A transverse harmonic wave on a string is described by y(x, t) = 3sin ( 36t + 0.018x + π/4) 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 problem, we need to analyze the given wave equation and determine which of the provided statements is incorrect. The wave is described by the equation:

\[ y(x, t) = 3 \sin(36t + 0.018x + \frac{\pi}{4}) \]

### Step 1: Identify the wave parameters
From the wave equation, we can identify the following parameters:
- Amplitude \( A = 3 \) cm
- Angular frequency \( \omega = 36 \) rad/s
- Wave number \( k = 0.018 \) rad/cm
- Phase constant \( \phi = \frac{\pi}{4} \)

### Step 2: Determine the direction of wave propagation
The general form of a wave traveling in the negative x-direction is given by:

\[ y(x, t) = A \sin(\omega t + kx + \phi) \]

Since the given wave equation has the term \( +0.018x \), it indicates that the wave is traveling in the negative x-direction.

### Step 3: 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 of \( \omega \) and \( k \):

\[ v = \frac{36 \, \text{rad/s}}{0.018 \, \text{rad/cm}} \]

To convert the speed from cm/s to m/s, we can use the conversion factor \( 100 \, \text{cm} = 1 \, \text{m} \):

\[ v = \frac{36}{0.018} \, \text{cm/s} = 2000 \, \text{cm/s} = 20 \, \text{m/s} \]

### Step 4: Calculate the frequency of the wave
The frequency \( f \) can be calculated using the relationship between angular frequency and frequency:

\[ \omega = 2\pi f \]

Rearranging gives:

\[ f = \frac{\omega}{2\pi} \]

Substituting \( \omega = 36 \):

\[ f = \frac{36}{2\pi} = \frac{18}{\pi} \, \text{Hz} \]

### Step 5: Analyze the statements
Now, we can analyze the provided statements:

1. The wave travels in the negative x-direction. **(True)**
2. The amplitude of the wave is 3 cm. **(True)**
3. The speed of the wave is 20 m/s. **(True)**
4. The frequency of the wave is \( \frac{9}{\pi} \) Hz. **(False)**

From our calculations, we found that the frequency is \( \frac{18}{\pi} \) Hz, not \( \frac{9}{\pi} \) Hz.

### Conclusion
The incorrect statement is option 4, which claims that the frequency of the wave is \( \frac{9}{\pi} \) Hz.

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