A transverse harmonic wave on a string is descrbed by `y(x, t) = 3.0 sin (36 t + 0.018 x + pi/4)` where x and y are in cm and t is in s. The position direction of x is from left to right.

Options

A. The wave is travelling from right to left
B. The speed of the wave is 20 m/s
C. Frequency of the wave is 5.7 Hz.
D. The least distance between two successive creasts in the wave is 2.5 cm.

Explanation

To solve the problem, we need to analyze the given wave equation and extract relevant information such as amplitude, angular frequency, wave number, wavelength, speed, and frequency of the wave.

The wave equation is given as:
\[ y(x, t) = 3.0 \sin(36t + 0.018x + \frac{\pi}{4}) \]

### Step 1: Identify the parameters from the wave equation
The general form of a transverse wave is:
\[ y(x, t) = A \sin(\omega t + kx + \phi) \]
where:
- \( A \) = amplitude
- \( \omega \) = angular frequency
- \( k \) = wave number
- \( \phi \) = phase constant

From the given equation, we can identify:
- Amplitude \( A = 3.0 \) cm
- Angular frequency \( \omega = 36 \) rad/s
- Wave number \( k = 0.018 \) rad/cm
- Phase constant \( \phi = \frac{\pi}{4} \)

### Step 2: Calculate the time period \( T \)
The angular frequency \( \omega \) is related to the time period \( T \) by the formula:
\[ \omega = \frac{2\pi}{T} \]
Rearranging gives:
\[ T = \frac{2\pi}{\omega} \]
Substituting \( \omega = 36 \):
\[ T = \frac{2\pi}{36} = \frac{\pi}{18} \text{ seconds} \]

### Step 3: Calculate the wavelength \( \lambda \)
The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]
Rearranging gives:
\[ \lambda = \frac{2\pi}{k} \]
Substituting \( k = 0.018 \):
\[ \lambda = \frac{2\pi}{0.018} \approx 349.1 \text{ cm} \]

### Step 4: Calculate the wave speed \( v \)
The wave speed \( v \) can be calculated using the relationship:
\[ v = \omega \cdot \frac{1}{k} \]
Substituting the values:
\[ v = \frac{36}{0.018} = 2000 \text{ cm/s} = 20 \text{ m/s} \]

### Step 5: Calculate the frequency \( f \)
The frequency \( f \) is the reciprocal of the time period \( T \):
\[ f = \frac{1}{T} = \frac{1}{\frac{\pi}{18}} = \frac{18}{\pi} \approx 5.73 \text{ Hz} \]

### Summary of Results
- Amplitude \( A = 3.0 \) cm
- Time Period \( T \approx 0.174 \) seconds
- Wavelength \( \lambda \approx 349.1 \) cm
- Wave Speed \( v = 20 \) m/s
- Frequency \( f \approx 5.73 \) Hz