the wave equation
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THE WAVE EQUATION. Alemayehu Adugna Arara Supervisor : Dr. J.H.M. ten Thije Boonkkamp November 04 , 2009. Outline. Occurence of the Wave Equation 1D Waves Spherical Waves Cylinderical Waves Supersonic Flow Past a body Revolution Initial value Problems in Two and Three Dimensions. - PowerPoint PPT PresentationTRANSCRIPT
THE WAVE EQUATION
Alemayehu Adugna Arara
Supervisor : Dr. J.H.M. ten Thije Boonkkamp
November 04 , 2009
Outline• Occurence of the Wave Equation• 1D Waves• Spherical Waves• Cylinderical Waves• Supersonic Flow Past a body Revolution• Initial value Problems in Two and Three
Dimensions
1. Occurrence of the Wave Equation
• Acoustics• Electromagnetism• Elasticity
22 2
2c
t
• Hyperpolic wave equation.
Acoustics• Linearized • Small disturbance about an equilibrium
state.• Body forces are neglected
, , 0.0 0
p p u • The initial disturbance has a uniform
entropy .p p
2 20 0 0 0, 'p p a a p
• From conservation of mass we have
Acoustics
0jj
ut x
• From balance momentum we also have
,i ji
ij i
u uu pF
t x x
• Then from these equation we have
0 00 , 0j i
j i
u u p
t x t x
Acoustics
,u •Introducing
• Hence2 20 ,tt a
0 0
00 2
0
,
,
,
ii
ux
p pt
a t
we have,
Linearized Supersonic Flow
• Disturbance is to remain small, – the motion should have to be very small or– the body must be very slender.
• Slender body moving with arbitrary constant velocity relates acoustics with aerodynamics.
1 2 3, ,x x Ut y x z x
• The velocity components in frame are
1 2 3( , , ), ii
U u u u ux
Linearized Supersonic Flow 1 2 3 1 2 3, , , , , , ,x x x t x y z x Ut x x
• Hence the equation wave for acoustics become
2
0
( 1) ,xx zz zz
UM M
a
• The velocity
0 0 .xp p U
( , , )x y zU
Electromagnetic Waves• Maxwell‘s equation
• And E satisfies the same equation.
10, ,
. 0, . 0,
B EE B
t t
B E
• Where B is the magnetic induction and E is the electric field. Therefore
22
2
1 1( ) ( ),
BB B
t
2. 1D Wave Equation2
tt xxc 2
0, ,x ct x ct
( ) ( ) ( ) ( ),f g f x ct g x ct
• The functions f and g follows from initial and boundary conditions
• For the intial value problem,
0 1( ), ( ), 0,tx x t x
0 0 1
1 1( ) ( ) ( )
2 2
x ct
x ct
x ct x ct dc
3. Spherical Waves• For waves symmetric about the origin
2 2
2 2 2
1 2, ( , )R t
c t R R R
• This can be written as2 2
2 2 2
1 ( ) ( ),
R R
c t R
( ) ( )f R ct g R ct
R R
• The general solution will be
3. Spherical Waves• For outgoing waves, the solution is
( )f R ct
R
• Standard form is to prescribe source is
• In acoustics, ∂φ/∂R is radial velocity and Q(t) is the flux of volume.
2
0( ) lim 4
RQ t R
R
1 ( / )
( ) 4 ( )4
Q t R cQ t f ct and
R
3. Spherical Waves• IVP, “balloon problem“ in acoustics:
00
, ,( ,0) 0, ( ,0) 0, 0.
0, ,t
PR R
R R t r
R R
0 0
00 0
0 0 0 02
( ) ( )( , )
( , ) [ '( ) '( )]
1 1( , ) [ ( ) ( )] [ '( ) '( )]
t
R
f R a t g R a tR t
R Ra
R t f R a t g R a tR
R t f R a t g R a t f R a t g R a tR R
• No source at the origin2
0lim 0R
RR
0 0, outp P p p p
3. Spherical Waves
0 000 0 0
0 0
20 0 0 0
2
0
1( ,0) ( ( ) ( )) 0 ( ) ( ) 0, 0
, 0 , 0( ,0) [ '( ) '( )] '( ) '( )
0, 0, .
( , ) [ ( ) ( )] [ '( ) '( )]
lim [ (
t
R
R
R f R g R f R g R RR
P PR R R R Ra
aR f R g R f R g RR
R R R R
R R t f R a t g R a t R f R a t g R a t
R fR
0 0 0) ( )] 0 ( ) ( ) 0, 0 (*)a t g a t f g a t
3. Spherical Waves2
00 0
0
2 20 0
0 0 0 0
0 0
20 0
0 0
0
, 02( ) ( ) (**)
, .
(*) (**)
, 0 , 04 2 4 2
( ) ( ) ( )
, , , .2 2
( ) ( ) 0
,
,4 2
( )
, | |2
PR D R R
af R g R
D R R
P D P DR R R R R R
a af R g R f R
D DR R R R
f g for
R
P DR R
af
DR
20
0 0
0
, 04 2
, ( )
, .2
P DR
ag
DR
3. Spherical Waves
t
R0
R+ a0 t=R0
R0 /2a0
R0 /a0
R0 /20R
R- a0 t=R0
R- a0 t=0
A
C D
E
B
3. Spherical Waves
0 0 0 0 00, (0 / 2 )R a t r a t R t R a
2 20 0 0 0
0 0 0 0
00 0 0
0 00
1 1( , ) ( ) ( ) ( ) ( )
4 2 4 2
( 4 )4
( , )t t
P D P DR t f R a t g R a t R a t R a t
R R a a
P PtRa t
R a
PR t p p P
Region A:
Region B:0 0R a t R
0
( , ) 0
( , ) 0, 0t
R t
R t p p
3. Spherical WavesRegion C:
0 0 0 0 0, 0,R a t R R a t R a t R
00 0 0
0
0, (0 )R
R a t R a t R ta
0
00
( , )
( , ) ,t
PtR t
PR t p p P
Region D:
20
0 0
00 0
0
( , ) ( )4
( )( , ) ( ),
2 2t
PR t a t R
a R
P a t RPR t a t R p p
R R
Region E: 0 0 00,R a t R a t R
20
0 0
00 0
0
( , ) ( )4
( )( , ) ( ),
2 2t
PR t R a t
a R
P R a tPR t R a t p p
R R
3. Spherical Waves• Time Evolution of pressure difference
0 0
0 0 00 0 0 0 0 0
0
0 0
, 0
0 0 , (1 ),2 2 2
0, .
P R R a t
R R a tPt a t p p R a t R R a t
a RR R a t
0 01
0 0
02
0 0
2,
2
2
R a tPp
R a t
RPp
R a t
00
0
, 00,
0,
P R Rt p p
R R
R
P
p-p0
p1
p2
0 R0-a0t R0+a0tR0
3. Spherical Waves
0 0
0 0 0 00 0 0 0 0 0 0
0 0
0 0
, 0
, (1 ),2 2 2
0, .
P R R a t
R R R a tPt a t R p p R a t R R a t
a a RR R a t
0 01
0 0
02
0 0
20,
2
0.2
R a tPp
R a t
RPp
R a t
R
P
p-p0
p1
p2
0 R0-a0t R0+a0tR0
3. Spherical Waves
00 00
00
0 0
(1 ), 0, 2
0, .
a tPR R a tR
t p p Ra
R R a t
R
p-p0
p2
0 R0+a0tR0
02
0 02
RPp
R a t
4. Cylinderical Waves• The sources are uniformily distrubeted on the z axis
with a uniform strength q(t) per unit length. • Total disturbance is
2 2
0
1 ( / ) 1 ( / )( , ) , .
4 2
q t R c dz q t R c dzr t R r z
R R
r
R
q(t)dz-
z
4.Cylinderical Waves• Various forms of this solution are valuable.
0
1sinh , cosh ( cosh )
2
rLet z r R r q t d
c
/22
2 2 2 2
1 ( ), ( ) , ( , )
2 ( ) /
t r cR r q dt z c t r t
c c t r c
2
0
1 1( ) sinh '( cosh )
2
lim sinh '( cosh ) .2
rr r tt
d c rc q t d
r d r c
c rq t
r c
• If q‘(t) →0 suficiently fast as t→-∞.
4. Cylinderical Waves
( / )cosh
0
1.
2i r c i te d e
cosh ( ) / ,Substituting c t r
/
2 2 20
/
2 2 20
1 1 '( )'( cosh ) ,
2 2 ( ) /
1 1 '( )cosh '( cosh ) .
2 2 ( ) /
t r c
t
t r c
r
r q dq t d
c t r c
r t q dq t d
c c r t r c
( ) ,i tq t e choosevery small
5. Supersonic Flow Past a body of Revolution
22
1, 1 .x t M
c
2 21, 1,xx rr rB B M
r
2 2 20
1 ( ), .
2 ( )
x Br qd x Br
x B r
r-distance from the flight path
x-distance from nose of the body
5. Supersonic Flow Past a body of Revolution
'( )( ) ( ).r xR x U on r R x
• Linearization • Body must be slender
• R‘(x) is small and Φx, are both small. '( ) ( ).r UR x on r R x
2
~ ( ) / 2 0,
( ) 2 ( ) '( ) '( ), ( ) ( ).
rwehave q x r as r
q x UR x R x US x S x R x
• If the body shape is given by r R x ,
5. Supersonic Flow Past a body of Revolution
2 2 20
'( ), 0.
2 ( )
x BrU S dx Br
x B r
• The components of the velocity perturbation are obtained by suitable modification
2 2 20
2 2 20
''( ),
2 ( )
( ) ''( ).
2 ( )
x Br
x
x Br
r
U S d
x B r
U x S d
r x B r
6. Initial Value Problem in Two and Three Dimensions
0 1( ), ( ), 0.tx x t
• We know from the spherical wave solution is(| | )
( , )| |
f x ctx t
x
• We consider(| | )
( , ) ( ) .| |
f x ctx t d
x
2
0 0 0
( , ) ( ) ( ) sin ,x t RI f R ct R dRd d
(sin cos ,sin sin ,cos ).I
6. Initial Value Problem in Two and Three Dimensions
• The initial source strength determining f acts only for an instant,
( ) ( ).f R ct R ct 2
0 0
( , ) ( )sin .x t ct x ctI d d
( )
1( , ) ,
S tx t dS
ct
0, 4 ( ) 0.t x as t
6. Initial Value Problem in Two and Three Dimensions
1• If we choose ( ) ( ) / 4 , we shall have solved
the special initial value problem
x x
10, ( ) 0.t x as t
1( )
1( , ) .
4 S tx t dS
ct
1 1 12 2 ( )
1( , ) [ ], [ ]
4 S tx t ctM M dS
c t
6. Initial Value Problem in Two and Three Dimensions
( , )• ( , ) is a solution of wave equation.
x tx t
t
2 2
4 ( ),
0 0.
t
t tt
x
c as t
0 0( ), 0 0, ( ) ( ) / 4 ,tTo give x as t we must choose x x and take
0( )
1( , ) .
4 S tx t dS
t
• The general initial value therefore is
0 1 0 1( ) ( )
1 1( , ) [ ] [ ].
4 4S t S tx t dS dS ctM ctM
t ct ct t
2• One can show .i itt x xc
6. Initial Value Problem in Two and Three Dimensions
Two Dimesional Problem
0 02 2 ( )
1[ ]
4 S tM dS
c t
2 2 2 20 1 2 1 20 1 1 2 22 2 2 2( )
1 1 2 2
( , )1[ ] , ( ) : ( ) ( )
2 ( ) ( )t
dM t x x c t
ct c t x x
0 1 2 1 2 1 1 2 1 21 2 2 2 2 2 2 2 2 2
( ) ( )1 1 2 2 1 1 2 2
( , ) ( , )1 1( , , )
2 2( ) ( ) ( ) ( )t t
d dx x t
t c t x x c t x x
Conclusion
• The wave equations occurs in many problems.
• The solution strongly depends on the initial value and boundary conditions.
• The wave equations are easier to solve in odd dimensions.
• One can easily solve the IVP in two and three dimensions.
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