Question

A transverse wave on a string is described by \(y = 3\sin(36t + 0.018x + \pi/4)\), where \(x, y\) are in cm and \(t\) in seconds. The least distance between the two successive crests in the wave is ____ \text{cm}. (Nearest integer) (\(\pi = 3.14\))}

Answer 1

Correct Answer: 349

Step 1: Understanding the Question:
The distance between two successive crests of a wave is equal to its wavelength (\(\lambda\)). We can find this value from the wave number (\(k\)) given in the wave equation.
Step 2: Key Formula or Approach:
The standard equation for a wave is \(y = A\sin(\omega t + kx + \phi)\).
The relationship between wavelength and wave number is:
\[ \lambda = \frac{2\pi}{k} \]
Step 3: Detailed Explanation:
Compare the given equation \(y = 3\sin(36t + 0.018x + \pi/4)\) with the standard form:
Wave number, \(k = 0.018 \text{ rad/cm}\).
Now, calculate the wavelength \(\lambda\):
\[ \lambda = \frac{2\pi}{k} \]
\[ \lambda = \frac{2 \times 3.14}{0.018} \]
\[ \lambda = \frac{6.28}{0.018} \]
\[ \lambda = \frac{6280}{18} \]
\[ \lambda \approx 348.888 \text{ cm} \]
Rounding to the nearest integer, we get \(349 \text{ cm}\).
Step 4: Final Answer:
The least distance between two successive crests is \(349 \text{ cm}\).