class15lecturenotes

class15lecturenotes - --Class 15, Friday, February 12, 2010...

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- - Class 15, Friday, February 12, 2010 Reading: J~, ~ /~ ~ i~. ~ 2) Standing Waves in a medium open at both ends (open-open). A tube does not have to be closed in order to give rise to standing waves. A flute, or an organ pipe, are examples of tabes that are ope1!'Otl both ends, and yet can give rise to a standing wave pattern. How does this compare to the example described previously? In this case we need to enforce antiBades at both boundaries, because the ends are aIrowed to vibrate freely. This is in contrast to the closed-closed case, where the ends had to be held fixed, thus enforcing nodes. Writing down the equation that describes a standing wave: y(x,t) =2Asin(kx) cos(wt), we notice that at x=O, y(O,t)=O for all times, because the sine of zero is zero. This means that with sin(kx) we will never manage to adequately describe our new situation. If we, however, add an appropriate phase to the sine, we can reproduce an adequate description of the situation: y(x,t)= 2Asin(kx+n:/2) COS(Olt) Since sin(kx+n:l2)=cos(kx), our new expression for a standing wave in this case is y(x,t)= 2Acos(kx) cos(wt) Note that both sine and cosine are valid descriptions of a wave! You are free to choose whichever is best suited to reproduce the physical situation you are studying. We derive the normal modes below. ~UrdJ. . ,.t x=-L fivdi t .(=) /tn(~L)/ it MI~)(· (q) (k.L) = :r / Ie L:! t?; li; 2,/11"" ,/)/I *.L: nii ffsL= nil [--In'' if- e.i> ... J co> ,11= I. q Notice that this condition is identical to that derived for the closed-closed situation! The first three normal modes look as sketched below. Here you count the nodes in order to find out what order the normal mode you observe has. IS: /
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), : l.! .:: 2L I I xx n.. J , - 2L 3 :xxx 3) Standing Waves in a medium open on one end, closed on the other (open-closed) 01'-------. L A trumpet is an example in this class. This is a bit more complicated, because we need to enforce nodes on the closed end and anti nodes on the open end. A standing wave of the form y(x,t) =2Asin(kx) cos(mt) adequately meets our requirement to have a node at x=O. In order to meet the requirement
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class15lecturenotes - --Class 15, Friday, February 12, 2010...

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