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 topic is Wave Motion.
The question asks for the "least distance between two successive crests," which is the definition of the wavelength (\(\lambda\)) of the wave. We can extract this from the given wave equation.
Step 2: Key Formula or Approach:
The standard form of a traveling wave equation is \(y(x, t) = A\sin(\omega t + kx + \phi)\).
The wavelength \(\lambda\) is related to the wave number \(k\) by the formula:
\[ \lambda = \frac{2\pi}{k} \]
Step 3: Detailed Explanation:
We compare the given wave equation, \(y = 3\sin(36t + 0.018x + \pi/4)\), to the standard form.
By comparison, we can identify the wave number \(k\) as the coefficient of \(x\):
\(k = 0.018 \text{ rad/cm}\).
Now, we use the formula to calculate the wavelength \(\lambda\):
\[ \lambda = \frac{2\pi}{k} = \frac{2 \times 3.14}{0.018} \]
\[ \lambda = \frac{6.28}{0.018} = \frac{6280}{18} \approx 348.88 \text{ cm} \]
The question asks for the answer to be rounded to the nearest integer.
Rounding \(348.88 \text{ cm}\) gives \(349 \text{ cm}\).
Step 4: Final Answer:
The least distance between two successive crests is \(349 \text{ cm}\).