group 2 bhadouria, arjun singh glave, theodore dean han, zhe chapter 5. laplace transform chapter...
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Group 2
Bhadouria, Arjun SinghGlave, Theodore Dean
Han, Zhe
Chapter 5. Laplace TransformChapter 19. Wave Equation
Wave Equation
Chapter 19
Overview
• 19.1 – Introduction– Derivation– Examples
• 19.2 – Separation of Variables / Vibrating String– 19.2.1 – Solution by Separation of Variables– 19.2.2 – Travelling Wave Interpretation
• 19.3 – Separation of Variables/ Vibrating Membrane• 19.4 – Solution of wave equation
– 19.4.1 – d’Alembert’s solution– 19.4.2 – Solution by integral transforms
19.1 - Introduction
• Wave Equation
– Uses:• Electromagnetic Waves• Pulsatile blood flow• Acoustic Waves in Solids• Vibrating Strings• Vibrating Membranes
http://www.math.ubc.ca/~feldman/m267/separation.pdf
Derivation
u(x, t) = vertical displacement of the string from the x axis at position x and time tθ(x, t) = angle between the string and a horizontal line at position x and time tT(x, t) = tension in the string at position x and time t ρ(x) = mass density of the string at position x
http://www.math.ubc.ca/~feldman/m267/separation.pdf
Derivation
• Forces:
• Tension pulling to the right, which has a magnitude T(x+Δx, t) and acts at an angle θ(x+Δx, t) above horizontal
• Tension pulling to the left, which has magnitude T(x, t) and acts at an angle θ(x, t) below horizontal
• The net magnitude of the external forces acting vertically F(x, t)Δx
• Mass Distribution:
http://www.math.ubc.ca/~feldman/m267/separation.pdf
Derivation
Vertical Component of Motion
Divide by Δx and taking the limit as Δx → 0.
http://www.math.ubc.ca/~feldman/m267/separation.pdf
Derivation
http://www.math.ubc.ca/~feldman/m267/separation.pdf
Derivation
For small vibrations:
Therefore,
http://www.math.ubc.ca/~feldman/m267/separation.pdf
Derivation
Substitute into (2) into (1)
http://www.math.ubc.ca/~feldman/m267/separation.pdf
Derivation
Horizontal Component of the Motion
Divide by Δx and taking the limit as Δx → 0.
http://www.math.ubc.ca/~feldman/m267/separation.pdf
Derivation
• For small vibrations: and
Therefore,
http://www.math.ubc.ca/~feldman/m267/separation.pdf
Solution
For a constant string density ρ, independent of xThe string tension T(t) is a constant, andNo external forces, F
http://www.math.ubc.ca/~feldman/m267/separation.pdf
Separation of Variables; Vibrating String
19.2.1 - Solution by Separation of Variables
Scenario
u(x, t) = vertical displacement of a string from the x axis at position x and time tl = string length
Recall:(1)
Boundry Conditions:u(0, t) = 0 for all t > 0 (2)u(l, t) = 0 for all t > 0 (3)
Initial Conditions u(x, 0) = f(x) for all 0 < x <l (4)ut(x, 0) = g(x) for all 0 < x <l (5)
http://logosfoundation.org/kursus/wave.pdf
Procedure
There are three steps to consider in order to solve this problem: Step 1:• Find all solutions of (1) that are of the special form for some
function that depends on x but not t and some function that depends on t but not x.
Step 2:• We impose the boundary conditions (2) and (3).
Step 3: • We impose the initial conditions (4) and (5).
http://logosfoundation.org/kursus/wave.pdf
Step 1 – Finding Factorized Solutions
Let:Since the left hand side is independent of t the right hand side must also be independent of t. The same goes for the right hand side being independent of x. Therefore, both sides must be constant (σ).
http://logosfoundation.org/kursus/wave.pdf
Step 1 – Finding Factorized Solutions
(6)
http://logosfoundation.org/kursus/wave.pdf
Step 1 – Finding Factorized Solutions
Solve the differential equations in (6)
http://logosfoundation.org/kursus/wave.pdf
Step 1 – Finding Factorized Solutions
If , there are two independent solutions for (6)
If ,
http://logosfoundation.org/kursus/wave.pdf
Step 1 – Finding Factorized Solutions
Solutions to the Wave EquationFor arbitrary and arbitrary
For arbitrary
http://logosfoundation.org/kursus/wave.pdf
Step 2 – Imposition of Boundaries
For
Thus, this solution is discarded.
http://logosfoundation.org/kursus/wave.pdf
Step 2 – Imposition of Boundaries
For , when When
Therefore,
http://logosfoundation.org/kursus/wave.pdf
Step 2 – Imposition of Boundaries
Since , in order to satisfy An integer k must be introduced such that:
Therefore,
http://logosfoundation.org/kursus/wave.pdf
Step 2 – Imposition of Boundaries
Where, and are allowed to be any complex numbers and are allowed to be any complex numbers
http://logosfoundation.org/kursus/wave.pdf
Step 3 – Imposition of the Initial Condition
From the preceding:
which obeys the wave equation (1) and the boundary conditions (2) and (3), for any choice of and
http://logosfoundation.org/kursus/wave.pdf
Step 3 – Imposition of the Initial Condition
The previous expression must also satisfy the initial conditions (4) and (5):
(4’)
(5’)
http://logosfoundation.org/kursus/wave.pdf
Step 3 – Imposition of the Initial Condition
For any (reasonably smooth) function, h(x) defined on the interval 0<x<l, has a unique representation based on its Fourier Series:
(7)Which can also be written as:
http://logosfoundation.org/kursus/wave.pdf
Step 3 – Imposition of the Initial Condition
For the coefficients. We can make (7) match (4 ) by ′choosing and .
Thus .
Similarly, we can make (7) match (5 ) by choosing ′and
Thus
http://logosfoundation.org/kursus/wave.pdf
Step 3 – Imposition of the Initial Condition
Therefore,
(8)Where,
http://logosfoundation.org/kursus/wave.pdf
Step 3 – Imposition of the Initial Condition
The sum (8) can be very complicated, each term, called a “mode”, is quite simple. For each fixed t, the mode
is just a constant times . As x runs from 0 to l, the argument of runs from 0 to , which is k half–periods of sin. Here are graphs, at fixed t, of the first three modes, called the fundamental tone, the first harmonic and the second harmonic.
http://logosfoundation.org/kursus/wave.pdf
Step 3 – Imposition of the Initial Condition
The first 3 modes at fixed t’s.
http://logosfoundation.org/kursus/wave.pdf
Step 3 – Imposition of the Initial Condition
For each fixed x, the mode
is just a constant times plus a constant times . As t increases by one second, the argument, , of both and increases by , which is cycles (i.e. periods). So the fundamental oscillates at cps, the first harmonic oscillates at 2cps, the second harmonic oscillates at 3cps and so on. If the string has density ρ and tension T , then we have seen that . So to increase the frequency of oscillation of a string you increase the tension and/or decrease the density and/or shorten the string.
http://logosfoundation.org/kursus/wave.pdf
Example
Problem:
Example
Let l = 1, therefore,
It is very inefficient to use the integral formulae to evaluate and . It is easier to observe directly, just by matching coefficients.
Example
Separation of Variables; Vibrating String
19.2.2 - Travelling Wave Interpretation
Travelling Wave
Start with the Transport Equation:
where, u(t, x) – functionc – non-zero constant (wave speed)x – spatial variable
Initial Conditions
http://www.math.umn.edu/~olver/pd_/pdw.pdf
Travelling Wave
Let x represents the position of an object in a fixed coordinate frame. The characteristic equation:
Represents the object’s position relative to an observer who is uniformly moving with velocity c.
Next, replace the stationary space-time coordinates (t, x) by the moving coordinates (t, ξ).
http://www.math.umn.edu/~olver/pd_/pdw.pdf
Travelling Wave
Re-express the Transport Equation:
Express the derivatives of u in terms of those of v:
http://www.math.umn.edu/~olver/pd_/pdw.pdf
Travelling Wave
Using this coordinate system allows the conversion of a wave moving with velocity c to a stationary wave. That is,
http://www.math.umn.edu/~olver/pd_/pdw.pdf
Travelling Wave
For simplicity, we assume that v(t, ξ) has an appropriate domain of definition, such that,
Therefore, the transport equation must be a function of the characteristic variable only.
http://www.math.umn.edu/~olver/pd_/pdw.pdf
The Travelling Wave Interpretation
http://www.math.umn.edu/~olver/pd_/pdw.pdf
Travelling Wave
Revisiting the transport equation,
Also recall that:
http://www.math.umn.edu/~olver/pd_/pdw.pdf
Travelling Wave
At t = 0, the wave has the initial profile
• When c > 0, the wave translates to the right.
• When c < 0, the wave translates to the left.
• While c = 0 corresponds to a stationary wave form that
remains fixed at its original location.
http://www.math.umn.edu/~olver/pd_/pdw.pdf
Travelling Wave
As it only depends on the characteristic variable, every solution to the transport equation is constant on the characteristic lines of slope c, that is:
where k is an arbitrary constant. At any given time t, the value of the solution at position x only depends on its original value on the characteristic line passing through (t, x).
http://www.math.umn.edu/~olver/pd_/pdw.pdf
Travelling Wave
http://www.math.umn.edu/~olver/pd_/pdw.pdf
19.3 Separation of VariablesVibrating Membranes
• Let us consider the motion of a stretched membrane
• This is the two dimensional analog of the vibrating string problem
• To solve this problem we have to make some assumptions
Physical Assumptions
1. The mass of the membrane per unit area is constant. The membrane is perfectly flexible and offers no resistance to bending
2. The membrane is stretched and then fixed along its entire boundary in the xy plane. The tension per unit length T is the same at all points and does not change
3. The deflection u(x,y,t) of the membrane during the motion is small compared to the size of the membrane
Vibrating MembraneRef: Advanced Engineering Mathematics, 8th Edition, Erwin
Kreyszig
Derivation of differential equation
We consider the forces acting on the membraneTension T is force per unit lengthFor a small portion ∆x, ∆y forces are approximately T∆x and T∆yNeglecting horizontal motion we have vertical components on right and left side as T ∆y sin β and -T ∆y sin αHence resultant is T∆y(sin β – sin α)As angles are small sin can be replaced with tangents Fres = T∆y(tan β – tan α)
Fres = TΔy[ux(x+ Δx,y1)-ux(x,y2)]
Similarly Fres on other two sides is given by
Fres = TΔx[uy(x1, y+ Δy)-uy(x2,y)]
Using Newtons Second Law we get
Which gives us the wave equation:
…..(1)
yxuyyxuxTyxuyxxuyTt
uyx yyxx ,,,, 21212
2
2
2
2
22
2
2
y
u
x
uc
t
u
Vibrating Membrane: Use of double Fourier series
• The two-dimensional wave equation satisfies the boundary condition(2) u = 0 for all t ≥ 0 (on the boundary of membrane)
• And the two initial conditions(3) u(x,y,0) = f(x,y) (given initial displacement f(x,y)And (4)
),(0
yxgt
u
t
Separation of Variables
• Letu(x,y,t) = F(x,y)G(t) …..(5)
• Using this in the wave equation we have
• Separating variables we get
GFGFcGF yyxx
2
22
1
yyxx FFFGc
G
• This gives two equations: for the time function G(t) we have
…..(6)And for the Amplitude function F(x,y) we have
…..(7)which is known as the Helmholtz equation
• Separation of Helmholtz equation:F(x,y) = H(x)Q(y) …..(8)
• Substituting this into (7) gives
02
GG
02 FFF yyxx
HQ
dy
QdHQ
dx
Hd 22
2
2
2
• Separating variables
• Giving two ODE’s(9)
And (10)
where
222
2
2
2 11kQ
dy
Qd
Qdx
Hd
H
022
2
Hkdx
Hd
022
2
Qpdy
Qd
222 kp
Satisfying boundary conditions
• The general solution of (9) and (10) areH(x) = Acos(kx)+Bsin(kx) and Q(y) = Ccos(py)
+Dsin(py)Using boundary condition we get H(0) = H(a) = Q(0) = Q(b) = 0which in turn givesA = 0; k = mπ/a; C = 0; p = nπ/bm,n Ε integer
• We thus obtain the solutionHm(x) = sin (mπx/a) and Qn(y) = sin(nπy/b)
• Hence the functions(11)Fmn(x) = Hm(x)Qn(y) = sin(mπx/a)sin (nπy/b)
Turning to time functionAs p2 = ν2-k2 and λ=cν we haveλ = c(k2+p2)1/2
Hence λmn = cπ(m2/a2+n2/b2)1/2 …..(12)
Therefore …(13)
b
yn
a
xmtBtBtyxu mnmnmnmnmn
sinsinsincos,, *
Solution of the Entire Problem:Double Fourier Series
…..(14)
Using (3)
1 1
*
1 1
sinsinsincos
),,(),,(
m nmnmnmnmn
m nmn
b
yn
a
xmtBtB
tyxutyxu
1 1
)15(),(sinsin0,,m n
mn yxfb
yn
a
xmByxu
• Using Fourier analysis we get the generalized Euler formula
And using (4) we obtain
)16(sinsin),(4
0 0
b a
mn dxdyb
yn
a
xmyxf
abB
)17(sinsin),(4
0 0
* b a
mnmn dxdy
b
yn
a
xmyxg
abB
Example
• Vibrations of a rectangular membraneFind the vibrations of a rectangular membrane of sides a = 4 ft and b = 2 ft if the Tension T is 12.5 lb/ft, the density is 2.5 slugs/ft2, the initial velocity is zero and the initial displacement is
ft241.0, 22 yyxxyxf
Solution]sec/ft[55.2/5.12/ 222 Tc
0 y)g(x, as0* mnB
oddnmnm
evennm
yxynxm
yyxxBmn
,426.0
,0
dd2
sin4
sin241.024
4
33
2
0
4
0
22
Which gives
nm odd
ynxmtnm
nmtyxu
,
2233 2
sin4
sin44
5cos
1426.0,,
Ref: Advanced Engineering Mathematics, 8th Edition, Erwin Kreyszig
19.4 Vibrating String Solutions19.4.1 d’Alembert’s Solution
• Solution for the wave equation
can be obtained by transforming (1) by introducing independent variables
)1(2
22
2
2
x
uc
t
u
)2(, ctxzctxv
• u becomes a function of v and z.• The derivatives in (1) can be expressed as
derivatives with respect to v and z.
• We transform the other derivative in (1) similarly to get
zvxzxvx uuzuvuu
zzvzvv
xzzvxxzvxzvxx
uuu
zuuvuuuuu
2
zzvzvvtt uuucu 22
• Inserting these two results in (1) we get
which gives
• This is called the d’Alembert’s solution of the wave equation (1)
)3(02
vz
uuvz
)4()()(),(
(2) from
)()(
ctxctxtxu
zvu
D’Alembert’s solution satisfying initial conditions
)6(0,
)5(0,
xgxu
xfxu
t
)8()(')('0,
)7()()(0,
xgxcxcxu
xfxxxu
t
Dividing (8) by c and integrating we get
)()()( where
)9()(1
)()()(
000
0
0
xxxk
dssgc
xkxxx
x
• Solving (9) with (7) gives
• Replacing x by x+ct for φ and x by x-ct for ψ we get the solution
x
x
x
x
xkdssgc
xfx
xkdssgc
xfx
0
0
)(2
1)(
2
1)(
2
1)(
)(2
1)(
2
1)(
2
1)(
0
0
ctx
ctx
dssgc
ctxfctxftxu )(2
1
2
1),(
19.4.2 Solution by integral transforms
Laplace TransformSemi Infinite string
Find the displacement w(x,t) of an elastic string subject to:
(i) The string is initially at rest on the x axis(ii) For time t>0 the left end of the string is
moved by(iii)
otherwise0
2t0 ifsin)(),0(
ttftw
0for t0),(lim
txwx
Solution
• Wave equation:• With f as given and using initial conditions
• Taking the Laplace transform with respect to t
2
22
2
2
x
wc
t
w
0
0)0,(
0
tt
w
xw
2
22
0
22
2
0,x
wLc
t
wxswwLs
t
wL
t
),(),(2
2
02
2
02
2
2
2
txwLx
dttxwex
dtx
we
x
wL stst
•We thus obtain
0 thus2
2
2
2
2
222
Wc
s
x
W
x
WcWs
•Which gives csxcsx esBesAsxW // )()(),(
• Using initial condition
• This implies A(s) = 0 because c>0 so esx/c
increases as x increases.• So we have W(0,s) = B(s)=F(s)• So W(x,s)=F(s)e-sx/c
• Using inverse Laplace we get
0),(lim),(lim),(lim00
dttxwedttxwesxW
x
stst
xx
otherwise zero and
2 ifsin),(
c
xt
c
x
c
xttxw
Travelling wave solutionRef: Advanced Engineering Mathematics, 8th Edition, Erwin
Kreyszig
References
• H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. 1st Edition., 2011, XIV, 600 p. 9 illus. 10.3
• R. Baber. The Language of Mathematics: Utilizing Math in Practice. Appendix F
• Poromechanics III - Biot Centennial (1905-2005)• http://www.math.ubc.ca/~feldman/m267/
separation.pdf• http://logosfoundation.org/kursus/wave.pdf• http://www.math.umn.edu/~olver/pd_/pdw.pdf
• Advanced engineering mathematics, 2nd edition, M. D. Greenberg
• Advanced engineering mathematics, 8th edition, E. Kreyszig
• Partial differential equations in Mechanics, 1st edition, A.P.S. Selvadurai
• Partial differential equations, Graduate studies in mathematics, Volume 19, L. C. Evans
• Advanced engineering mathematics, 2nd edition, A.C. Bajpai, L.R. Mustoe, D. Walker