Physics 53Wave Motion 2If at first you don't succeed, try, try again. Then quit. No use being a damn fool about it.! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! — W.C. FieldsWaves in two or three dimensionsOur 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 = 2π/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) = Acos(k ⋅ r −ωt +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) =Psource4πr2.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 loudnessOur 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 2fortunately 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 I0 is chosen (one usually picks I0= 10−12 W/m2, approximately the faintest audible sound). The received intensity is then converted to the sound loudness β, measured in db according to the definitionLoudness (in db) β= 10log10II0⎛⎝⎜⎞⎠⎟A sound of intensity 1 W/m2 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.The decibel scale is also used to describe amplification or attenuation of signals in electronic equipment. In those cases the relevant variable is power, not intensity.The Doppler effectConsider a source emitting harmonic waves along a straight line toward a receiver located on that line. We are interested in the effects of motion along the line of the source, receiver, or both.If the source is stationary relative to the medium, the waves it emits have equally spaced wave crests. However, if the source is moving the wave crests in front of it are crowded closer together, while those behind it are spaced farther apart. Relative to the medium, the wavelength is smaller in front of the source and larger behind it. A stationary receiver in front of the moving source will detect more wave crests per second than if the source had been stationary, so the frequency received will be higher than that of the source. A receiver placed behind the moving source will detect a lower frequency. If the source is stationary but the receiver is moving, similar things happen. A receiver moving toward the source receives more wave crests per second than a receiver at rest. The frequency received is thus higher. Conversely, if the receiver moves away from the source the frequency received is lower. These phenomena constitute the Doppler effect. It is a general property of waves.PHY 53! 2! Wave Motion 2For waves in a medium (such as sound) there is a simple formula relating the received frequency fR to the source frequency fS:Doppler effect (waves in a medium) fR= fSv − vRv − vSHere v is the wave speed, vR is the velocity of the receiver, and vS is the velocity of the source (all relative to the medium). The drawing shows the case for which both vR and vS are positive numbers. Other cases are handled by making one or both of these speeds negative.Note that as vS→ v the received frequency becomes infinite. The entire wave collapses into a single pulse, called a shock wave. The “bow wave” created by a boat moving faster than the speed of water waves is a familiar example, as is the “sonic boom” caused by an object traveling faster than the speed of sound.The formula given is for “mechanical” waves resulting from motion of particles in a medium. Light is different, in that there is no medium supporting the wave, only electric and magnetic fields. The Doppler effect still occurs, but the formula is somewhat different. If the speeds of source and receiver are small compared to the speed of light waves (c), the formula given above can be used as an approximation.It is through the Doppler effect for light from distant galaxies that we know the universe is expanding in a way consistent with the “big bang” model.Interference of harmonic wavesNow we look at the situation where two one-dimensional harmonic waves exist simultaneously in the same medium. At first we assume they travel in the same direction, and to simplify things further we will assume they have the same amplitude. They are described by the wavefunctions! y1(x,t) = Acos(k1x −ω1t)y2(x,t) = Acos(k2x −ω2t +φ)To find the total wavefunction, given by y = y1+ y2, we use a trigonometric identity: S R vS vR Medium at restPHY 53! 3! Wave Motion 2! cosα+ cosβ= 2cosα+β2⎛⎝⎜⎞⎠⎟⋅cosα−β2⎛⎝⎜⎞⎠⎟.For our case this gives! y(x,t) = 2A ⋅cos12Δk ⋅ x − Δω⋅ t −φ( )⋅ cos kavx −ωavt +12φ( )!where we have introduced the notations! Δk = k2− k1, Δω=ω2−ω1! kav=12(k1+ k2), ωav=12(ω1+ω2)This somewhat complicated formula for y(x,t) gives the general solution to our problem. We will examine some