Particles on a string vibrate with amplitude of 6cm speed of wave is 300 m/s and angular frequency of oscillations is 245. Find wave equation of wave is travelling along positive x-direction.

Options

A. `y = 0.06 sin(245t-49x/60)`
B. `y = 0.06 sin(245t+49x/60)`
C. `y = 0.06 sin(245t-300x)`
D. `y = 0.06 sin(245t+300x)`

Answer

To find the wave equation for a wave traveling along the positive x-direction, we can follow these steps:

### Step 1: Identify the given parameters
- Amplitude \( A = 6 \, \text{cm} = 0.06 \, \text{m} \) (converted to meters)
- Speed of the wave \( V = 300 \, \text{m/s} \)
- Angular frequency \( \omega = 245 \, \text{rad/s} \)

### Step 2: Write the standard wave equation
The standard wave equation for a wave traveling in the positive x-direction is given by:
\[ y(x, t) = A \sin(\omega t - kx) \]
where \( k \) is the wave number.

### Step 3: Calculate the wave number \( k \)
The wave number \( k \) can be calculated using the relationship between wave speed, angular frequency, and wave number:
\[ V = \frac{\omega}{k} \implies k = \frac{\omega}{V} \]
Substituting the values we have:
\[ k = \frac{245 \, \text{rad/s}}{300 \, \text{m/s}} = \frac{245}{300} \, \text{rad/m} \]

### Step 4: Substitute the values into the wave equation
Now we can substitute the values of \( A \), \( \omega \), and \( k \) into the wave equation:
\[ y(x, t) = 0.06 \sin(245t - \frac{245}{300} x) \]

### Step 5: Simplify the equation
To simplify the equation, we can express \( k \) as:
\[ k = \frac{49}{60} \, \text{rad/m} \]
Thus, the wave equation becomes:
\[ y(x, t) = 0.06 \sin(245t - \frac{49}{60} x) \]

### Final Wave Equation
The final wave equation is:
\[ y(x, t) = 0.06 \sin(245t - \frac{49}{60} x) \]