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# waves2 - Physics 53 Wave Motion 2 If at rst you don't...

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Physics 53 Wave Motion 2 If at first you don't succeed, try, try again. Then quit. No use being a damn fool about it. — W.C. Fields Waves in two or three dimensions Our description so far has been confined to waves in which the energy moves only along one line. For waves in a string this is good enough, but energy in sound, water and light waves generally spreads out in two or three dimensions. The main new concepts needed in more dimensions are these: The direction of a harmonic wave is specified by giving a wave vector k . Its direction is that of the energy flow; its magnitude is k = 2 π / λ , as in the one-dimensional case. The wavefunction at a position r relative to a small source then takes the form Φ ( r , t ) = A cos( k r ω t + φ ) . The energy spreads out in space over larger and larger areas, so the intensity (power per unit area) must decrease with distance from the source. In the simple case where the energy spreads equally in all directions, the intensity at distance r from the source is equal to the power emitted by the source divided by the area of a sphere of radius r : I ( r ) = P source 4 π r 2 . A wave of this type is a spherical wave . Since intensity is proportional to the square of the amplitude, the amplitude of a spherical wave must fall off with distance as 1/ r . We are often dealing with the situation where the waves are received by a relatively small detector (ear, eye, or whatever) which is at a large distance from the source. This detector samples only a very small part of the spherical wave, so the curvature of the wavefront is negligible. A good approximation in that case is to treat the waves as one dimensional, moving directly away from the source with constant amplitude. This is the “plane-wave” approximation. Decibel scale of loudness Our perception of loudness of a sound is based on the response of our ears to the intensity of the waves entering them. Sound intensities vary over a vast range, but PHY 53 1 Wave Motion 2

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fortunately our ears respond (approximately) to the logarithm of the intensity. For this reason, a logarithmic scale of intensities is commonly used for the loudness of sound. The standard scale is based on a unit called a decibel (db). An arbitrary reference intensity I 0 is chosen (one usually picks I 0 = 10 12 W/m 2 , approximately the faintest audible sound). The received intensity is then converted to the sound loudness β , measured in db according to the definition Loudness (in db) β = 10log 10 I I 0 A sound of intensity 1 W/m 2 is painful to most hearers. This is a loudness level β = 120 db. It is typical for the sound near the stage in a rock concert. The decibel name comes from “deci”, meaning one-tenth, and “bel”, a unit named after A.G. Bell, the inventor of the telephone.
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waves2 - Physics 53 Wave Motion 2 If at rst you don't...

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