To determine the time it takes for the wave to go through half an oscillation, we need to find the time at which the cosine function reaches its halfway point, which is when the argument of the cosine function equals π/2.
In the given equation, y(x,t) = 3cos(9x - 14t + 2), we can see that the argument of the cosine function is (9x - 14t + 2).
Setting the argument equal to π/2:
9x - 14t + 2 = π/2
To find the time it takes for half an oscillation, we need to solve for t.
14t = 9x + 2 - π/2
t = (9x + 2 - π/2) / 14
Therefore, the time it takes for the wave to go through half an oscillation is given by (9x + 2 - π/2) / 14 seconds.