A travelling wave pulse is given by
`y=(4)/(3x^(2)+48t^(2)+24xt+2)`
where x and y are in metre and t is in second. The velocity of wave is :-

Options

A. 4m/s
B. 2 m/s
C. 8 m/s
D. 12 m/s

Answer

To find the velocity of the traveling wave pulse given by the equation:

\[ y = \frac{4}{3x^2 + 48t^2 + 24xt + 2} \]

we will follow these steps:

### Step 1: Rewrite the denominator
We start by rewriting the denominator to identify the form of the wave equation. The denominator is:

\[ 3x^2 + 48t^2 + 24xt + 2 \]

### Step 2: Factor out the common term
We can factor out a 3 from the first three terms in the denominator:

\[ 3(x^2 + 8xt + 16t^2) + 2 \]

### Step 3: Complete the square
Next, we need to complete the square for the expression \(x^2 + 8xt + 16t^2\). This can be rewritten as:

\[ (x + 4t)^2 \]

Thus, we have:

\[ 3((x + 4t)^2) + 2 \]

### Step 4: Substitute back into the equation
Now substituting this back into the original equation gives us:

\[ y = \frac{4}{3((x + 4t)^2) + 2} \]

### Step 5: Identify the wave parameters
From the rewritten equation, we can see that it resembles the general form of a wave equation. The term \(x + 4t\) indicates that the wave is traveling in the positive x-direction.

### Step 6: Determine the wave velocity
The velocity \(v\) of the wave can be determined from the coefficient of \(t\) in the term \(x + 4t\). The coefficient of \(t\) is 4, and the coefficient of \(x\) is 1. The velocity \(v\) of the wave is given by the formula:

\[ v = \frac{\text{coefficient of } t}{\text{coefficient of } x} \]

Substituting the values, we get:

\[ v = \frac{4}{1} = 4 \text{ m/s} \]

### Final Answer
Thus, the velocity of the wave is:

\[ \boxed{4 \text{ m/s}} \]

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