While walking along the shore at your beachfront home, you notice that there are two narrow gaps in the breakwater, the wall that protects the shore from the waves. These gaps are 9.0 m apart and the breakwater is 12.0 m from the shore and parallel to it. You go to the shore directly opposite the midpoint between these gaps. As you walk along the shore, the first point where no waves reach you is 1.7 m from your starting point. Out beyond the breakwater you observe that there are ten wave crests in 18 s. To me, this seems to be a straightforward application of the formula r2-r1 = (m+1/2)lambda. I found the path difference (r2-r1) by simple geometry (1.19m), then I set m=0 since 1.7m is the first point of destructive interference. The answer I have is lambda = 2.38m, and it is wrong for some reason. I also do not understand the significance of the given frequency, and I have not used it in my calculations. I would love to know what I'm missing here. Thanks!
Something here does not compute. You calculated lambda and said you got the wrong answer. Lambda is the distance between wave crests, and your result looked OK to me. The only reason to give the frequency would be to then ask you to find velocity. I am assuming the waves are arriving with crests parallel to the breakwater and that the midpoint between the gaps is a wave maximum, but when you said you had the wrong lambda it made me wonder.
Conclusion: I am not missing anything. I found out today that the online answer is incorrect, and my answer is correct
That will do it. I was imagining various possibilities, like the slowing if the waves as they approach the shore reducing the wavelength, but clearly the problem did not give enough information to treat that complication, and the fact that they were asking for the wavelength dismissed that possibility.