The maximum transverse velocity and maximum transverse acceleration of a harmonic wave in a one - dimensional string are `1 ms^(-1) and 1 ms^(-2)` respectively. The phase velocity of the wave is `1 ms^(-1)`. The waveform is

Options

A. `sin (x - t)`
B. `sin (x - 2t)`
C. `sin (x/2 - t)`
D. `sin (x - t/2)`

Answer

To solve the problem step by step, we will derive the equation of the harmonic wave based on the given parameters: maximum transverse velocity, maximum transverse acceleration, and phase velocity.

### Step 1: Identify Given Values
We are given:
- Maximum transverse velocity, \( V_{\text{max}} = 1 \, \text{m/s} \)
- Maximum transverse acceleration, \( A_{\text{max}} = 1 \, \text{m/s}^2 \)
- Phase velocity, \( v = 1 \, \text{m/s} \)

### Step 2: Relate Maximum Velocity and Maximum Acceleration to Angular Frequency and Amplitude
The relationships for maximum velocity and maximum acceleration in a harmonic wave are given by:
\[ V_{\text{max}} = \omega A \]
\[ A_{\text{max}} = \omega^2 A \]
where:
- \( \omega \) is the angular frequency,
- \( A \) is the amplitude.

### Step 3: Calculate Angular Frequency (\( \omega \))
From the maximum velocity equation:
\[ \omega = \frac{V_{\text{max}}}{A} \]
From the maximum acceleration equation:
\[ A = \frac{A_{\text{max}}}{\omega^2} \]

### Step 4: Substitute to Find \( \omega \)
Dividing the two equations:
\[ \frac{A_{\text{max}}}{V_{\text{max}}} = \frac{\omega^2 A}{\omega A} \]
This simplifies to:
\[ \frac{A_{\text{max}}}{V_{\text{max}}} = \omega \]
Substituting the known values:
\[ \omega = \frac{1 \, \text{m/s}^2}{1 \, \text{m/s}} = 1 \, \text{rad/s} \]

### Step 5: Calculate Amplitude (\( A \))
Now substituting \( \omega \) back into the maximum velocity equation:
\[ 1 = 1 \cdot A \implies A = 1 \, \text{m} \]

### Step 6: Calculate Wave Number (\( k \))
The phase velocity \( v \) is given by:
\[ v = \frac{\omega}{k} \]
Rearranging gives:
\[ k = \frac{\omega}{v} \]
Substituting the known values:
\[ k = \frac{1 \, \text{rad/s}}{1 \, \text{m/s}} = 1 \, \text{rad/m} \]

### Step 7: Write the Wave Equation
The general form of the wave equation is:
\[ y(x, t) = A \sin(kx - \omega t + \phi) \]
Assuming the wave starts from the origin (\( \phi = 0 \)):
\[ y(x, t) = 1 \sin(1 \cdot x - 1 \cdot t) = \sin(x - t) \]

### Final Answer
The waveform is:
\[ y(x, t) = \sin(x - t) \]