A harmonically moving transverse wave on a string has a maximum particle velocity and acceleration of 3 m/s and `90 m//s^(2)` respectively. Velocity of the wave is `20 m//s`. Find the waveform.

Explanation

To find the waveform of a harmonically moving transverse wave on a string, we will follow these steps:

### Step 1: Understand the given parameters
We are given:
- Maximum particle velocity, \( V_{\text{max}} = 3 \, \text{m/s} \)
- Maximum particle acceleration, \( A_{\text{max}} = 90 \, \text{m/s}^2 \)
- Velocity of the wave, \( v = 20 \, \text{m/s} \)

### Step 2: Relate maximum particle velocity and acceleration to amplitude and angular frequency
The maximum particle velocity is given by the formula:
\[ V_{\text{max}} = A \omega \]
where \( A \) is the amplitude and \( \omega \) is the angular frequency.

The maximum particle acceleration is given by:
\[ A_{\text{max}} = A \omega^2 \]

### Step 3: Set up the equations
From the first equation, we have:
\[ A \omega = 3 \quad \text{(1)} \]
From the second equation, we have:
\[ A \omega^2 = 90 \quad \text{(2)} \]

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

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

### Step 6: Use wave velocity to find wave number \( k \)
The wave velocity is related to \( \omega \) and \( k \) by:
\[ v = \frac{\omega}{k} \]
Rearranging gives:
\[ k = \frac{\omega}{v} \]
Substituting the known values:
\[ k = \frac{30}{20} = 1.5 \, \text{m}^{-1} \]

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

### Final Waveform Equation
Thus, the equation of the waveform is:
\[ y(x, t) = 0.1 \sin(1.5x + 30t + \phi) \]