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Problems for the 1D Wave Equation
Matthew J. Hancock
Fall 2005
1
Problem 1
(i) Suppose that an &in±nite string²has an initial displacement
u
(
x;
0) =
f
(
x
) =
8
>
<
>
:
x
+ 1
;
&
1
±
x
±
0
1
&
2
x;
0
±
x
±
1
=
2
0
;
x <
&
1
and
x >
1
=
2
and zero initial velocity
u
t
(
x;
0) = 0
. Write down the solution of the wave equation
u
tt
=
u
xx
with ICs
u
(
x;
0) =
f
(
x
)
and
u
t
(
x;
0) = 0
using D³Alembert³s formula. Illustrate the nature
of the solution by sketching the
ux
pro±les
y
=
u
(
x; t
)
of the string displacement for
t
=
0
;
1
=
2
;
1
;
3
=
2
.
(ii) Repeat the procedure in (i) for a string that has zero initial displacement but is given
an initial velocity
u
t
(
x;
0) =
g
(
x
) =
8
>
<
>
:
&
1
;
&
1
±
x <
0
1
;
0
±
x
±
1
0
;
x <
&
1
and
x >
1
2
Problem 2
(i) For an in±nite string (i.e. we don³t worry about boundary conditions), what initial
conditions would give rise to a purely forward wave? Express your answer in terms of the
1
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Question 1
[20 points total]
Suppose you shake the end of a rope of dimensionless length 1 at a certain frequency
!
.
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This note was uploaded on 01/06/2011 for the course MATH 3333 taught by Professor Staff during the Spring '08 term at University of Houston.
 Spring '08
 Staff
 Equations

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