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standing_waves

# standing_waves - STANDING WAVES STRING INSTRUMENTS PLUS...

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Unformatted text preview: STANDING WAVES/ STRING INSTRUMENTS PLUS PHASE CHANGE DURING REFLECTION VI —> x=0 —) t i We have learned that if two strings meet at F0, then an incident wave If 2 A,Sin (xx — cut) Where 3:: v Will, on arriving at x=0, give rise to two waves Reﬂected K = A,.Sin (ICC + 0)!) and Transmitted X = AlSz'n (K’ 96— (0t) Where K, = v’ k Note: FREQUENCY DOES NOT CHANGE Further, ﬁ_ v—v’ Ai _ v+v’ ﬁ_ 212' A. _ v+v' ! Interesting situation arises if v’ —> 0 , that is, string on the right is like a ‘wall” or equivalently the end of the string on left is “ﬁxed” at x=0. In that case K = A]. Sin(l<x~ cot) A J‘ :1 _ ’ SO’ If. : AiSin(Ioc+ cot) A. I Now we have two waves on the string at the same time and to handle it, we use the principle of SUPERPOSITION. Since a wave is just a disturbance or a deviation, it is perfectly legitimate to have many simultaneous disturbances at the same point in space. The net effect is that one must algebraically add all of the disturbances D = Z , where D, = A,Sz'n(K,x 4—- (0,11) and (0,. _: v K. I So, that total wave will be Y = 17 + K : A, Sin (Kx — cot) + A]. Sin(Kx — cut) using the trigonometric identity \$17109, i (92) : Sinél] C03192 i C0561 S17162 we get y = 2A, Sin/(x Coswt ' 2 = 2A, Sin%Coswt and you see that y=0 if —/’l —3/1 x — 0,7; 1,7,etc. That is, there is NO MOTTION AT ALL AT SOME POINTS OF THE STRING. These points are called NODES. In between two nodes, that is, at ~/t —3/t 5/1 x=—,—,—,etc. 4 4 4 The string vibrates with twice the amplitude. These points are termed ANTINODES. This is how the string will look where the ' i and r waves ware both present. The case of most interest arises when the wire is ﬁxed at both ends (as in musical instruments). Because of what we learned above, there must be a node at either end and there must xi be a node every 5 as well. This requires that the wire can vibrate in only certain speciﬁc MODES such as: FIRST HARMONIC, n= 1 £_ 2_L SECOND HARMONIC, n=2 That is, the wavelengths in of the modes must obey min = L 2 01" 2L 1,. = — n n=1,2,3, etc. or in words, only those modes can occur in which there is an integer number of “half wavelengths” ﬁtting on the wire. The modes with n 2 2 are called Harmonics of the fundamental mode. That word comes from musical ethos. Next, And this Equation describes all string instruments. To be precise: 1) When you tighten a string, the note goes “up” because v increases for a given it (length). 2) The shorter the string the higher the note. 3) If you look inside a piano you will notice that the lowest notes have very thick strings. Here, a high ,u is used to reduce v and thereby lower f. 4) If you are “playing” a single string on the sitar or guitar you must move close to the lower end to get a higher note as this reduces the length of the string where you are plucking. 5) If you pull the string sideways you can get subtle variations in the frequency. This is most often used by sitar players. It works because you can vary the tension by small amounts. Such subtle variations are also accomplished by imaginative bowing of the Violin/Viola/bass/ﬁddle. APPENDIX PHASE CHANGES ON REFLECTION When v' =0 A If: A,Sin(/cx— (at) 7": X. = Irma“ wt) " Note that reﬂected wave is “born” when incident wave arrives at x=0. We can compare the phases at x: 0 ‘ X: AiSin(—a)t) =—A,.Sin cot =+ A.Sin(a)t+ 7r) KzAiSz'n wt So you see that during reﬂection at a ﬁxed end there is a phase change of 7:. If a “crest” arrives, it leaves as a “trough” and vice versa. The other extreme case If 12’» v iN 1.4L_ 2 Ai : ,Ai — . _ , 1 2 2F Srnce energy transport is 77: EA (0 7 and v’ >> v, nTis very small. That is very little energy is transmitted into the wire on the right. For wire on the left at x=0, A, = ~A, hence 1/; = — AiSinwt also if. = ~ AiSinwt So, no change of phase in this case. When a crest arrives it leaves as a crest. Summary Reﬂection at a “ﬁxed” end 9 phase change of it . ‘ Reﬂection at an “open” end -> No phase change. (We return to this in more detail later). ...
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standing_waves - STANDING WAVES STRING INSTRUMENTS PLUS...

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