Suppose a tuning fork generates sound waves with a frequency of 100 Hz. The waves travel in opposite directions along a hallway, are reflected by end walls, and return. The hallway is 47.0 m long and the tuning fork is located 14.0 m from one end. What is the phase difference between the two reflected wave at the point of the tuning fork.
For this question, I was able to solve using the path length difference. However I tried to solve it using wave equation,
i.e. $y_1 = A \cos(kx - wt),\ y_2 = A \cos(kx + wt)$
The two waves travel in opposite directions, so $w$ has different signs. I was planning to compute the $kx-w$t term for $y_1$ and $y_2$, then the phase difference will be the difference of those two phases, but I was unable to get the right answer. I think the reflection shift the phases of $y_1$ and $y_2$ by $\pi$, so the effect of reflection can be omitted.
Can anyone please explain what $x$ and $t$ should be when the two reflected waves meet at the point of tuning fork?
Thank you very much for the help.