A wave equation which gives the displacement along the y-direction is given by `y = 10^(-4) sin(60t + 2x)` where `x and y` are in meters and `t` is time in seconds. This represents a wave

Options

A. Travelling with a velocity of `30 m//s` in the negative x-direction
B. of wavelength `pi` metres
C. of frequency `30//pi` Hertz
D. of amlitude `10^(-4)m` travelling along the negative x-direction

Explanation

To solve the problem, we need to analyze the given wave equation and extract relevant information to verify the statements provided. The wave equation given is:

\[ y = 10^{-4} \sin(60t + 2x) \]

### Step 1: Identify the Amplitude
The amplitude \( A \) of the wave can be directly identified from the equation.

**Solution:**
The amplitude \( A \) is given as:
\[ A = 10^{-4} \, \text{meters} \]

### Step 2: Determine the Angular Frequency
The angular frequency \( \omega \) can be found from the coefficient of \( t \) in the sine function.

**Solution:**
From the equation, we have:
\[ \omega = 60 \, \text{radians/second} \]

### Step 3: Calculate the Frequency
The frequency \( f \) can be calculated using the relationship between angular frequency and frequency:
\[ f = \frac{\omega}{2\pi} \]

**Solution:**
Substituting the value of \( \omega \):
\[ f = \frac{60}{2\pi} = \frac{30}{\pi} \, \text{Hertz} \]

### Step 4: Determine the Wave Number
The wave number \( k \) can be found from the coefficient of \( x \) in the sine function.

**Solution:**
From the equation, we have:
\[ k = 2 \, \text{radians/meter} \]

### Step 5: Calculate the Wavelength
The wavelength \( \lambda \) is related to the wave number by the formula:
\[ \lambda = \frac{2\pi}{k} \]

**Solution:**
Substituting the value of \( k \):
\[ \lambda = \frac{2\pi}{2} = \pi \, \text{meters} \]

### Step 6: Calculate the Wave Velocity
The wave velocity \( v \) can be calculated using the relationship:
\[ v = \frac{\omega}{k} \]

**Solution:**
Substituting the values of \( \omega \) and \( k \):
\[ v = \frac{60}{2} = 30 \, \text{meters/second} \]

### Step 7: Determine the Direction of Propagation
To determine the direction of wave propagation, we analyze the signs of the coefficients of \( t \) and \( x \).

**Solution:**
Since both coefficients of \( t \) and \( x \) are positive, the wave travels in the negative x-direction.

### Summary of Results
- Amplitude \( A = 10^{-4} \, \text{meters} \)
- Frequency \( f = \frac{30}{\pi} \, \text{Hertz} \)
- Wavelength \( \lambda = \pi \, \text{meters} \)
- Wave velocity \( v = 30 \, \text{meters/second} \)
- Direction: Negative x-direction

### Conclusion
All provided statements (A, B, C, D) are true based on our calculations.

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