A pulse is propagating on a long stretched string along its length taken as positive x-axis. Shape of the string at t = 0 is given by ` y = sqrt(a^2 - x^2)`when `|x| lt= a = 0` when `|x| gt= a ` . What is the general equation of pulse after some time 't' , if it is travelling along positive x-direction with speed V?

Options

A. `y(x,t) = sqrt(a^2 - (x + Vt)^2) " when " |x + Vt| lt= a = x + Vt " when " | x + Vt| gt= a `
B. `y(x,t) = sqrt(a^2 + (x - Vt)^2) " when " |x - Vt| lt= a = a " when " |x + Vt| gt= 0 `
C. `y(x,t) = sqrt(a^2 - (x - Vt)^2) " when " |x - Vt| lt= a = 0 " when " |x - Vt| gt= a `
D. `y(x,t) = sqrt(a^2 + (x + Vt)^2) " when " |x + Vt| lt= a = a " when " (x + Vt) gt= a `

Answer

To find the general equation of the pulse after some time \( t \), we start with the initial shape of the pulse given by:

\[ y = \sqrt{a^2 - x^2} \quad \text{for } |x| \leq a \]
\[ y = 0 \quad \text{for } |x| > a \]

This shape represents a semicircle with radius \( a \) centered at the origin.

### Step 1: Understand the propagation of the pulse
The pulse is traveling in the positive x-direction with speed \( V \). The general form of a wave traveling in the positive x-direction can be expressed as:

\[ y(x, t) = f(x - Vt) \]

where \( f \) is the shape of the wave at \( t = 0 \).

### Step 2: Substitute the initial shape into the wave equation
From the problem, we know that at \( t = 0 \):

\[ f(x) = \sqrt{a^2 - x^2} \quad \text{for } |x| \leq a \]
\[ f(x) = 0 \quad \text{for } |x| > a \]

Thus, substituting \( x - Vt \) into \( f \):

\[ y(x, t) = \sqrt{a^2 - (x - Vt)^2} \quad \text{for } |x - Vt| \leq a \]
\[ y(x, t) = 0 \quad \text{for } |x - Vt| > a \]

### Step 3: Define the conditions for the pulse
The conditions for the pulse to exist can be expressed as:

\[ |x - Vt| \leq a \]

This means:

\[ -a \leq x - Vt \leq a \]

### Step 4: Rewrite the conditions
Rearranging the inequalities gives:

\[ Vt - a \leq x \leq Vt + a \]

### Final Equation
Thus, the general equation of the pulse after some time \( t \) is:

\[ y(x, t) = \sqrt{a^2 - (x - Vt)^2} \quad \text{for } Vt - a \leq x \leq Vt + a \]
\[ y(x, t) = 0 \quad \text{for } |x - Vt| > a \]