A transverse harmonic wave on a string is described by y(x, t) = 3sin ( 36t + 0.018x + ~) where x and Y are in cm and t is in s. Which of the following statements is incorrect? (

Options

A. The waves is travelling in negative x-direction
B. The amplitude of the wave is 3 cm
C. The speed of the wave is `20ms^(-1)`
D. The frequency of the wave is`9/pi`Hz.

Explanation

To solve the question regarding the transverse harmonic wave described by the equation \( y(x, t) = 3 \sin(36t + 0.018x + \pi/4) \), we need to analyze the given statements about the wave and determine which one is incorrect.

### Step-by-Step Solution:

1. **Identify the Wave Equation Parameters**:
The wave equation is given in the form:
\[ y(x, t) = A \sin(\omega t + kx + \phi) \]
where:
- \( A = 3 \) cm (amplitude)
- \( \omega = 36 \) rad/s (angular frequency)
- \( k = 0.018 \) rad/cm (wave number)
- \( \phi = \frac{\pi}{4} \) (phase angle)

2. **Determine the Direction of Propagation**:
The general form of the wave indicates that if the equation is \( y = A \sin(\omega t + kx + \phi) \), the wave travels in the negative x-direction. Since we have a positive sign between \( \omega t \) and \( kx \), the wave travels in the negative x-direction.

3. **Calculate the Amplitude**:
The amplitude of the wave is given directly from the equation as:
\[ A = 3 \text{ cm} \]
This statement is correct.

4. **Calculate the Speed of the Wave**:
The speed \( v \) of the wave can be calculated using the formula:
\[ v = \frac{\omega}{k} \]
Substituting the values:
\[ v = \frac{36 \text{ rad/s}}{0.018 \text{ rad/cm}} = 2000 \text{ cm/s} = 20 \text{ m/s} \]
This statement is also correct.

5. **Calculate the Frequency of the Wave**:
The frequency \( f \) is related to the angular frequency \( \omega \) by the formula:
\[ \omega = 2\pi f \]
Rearranging gives:
\[ f = \frac{\omega}{2\pi} = \frac{36}{2\pi} = \frac{18}{\pi} \text{ Hz} \]
The statement claims the frequency is \( \frac{9}{\pi} \), which is incorrect.

### Conclusion:
The incorrect statement is that the frequency of the wave is \( \frac{9}{\pi} \) Hz. The correct frequency is \( \frac{18}{\pi} \) Hz.