Question

A wave in a string has an amplitude of $2\, cm$. The wave travels in the +ve direction of $x$ -axis with a speed of $128\, m / s$ and it is noted that 5 complete waves fit in $4\, m$ length of the string. The equation describing the wave is

• A. y = (0.02) m sin (15.7 x -2010t)
• B. y = (0.02) m sin (15.7 x+ 2010t)
• C. y= (0.02) m sin (7.85 x- 1005t)
• D. y= (0.02) m sin (7.85 x + 1005t)

Answer 1

The Correct Option is C

 To find the equation that describes the given wave, let's analyze the information provided:

1. The amplitude of the wave is \(2\, \text{cm} = 0.02\, \text{m}\).
2. The wave speed (\(v\)) is \(128\, \text{m/s}\).
3. Five complete waves fit in a \(4\, \text{m}\) length of the string.

To write the equation of a wave, we follow the general formula:

\(y = A \sin(kx - \omega t)\)

where:

• \(A\) is the amplitude.
• \(k\) is the wave number, which is calculated as \(k = \frac{2\pi}{\lambda}\), with \(\lambda\) being the wavelength.
• \(\omega\) is the angular frequency, calculated as \(\omega = 2\pi \cdot f\).

Let's determine the wavelength (\(\lambda\)) first:

The length of the string that contains 5 complete waves (5 wavelengths) is \(4\, \text{m}\).

Thus, the wavelength \(\lambda\) is:

\(\lambda = \frac{4\, \text{m}}{5} = 0.8\, \text{m}\)

Now calculate the wave number \(k\):

\(k = \frac{2\pi}{\lambda} = \frac{2\pi}{0.8} = 7.85\, \text{m}^{-1}\)

Next, calculate the angular frequency \(\omega\) using the wave speed formula:

\(v = \frac{\omega}{k}\), solving for \(\omega\) gives:

\(\omega = vk = 128\, \text{m/s} \times 7.85\, \text{m}^{-1} = 1005\, \text{rad/s}\)

Therefore, the wave equation becomes:

\(y = 0.02 \sin(7.85x - 1005t)\)

Matching this with the given options, the correct answer is:

 

y= (0.02) m sin (7.85 x- 1005t)

 

This completes our step-by-step analysis, verifying all calculations with the wave properties provided.