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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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 Fall '11
 Bhagat
 Physics, Musical instrument, Fundamental physics concepts, Wire

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