If at `t = 0`, a travelling wave pulse in a string is described by the function,
`y = (10)/((x^(2) + 2 ))`
Hence, `x and y` are in meter and `t` in second. What will be the wave function representing the pulse at time `t`, if the pulse is propagating along positive x-axix with speed `2 m//s`?

Explanation

To solve the problem step by step, we need to find the wave function representing the pulse at time `t`, given the initial wave function and the speed of propagation. Let's break it down:

### Step 1: Understand the Initial Wave Function
The initial wave function at `t = 0` is given by:
\[ y = \frac{10}{x^2 + 2} \]
This describes the shape of the wave pulse at the initial time.

### Step 2: Identify the Wave Speed
The problem states that the wave pulse is propagating along the positive x-axis with a speed of `2 m/s`.

### Step 3: Determine the Wave Number (k) and Angular Frequency (ω)
The general form of a traveling wave can be expressed as:
\[ y(x, t) = f(x - vt) \]
where `v` is the speed of the wave.

Given that:
- \( v = 2 \, \text{m/s} \)

We can relate the wave speed to the wave number `k` and angular frequency `ω` using the formula:
\[ v = \frac{\omega}{k} \]

### Step 4: Find the Wave Number (k)
From the initial wave function, we can identify that the coefficient of \( x^2 \) in the denominator gives us the wave number `k`. The initial function can be rewritten as:
\[ y = \frac{10}{x^2 + 2} \]
At \( t = 0 \), the coefficient of \( x^2 \) is `1`, which means:
\[ k^2 = 1 \]
Thus:
\[ k = 1 \, \text{(since k cannot be negative)} \]

### Step 5: Find the Angular Frequency (ω)
Using the wave speed:
\[ v = 2 = \frac{\omega}{k} \]
Substituting \( k = 1 \):
\[ 2 = \frac{\omega}{1} \]
Thus:
\[ \omega = 2 \, \text{rad/s} \]

### Step 6: Write the General Wave Function
The general form of the wave function is:
\[ y(x, t) = f(x - vt) \]
Substituting \( v = 2 \):
\[ y(x, t) = f(x - 2t) \]

### Step 7: Substitute the Initial Function
To find the wave function at time `t`, we replace \( x \) in the initial function with \( (x - 2t) \):
\[ y(x, t) = \frac{10}{(x - 2t)^2 + 2} \]

### Final Answer
Thus, the wave function representing the pulse at time `t` is:
\[ y(x, t) = \frac{10}{(x - 2t)^2 + 2} \]

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