Question

A plane sound wave is travelling in a medium. In reference to a frame A, its equation is: $y=acos[\omega t-kx]$.
With reference to a frame B, moving with a constant velocity v in the direction of propagation of the wave, equation of the wave cannot be

• A. $y=acos\left[\right(\omega +kv)t-kx]$
• B. $y=-acos\left[\right(\omega -kv)t-kx]$
• C. $y=acos\left[\right(\omega -kv)t-kx]$
• D. $y=acos[\omega +kv)t+kx]$

Answer 1

Answer: (A), (B), (D)

$y=acos[\omega +kv)t+kx]$

Suppose at an instant t, x-coordinate of a point in reference to a moving frame is $x_0.$ Since, at this moment origin of moving frame is at distance vt from origin of the fixed reference frame, therefore, the actual x-coordinate of the point will be equal to $(vt+x_0)$.
Substituting this value of x in the given equation we get
$y=acos[\omega t-k(vt+x_o\left)\right]$
or $y=acos\left[\right(\omega -kv)t-k{x}_o]$
Hence, option (c) is correct while all other options (a), (b), (d) are not possible.

Why this question? Caution: The magnitude of relative velocity is the subtraction of the magnitudes of two velocities if the bodies/energies/frames move in the same direction.