $5\lambda = 4 	ext{ m} \implies \lambda = \frac{4}{5} = 0.8 	ext{ m}$.
3. **Calculate the Wave Number ($k$):** The wave number is $k = \frac{2\pi}{\lambda}$ .
   $$k = \frac{2\pi}{0.8} = 2.5\pi \approx 2.5 	imes 3.14 = 7.85 	ext{ rad/m}$$
4. **Calculate the Angular Frequency ($\omega$):** The wave speed is $v = \frac{\omega}{k}$, so $\omega = v 
cdot k$ .
   $$\omega = 128 	imes 7.85 = 1004.8 \approx 1005 	ext{ rad/s}$$
5. **Determine the Wave Equation:** The general equation for a wave travelling in the positive x-direction is $y = A\sin(kx - \omega t)$ or $y = A\sin(\omega t - kx)$ . 
   Looking at the options, we use the form $y = A\sin(kx - \omega t)$. Substituting the values, we get:
   $$y = 0.02 \sin(7.85x - 1005t)$$

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