A progressive wave has a shape (or waveform) given by the equation, `y = (2)/((x^2 - 6x + 14)^(3//2))`, at the instant time t = 1. Express the wave equation in terms of time t,

Options

A. `y = (2)/([5 + (x - 3t)^2]^(3//2))`
B. `y = (2)/([3 + (x - 3t)^2]^(3//2))`
C. `y = (2)/([3 + (x - 3t)^2]^(1//2))`
D. `(2)/([3 + (2x - 3t)^2]^(1//2))`

Answer

To express the wave equation in terms of time \( t \), we start with the given waveform equation:

\[ y = \frac{2}{(x^2 - 6x + 14)^{3/2}} \]

### Step 1: Identify the form of the wave equation
The equation represents a progressive wave, which can generally be expressed in the form:

\[ y = f(x - vt) \]

where \( v \) is the wave speed.

### Step 2: Rewrite the expression in terms of \( x - vt \)
We need to express \( x \) in terms of \( t \). To do this, we can assume a wave traveling in the positive x-direction. Let’s set:

\[ x - vt = x - 3t \]

Here, we assume \( v = 3 \) for the wave speed.

### Step 3: Substitute \( x - 3t \) into the waveform equation
Now, we will substitute \( x \) with \( (x + 3t) \):

\[ y = \frac{2}{((x + 3t)^2 - 6(x + 3t) + 14)^{3/2}} \]

### Step 4: Simplify the expression
Now, we simplify the expression inside the parentheses:

1. Expand \( (x + 3t)^2 \):
\[ (x + 3t)^2 = x^2 + 6xt + 9t^2 \]

2. Substitute into the equation:
\[ y = \frac{2}{(x^2 + 6xt + 9t^2 - 6x - 18t + 14)^{3/2}} \]

3. Combine like terms:
\[ y = \frac{2}{(x^2 - 6x + 14 + 6xt + 9t^2 - 18t)^{3/2}} \]

### Final Expression
Thus, the wave equation in terms of time \( t \) is:

\[ y = \frac{2}{(x^2 - 6x + 14 + 6xt + 9t^2 - 18t)^{3/2}} \]