A plane sound wave is travelling in a medium. In reference to a frame A, its equation is y=a cos `(omegat-kx)`. Which refrence to frame B, moving with a constant velocity v in the direction of propagation of the wave, equation of the wave will be

Options

A. `y =a cos[ omegat+kvt-kx]`
B. `y =-a cos[omegat-kvt-kx]`
C. `y=acos[omegat-kvt-kx]`
D. `y =acos[ omegat+ kv t +kx]`

Explanation

To solve the problem, we need to find the equation of a plane sound wave in a new reference frame (frame B) that is moving with a constant velocity \( v \) in the direction of wave propagation. The original wave equation in frame A is given as:

\[ y = A \cos(\omega t - kx) \]

### Step 1: Understand the relationship between the two frames
In frame A, the wave travels with a velocity \( v_{\text{wave}} = \frac{\omega}{k} \). In frame B, which is moving with velocity \( v \) in the same direction as the wave, the relative velocity of the wave with respect to frame B will be:

\[ v_{\text{relative}} = v_{\text{wave}} - v \]

### Step 2: Substitute the relative velocity into the wave equation
The equation of the wave in frame B can be expressed as:

\[ y' = A \cos(\omega t' - kx') \]

where \( t' \) and \( x' \) are the time and position in frame B. Since frame B is moving with velocity \( v \), we can relate the coordinates in frame A to those in frame B as follows:

\[ x' = x - vt \quad \text{and} \quad t' = t \]

### Step 3: Rewrite the wave equation using the new coordinates
Substituting \( x' \) into the wave equation gives:

\[ y' = A \cos(\omega t - k(x - vt)) \]

### Step 4: Simplify the equation
Expanding the equation:

\[ y' = A \cos(\omega t - kx + kvt) \]

This can be rearranged as:

\[ y' = A \cos((\omega + kv)t - kx) \]

### Final Equation
Thus, the equation of the wave in frame B becomes:

\[ y' = A \cos\left(\left(\omega - kv\right)t - kx\right) \]

### Conclusion
The new equation of the wave in frame B is:

\[ y' = A \cos\left(\omega t - kvt - kx\right) \]