ch35-p087 - 87. The wave that goes directly to the receiver...

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Using the binomial theorem, with D 2 large and a 2 + x 2 small, we approximate this expression: () 2 2 /2 . LDa x D ≈++ The distance traveled by the direct wave is LD a x 1 2 2 =+ b g . Using the binomial theorem, we approximate this expression: 2 1 x D ≈+− Thus, LLD aa x x D D x x D ax D 21 22 2 2 2 2 2 −≈+ ++ −− −+ = . Setting this equal to m + 1 2 b g λ , where m is zero or a positive integer, we find xm D a 1 2 2 b gb g λ . 87. The wave that goes directly to the receiver travels a distance L 1 and the reflected wave travels a distance L 2 . Since the index of refraction of water is greater than that of air this last wave suffers a phase change on reflection of half a wavelength. To obtain constructive interference at the receiver, the difference L 2 L 1 must be an odd multiple of a half wavelength. Consider the diagram below. The right triangle on the left, formed by
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