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2008-11-07

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2008-11-07

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Lecture Notes November 7th Waves Traveling Wave y(x,t)=y(x,0)=Asin(2?/?)x=Asinkx where k=2?/? aka wave number Dashed curve shows wave after time t elapsed On dashed curve, point 1 at x used to be at point 2 at (x±vt) Y(x,t)=Y(x-vt,0) Y(x,t)=Asink(x-vt) Any function of x-vt is a traveling wave with velocity v. Asink(x-vt)=Asin(kx-kvt) remember-(kv=?) => y(x,t)=Asin(kx-?t) V=?/k kv=(2?/?)v remember: ?f=v => v/?=f => kv=2?f=? Reflection of a Transmission When light hits pane of glass, some is reflected, some passes through Interference When two waves interact, simply add their amplitudes to find the resulting wave Constructive Interference Adding the two waves above yields the wave to the right. Destructive Interference Adding these two waves yields a straight line (every point cancels out) Example Constructive interference if L=n? Destructive interference if L=(n+1/2)? Standing Waves L=?/2 L=2?/2 => L=n?/2 Plucked Violin String Fundamental aka 1st Harmonic: 2nd Harmonic aka 1st Overtone: L=n(?n/2) ?n=2L/n=?1/n Recall: v=?(FT/(m/l)) Recall ?f=v?f=v/? fn=v/?=nv/?=nf1 fn=f1*n f1=v/?1 Y1=Asin(kx-?t) Y2=Asin(kx+?t) Y1+Y2=Asin kx cos ?t Recall: sin ?1 + sin ?2 = 2sin(?1+?2/2)Cos(?1-?2/2) Y1+Y2=B(x) cos ?t which is SHM B(x) equals 2Asinkx This is SHM with a variable amplitude given by B(x)