A wave pulse is described by `y(x, t) = ae^-(bx - ct)^(2)`, where a,b,and c are positive constants. What is speed of this wave?

Explanation

To find the speed of the wave described by the equation \( y(x, t) = ae^{-(bx - ct)^2} \), we can follow these steps:

### Step 1: Identify the wave equation form
The given wave pulse can be expressed in a standard wave form. The general form of a wave traveling in one dimension is given by:
\[ y(x, t) = f(x - vt) \]
where \( v \) is the speed of the wave.

### Step 2: Rewrite the wave equation
From the given equation \( y(x, t) = ae^{-(bx - ct)^2} \), we can identify the argument of the exponential function:
\[ -(bx - ct)^2 \]
This can be rewritten as:
\[ -(b(x - \frac{c}{b}t))^2 \]
This shows that the wave is moving in the positive x-direction.

### Step 3: Identify the coefficients
From the expression \( bx - ct \), we can identify:
- The coefficient of \( x \) is \( b \)
- The coefficient of \( t \) is \( c \)

### Step 4: Calculate the speed of the wave
The speed \( v \) of the wave can be calculated using the relationship:
\[ v = \frac{\text{Coefficient of } t}{\text{Coefficient of } x} \]
Thus, substituting the coefficients we identified:
\[ v = \frac{c}{b} \]

### Final Answer
The speed of the wave is:
\[ v = \frac{c}{b} \]

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