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Physics IV Homework due 3/2
It is well known that the tone quality (harmonic content) of a vibrating string depends on
where and how it is plucked.
Consider a string of length
L
which is fixed at both ends
and which is plucked in the center.
Consider a pluck with corresponds to a simple
displacement with no initial velocity of the string.
The wave equation is:
2
2
2
2
2
x
f
c
t
f
∂
∂
=
∂
∂
where
y=f(x,t)
is the displacement of the string.
The initial conditions are
f(0<x<L/2,0)=2Hx/L
;
f(L/2<x<L,0)=2H(Lx)/L
and
0
)
0
,
(
=
∂
∂
t
x
f
.
1) Use a Fourier sine series to represent the wave function and find the natural
frequencies of the string.
2) What time dependence must the
n
f
have in order to satisfy both the wave equation and
the initial condition on the time derivative?
3) Perform the projection integrals on the initial displacement to determine the amplitude
of the
n
f
’s.
Hint; integrate by parts when you have the integral of x*sin(kx).
4) Write down the full solution to the wave equation
f(x,t)
which satisfies the initial
conditions.
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This note was uploaded on 03/30/2008 for the course 029 030 taught by Professor Skiff during the Spring '08 term at University of Iowa.
 Spring '08
 SKIFF

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