You are at sea on a foggy night. You are trying to find out how far you are from

Question

You are at sea on a foggy night. You are trying to find out how far you are from the shore, but the fog is too thick to see anything. After a certain time of aimless sailing, you dimly hear two separate foghorns on your port (left) side, perpendicular to your direction of travel. Each foghorn emits one short blast of sound at a pitch of 120 Hz every 2.00 s exactly, and each sounds about as loud as the other. Looking at your map, you see two foghorn locations marked plausibly near what you guess your location to be, and on the map the foghorns are 1700 m apart, flanking the entrance to a harbor. At a certain time, you notice that you hear the blasts from the foghorns simultaneously. After you have sailed at a steady heading for 22 min at a speed of 8.8 km/h (as measured by your boat's speedometer), you hear the foghorns exactly out of phase (one honks, then the other, then the first, etc.). Roughly how far are you from the foghorns now? (Hint: Treat this as a two-slit interference problem.

Explanation / Answer

given, distance between the froghorns, d = 1700 m
frequency, f = 120 Hz
distance moved parallel to the two froghorns, s = 8.8 km/h * 22 min = 2.444 m/s * 22*60 s = 3226.608 m

so, let initially the boat be at a distacne D from the both froghorns located at perpendicular bisector of the horns
then after moving distance s

path difference of the distance from the two froghorns = sqroot(d^2 + (1700/2 + 3226.608)^2) - sqroot(d^2 + (1700/2 - 3226.608)^2)
path diff, p = sqroo(d^2 + 16618732.79) - sqroot(d^2 + 5648265.546)
but at this path difference, destructive interference happens for the first time
so, p = lambda/2
but lambda*f = v = 334 m/s ( speed of sound in air)
so lambda = 334/120 = 2.7833 m
hence p = 1.391667 m = sqroo(d^2 + 16618732.79) - sqroot(d^2 + 5648265.546)
1.9367 = d^2 + 16618732.79 + d^2 + 5648265.546 - 2*( sqroo(d^2 + 16618732.79))(sqroot(d^2 + 5648265.546))
d^2 + 11133498.2 = ( sqroo(d^2 + 16618732.79))(sqroot(d^2 + 5648265.546))
d^4 + 1.23954*10^14 + 22266996.4d^2 = d^4 + 22266998.34 d^2 + 9.386701*10^13

1.23954*10^14 + 22266996.4d^2 = 22266998.34 d^2 + 9.386701*10^13
1.94d^2 = 3.008699*10^13
distance form froghorns
d = 3938.116013 km

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