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The solution to a wave equation in one dimension using various methods, including the use of trigonometric identities and the d'alembert solution. The document also discusses the behavior of wave phenomena and the importance of boundary conditions in numerical solutions.
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Larry Caretto Mechanical Engineering 501B Seminar in EngineeringSeminar in Engineering AnalysisAnalysis March 2, 2009
2
Overview
3
Midterm Exam
Review Wave Equation
2 2
2 2
2 2
=−λ ∂
x
X x t X x
Tt c Tt
Result is function of t equal to function of x [ sin( ) cos( )][ sin( ) cos( )]
A ct B ct C x D x
uxt TtX x
5
Review General Solution
∑
∞
=
1
( ,) sin cos sin n
n n L
nx L
nct B L
nct uxt A π π π
∫ ⎟⎠
⎞ ⎜ ⎝
⎛ π π
=
L m (^) L dx gx mx m
A 0
⎠
⎞ ⎜ ⎝
=^ L ⎛ m (^) L dx f x mx L
B 0
(^2) ()sin π
x Lt
x
u c t
u
0 , 0
2
2 2 2
2
≤ ≤ ≥
∂
∂
∂
6
Review Use of Trig Identities
∑
∞
= ⎪⎩
⎪ ⎨
⎧ ⎥ ⎦
⎤ ⎢ ⎣
⎡ ⎟ ⎠
⎜ ⎞ ⎝
⎟− ⎛^ π + ⎠
⎜ ⎞ ⎝
= ⎛^ π − 1
( ) cos ( ) cos 2
1 (,) n
n (^) L
n x ct L
uxt A n x ct
∑
∞
=
1
( ,) sin cos sin n
n n L
nx L
nct B L
nct uxt A π π π
⎪⎭
⎪ ⎬
⎫ ⎥ ⎦
⎤ ⎢ ⎣
⎡ ⎟ ⎠
⎞ ⎜ ⎝
⎞ ⎜ ⎝
n x ct L
B n x ct n sin ( ) sin ( )
sin y cos y sin z
7
Review D’Alembert Solution
2
2 2 2
2
x
u c t
u ∂
∂
∂
−
= + + − +
x ct
xct
g d c
uxt f x ct f x ct (ν ) ν 2
( ) ( )^1 2
(,)^1
8
Review D’Alambert II
9
Review Wave Propagation
-5 -4 -3 -2 -1 0 1 2 3 4 5 distance, x
time, t
t = 0 ct=1ct = 2 ct = 3 x+ct x+ct x-ctx-ct
10
Review Boundaries
∑
∞
=
1
sin
n
n (^) L
n x B
ux f x
-
-0.
0
0.
1
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x
f(x) Actual solution Periodic extensions
⎥ ⎦
⎤ ⎢ ⎣
⎡ ⎟ ⎠
⎜ ⎞ ⎝
⎟− ⎛ ⎠
⎜ ⎞ ⎝
⎟− ⎛ ⎠
⎜ ⎞ ⎝
= ⎛ 5
3 sin 5
2 sin 2 2 sin 20 2 2
π π π π
m m m m Bm
11
Review Time Evolution
f x ct f x ct u xt
-
-0.
0
0.
1
-2 -1.5 -1 -0.5 (^0) x 0.5 1 1.5 2
f(x)
Initial Condition
f(x+ct)/2 = f(x+0.4)/2 f(x-ct)/2 = f(x-0.4)/
-0.
0
0.
1
0 0.2 0.4 0.6 0.8 1 x
f(x+ct)
ct = 0.
-0.
0
0.
1
0 0.2 0.4 0.6 0.8 1 x
f(x-ct) ct = 0.
-0.
0
0.
1
0 0.2 0.4 0.6 0.8 1 x
u(x,t) ct = 0.
Phase behavior of sine function causes initial wave form to be reflected at bound- aries
19
Behavior of Equation Types
Right-running charac- teristic (dy/dx > 0 )
Left-running charac- teristic (dy/dx < 0 )
x 1 , y 1
Initial Condition Curve
Domain of dependence
20
Hyperbolic Equations
21
Elliptic PDEs
Parabolic PDEs
23
Example Question
24
Work on Homework Problem
⎧ π− π ≤ ≤π
∂
∂ = = =
= = ∂
∂
∂ = x x
x x t
u u t uLt ux
c x
u c t
u t^0.^012
0 1 0
2
2 2 2
2
[ sin( ) cos( )][ sin( ) cos( )]
A ct B ct C x D x
uxt TtX x λ + λ λ + λ
25
Work on Homework Problem
⎧ π− π ≤ ≤π
∂
∂ = = =
= = ∂
= ∂ ∂
∂ = x x
x x t
u
u t uLt ux
c x
c u t
u t^0.^012
0 1 0
2
2 2 2
2
[ sin( ) cos( )][ sin( ) cos( )]
A ct B ct C x D x
uxt TtX x λ + λ λ + λ
26
Work on Homework Problem
u ( x , t )= [ An sin( λ n ct )+ Bn cos( λ nct )] sin( λ nx )
⎞ ⎜ ⎝
=^ L ⎛ m (^) L dx gx mx mc
A 0
(^2) ()sin π
⎟ ⎠
⎞ ⎜ ⎝
=^ L ⎛ m (^) L dx f x mx L
B 0
(^2) ()sin π
27
Work on Homework Problem
∑
∞
=
1
( ,) sin( )sin( ) n
⎞ ⎜ ⎝
=^ L ⎛ m (^) L dx
mx gx mc
A 0
()sin 2 π π
⎠
⎞ ⎜ ⎝
⎞ ⎜ ⎝
= ⎛ π π
π π π π
π π 2
2
0
(^2 0). 01 sin (^20). 01 sin dx L
x mx m c
dx L
x mx mc
Am
Multidimensional Equations
29
Multidimensional Laplace
0
2 ∇ u =
φ
φ φ θ φ
θ
∂
∂
∂
∂
∂
∂ ⎟+ ⎠
⎜ ⎞ ⎝
⎛ ∂
∂ ∂
∂ ∇ =
∂
∂
∂
∂ ⎟+ ⎠
⎜ ⎞ ⎝
⎛ ∂
∂ ∂
∂ ∇ =
∂
+∂ ∂
+∂ ∂
∇ =∂
u r
u r
u r r
u r r r Sphere u
z
u u r r
u r r r Cylindrical u
z
u y
u x
Cartesian u u
2 2
2 2 2
2 2 2 2 2 2
2
2 2
2 2 2
2
2 2
2 2
2 2
1 cot sin
1 1
1 1
30
Multidimensional Diffusion
u t
u (^) 2 = ∇ ∂
∂
φ
φ φ θ φ
θ
∂
∂
∂
∂
∂
∂ ⎟+ ⎠
⎜ ⎞ ⎝
⎛ ∂
∂ ∂
∂ ∇ =
∂
∂
∂
∂ ⎟+ ⎠
⎜ ⎞ ⎝
⎛ ∂
∂ ∂
∂ ∇ =
∂
+∂ ∂
+∂ ∂
∇ =∂
u r
u r
u r r
u r r r Sphere u
z
u u r r
u r r r Cylindrical u
z
u y
u x
Cartesian u u
2 2
2 2 2
2 2 2 2 2 2
2
2 2
2 2 2
2
2 2
2 2
2 2
1 cot sin
1 1
1 1
37
Initial Condition
⎟ ⎠
⎞ ⎜ ⎝
⎛ ⎟ ⎠
⎞ ⎜ ⎝
= ⎛
⎥⎥⎦
⎤ ⎢⎢⎣
⎡ (^) ⎟ ⎠ ⎜ ⎞ ⎝ ⎟+⎛ ⎠ ⎜ ⎞ ⎝ −⎛ H
my L
u xyt C e nx n m
nL mH t nm
α π π π π ( , ,) sin sin 1 1
2 2
⎟ ⎠
⎞ ⎜ ⎝
⎛ ⎟ ⎠
⎞ ⎜ ⎝
= = ⎛
∞ = H
my L
u xy f xy C nx n m nm ( , , 0 ) (, ) sin π^ sin^ π 1 1
Initial Condition Gives Cmn
22
sin sin
sin sin sin sin
(, )sin sin
0
2 0
2
0 0 1 1
0 0
dx C H^ L L
dy px H
C qy
dxdy H
qy L
px H
my L
nx C
dxdy H
qy L
px f xy
pq
H L pq
H L n m nm
H L
⎟^ = ⎠
⎞ ⎜ ⎝
⎛ ⎥ ⎦
⎤ ⎢ ⎣
⎡ ⎟ ⎠
⎞ ⎜ ⎝
= ⎛
⎟ ⎠
⎜ ⎞ ⎝
⎟ ⎛ ⎠
⎜ ⎞ ⎝
⎟ ⎛ ⎠
⎜ ⎞ ⎝
⎟ ⎛ ⎠
⎜ ⎞ ⎝
⎛
⎟ = ⎠
⎜ ⎞ ⎝
⎟ ⎛ ⎠
⎜ ⎞ ⎝
⎛
π π
π π π π
π π
∫∫ ⎟⎠ ⎜ ⎞ ⎝ ⎟ ⎛^ π ⎠ ⎜ ⎞ ⎝ = ⎛^ π
H L pq (^) H dxdy qy L fxy px C HL 0 0
(^4) (,)sin sin