A transverse wave travelling in a string produce maximum transverse velocity of `3 m//s` and maximum transverse acceleration `90 m//s^(2)` in a particle. If the velocity of wave in the string is `20 m//s`. Datermine the equation of the wave ?

Explanation

To determine the equation of the wave, we will follow these steps:

### Step 1: Identify the given values
We have the following values given in the problem:
- Maximum transverse velocity, \( V_{max} = 3 \, \text{m/s} \)
- Maximum transverse acceleration, \( a_{max} = 90 \, \text{m/s}^2 \)
- Velocity of wave in the string, \( v = 20 \, \text{m/s} \)

### Step 2: Relate maximum velocity and maximum acceleration to amplitude and angular frequency
For a particle in simple harmonic motion (SHM), the maximum velocity and maximum acceleration can be expressed as:
- Maximum velocity:
\[ V_{max} = A \omega \]
- Maximum acceleration:
\[ a_{max} = A \omega^2 \]

Where:
- \( A \) is the amplitude
- \( \omega \) is the angular frequency

### Step 3: Set up equations for \( A \) and \( \omega \)
From the maximum velocity equation:
\[ A \omega = 3 \quad \text{(1)} \]
From the maximum acceleration equation:
\[ A \omega^2 = 90 \quad \text{(2)} \]

### Step 4: Divide equation (2) by equation (1)
Dividing equation (2) by equation (1) gives:
\[ \frac{A \omega^2}{A \omega} = \frac{90}{3} \]
This simplifies to:
\[ \omega = 30 \, \text{rad/s} \]

### Step 5: Substitute \( \omega \) back to find \( A \)
Now substituting \( \omega = 30 \) into equation (1):
\[ A \cdot 30 = 3 \]
Solving for \( A \):
\[ A = \frac{3}{30} = 0.1 \, \text{m} \]

### Step 6: Calculate wave number \( k \)
The wave number \( k \) is given by:
\[ k = \frac{\omega}{v} \]
Substituting the known values:
\[ k = \frac{30}{20} = 1.5 \, \text{m}^{-1} \]

### Step 7: Write the general wave equation
The general form of the wave equation is:
\[ y(x, t) = A \sin(kx - \omega t) \]
Substituting the values of \( A \), \( k \), and \( \omega \):
\[ y(x, t) = 0.1 \sin(1.5x - 30t) \]

### Final Equation of the Wave
Thus, the equation of the wave is:
\[ y(x, t) = 0.1 \sin(1.5x - 30t) \]

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