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, we need to derive the wave equation based on the given parameters: maximum transverse velocity, maximum transverse acceleration, and phase velocity. Let's break it down step by step.

### Step 1: Understand the given parameters
We have:
- Maximum transverse velocity, \( V_{\text{max}} = 1 \, \text{ms}^{-1} \)
- Maximum transverse acceleration, \( A_{\text{max}} = 1 \, \text{ms}^{-2} \)
- Phase velocity, \( v = 1 \, \text{ms}^{-1} \)

### Step 2: Use the formulas for maximum velocity and acceleration
In simple harmonic motion (SHM), the maximum transverse velocity and maximum transverse acceleration are given by:
- \( V_{\text{max}} = A \omega \)
- \( A_{\text{max}} = A \omega^2 \)

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

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

### Step 4: Solve for \( A \) and \( \omega \)
From equation (1), we can express \( A \) in terms of \( \omega \):
\[ A = \frac{1}{\omega} \quad \text{(3)} \]

Substituting equation (3) into equation (2):
\[ 1 = \left(\frac{1}{\omega}\right) \omega^2 \]
This simplifies to:
\[ 1 = \frac{\omega^2}{\omega} \implies 1 = \omega \]
Thus, we find:
\[ \omega = 1 \, \text{rad/s} \]

### Step 5: Find the amplitude \( A \)
Substituting \( \omega = 1 \) back into equation (3):
\[ A = \frac{1}{1} = 1 \, \text{m} \]

### Step 6: Calculate the wave number \( k \)
The phase velocity \( v \) is related to \( \omega \) and \( k \) by the equation:
\[ v = \frac{\omega}{k} \]
Rearranging gives:
\[ k = \frac{\omega}{v} \]
Substituting \( \omega = 1 \) and \( v = 1 \):
\[ k = \frac{1}{1} = 1 \, \text{m}^{-1} \]

### Step 7: Write the wave equation
The general form of the wave equation is:
\[ y = A \sin(kx - \omega t) \]
Substituting \( A = 1 \), \( k = 1 \), and \( \omega = 1 \):
\[ y = 1 \sin(1x - 1t) = \sin(x - t) \]

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