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# Lecture 10 - Wave Equations Engineering Mathematics Notes | EduRev

## Engineering Mathematics : Lecture 10 - Wave Equations Engineering Mathematics Notes | EduRev

``` Page 1

Wave Equations
Institute of Lifelong Learning, University of Delhi                                                                                         pg. 1

Subject: Math
Lesson: Wave Equations
Course Developer: Dr. Preeti Jain
College/Department: A.R.S.D. College, University of Delhi

Page 2

Wave Equations
Institute of Lifelong Learning, University of Delhi                                                                                         pg. 1

Subject: Math
Lesson: Wave Equations
Course Developer: Dr. Preeti Jain
College/Department: A.R.S.D. College, University of Delhi

Wave Equations
Institute of Lifelong Learning, University of Delhi                                                                                         pg. 2

Chapter : Wave Equations
? 1: Learning Outcomes
? 2: Introduction
? 3: Homogeneous Wave Equation
? 4: Initial Boundary Value Problem
? 5: Non- Homogeneous Boundary Conditions
? 6: Vibration of Finite Strings with Fixed Ends
? 7: Non- homogeneous Wave Equations
? 8: Riemann Method
? 9: Goursat Problem
? 10: Spherical Wave Equation
? Summary
? Exercises
? Glossary

Page 3

Wave Equations
Institute of Lifelong Learning, University of Delhi                                                                                         pg. 1

Subject: Math
Lesson: Wave Equations
Course Developer: Dr. Preeti Jain
College/Department: A.R.S.D. College, University of Delhi

Wave Equations
Institute of Lifelong Learning, University of Delhi                                                                                         pg. 2

Chapter : Wave Equations
? 1: Learning Outcomes
? 2: Introduction
? 3: Homogeneous Wave Equation
? 4: Initial Boundary Value Problem
? 5: Non- Homogeneous Boundary Conditions
? 6: Vibration of Finite Strings with Fixed Ends
? 7: Non- homogeneous Wave Equations
? 8: Riemann Method
? 9: Goursat Problem
? 10: Spherical Wave Equation
? Summary
? Exercises
? Glossary

Wave Equations
Institute of Lifelong Learning, University of Delhi                                                                                         pg. 3

1. Learning Outcomes:
After you have read this lesson , you will be able to understand, define
and simplify the homogeneous wave equation, non- homogeneous
wave equation, non- homogeneous boundary conditions, initial
boundary value problem, finite string problem with fixed ends,
Riemann problem, Goursat problem and spherical wave equation. You
should be able to differentiate between homogenous wave equations
and non- homogeneous wave equations. You will also be able to
understand why it is said that the solution of finite vibrating string
problem with fixed ends is more complicated then the problem of
infinite vibrating string. Moreover, you should be able to apply the
knowledge to solve various initial boundary problems which arises in
many practical problems.

2. Introduction:
Differential equations enables us to solve various problems which we
come across in many mathematical problems. Our main concern here
is on boundary value problems. Boundary value problems are
problems in differential equations in which certain conditions (which
we can also call as constraints) are imposed on the equations. After
finding the general solution of these problems, we apply these
additional condition to get a solution which also satisfies the boundary
conditions. We also come across the boundary value problems in many
physical quantities. Generally, these conditions are specified at the
extremes. Here, when we are dealing with wave equations, we will be
considering both the cases homogeneous wave equation and non-
homogeneous wave equation. If we define (initialize) the independent
variable at the lower boundary of the domain, we can often term
homogeneous wave equations as initial boundary value problems.
Page 4

Wave Equations
Institute of Lifelong Learning, University of Delhi                                                                                         pg. 1

Subject: Math
Lesson: Wave Equations
Course Developer: Dr. Preeti Jain
College/Department: A.R.S.D. College, University of Delhi

Wave Equations
Institute of Lifelong Learning, University of Delhi                                                                                         pg. 2

Chapter : Wave Equations
? 1: Learning Outcomes
? 2: Introduction
? 3: Homogeneous Wave Equation
? 4: Initial Boundary Value Problem
? 5: Non- Homogeneous Boundary Conditions
? 6: Vibration of Finite Strings with Fixed Ends
? 7: Non- homogeneous Wave Equations
? 8: Riemann Method
? 9: Goursat Problem
? 10: Spherical Wave Equation
? Summary
? Exercises
? Glossary

Wave Equations
Institute of Lifelong Learning, University of Delhi                                                                                         pg. 3

1. Learning Outcomes:
After you have read this lesson , you will be able to understand, define
and simplify the homogeneous wave equation, non- homogeneous
wave equation, non- homogeneous boundary conditions, initial
boundary value problem, finite string problem with fixed ends,
Riemann problem, Goursat problem and spherical wave equation. You
should be able to differentiate between homogenous wave equations
and non- homogeneous wave equations. You will also be able to
understand why it is said that the solution of finite vibrating string
problem with fixed ends is more complicated then the problem of
infinite vibrating string. Moreover, you should be able to apply the
knowledge to solve various initial boundary problems which arises in
many practical problems.

2. Introduction:
Differential equations enables us to solve various problems which we
come across in many mathematical problems. Our main concern here
is on boundary value problems. Boundary value problems are
problems in differential equations in which certain conditions (which
we can also call as constraints) are imposed on the equations. After
finding the general solution of these problems, we apply these
additional condition to get a solution which also satisfies the boundary
conditions. We also come across the boundary value problems in many
physical quantities. Generally, these conditions are specified at the
extremes. Here, when we are dealing with wave equations, we will be
considering both the cases homogeneous wave equation and non-
homogeneous wave equation. If we define (initialize) the independent
variable at the lower boundary of the domain, we can often term
homogeneous wave equations as initial boundary value problems.
Wave Equations
Institute of Lifelong Learning, University of Delhi                                                                                         pg. 4

3. Homogeneous Wave Equation:
The equation
2
0
tt xx
u c u ?? is the standard example of hyperbolic
equation. There are two real characteristic slopes at each point ? ? , xt .
This equation is popularly known as wave equation in one dimension
and describes the propagation (bi-directional) of waves with finite
speed c ? . The characteristics are therefore, curves in the real domain
of the problem. Oscillatory (not always periodic) behaviour in time.
Here time reversible is permissible as it reverses the direction of wave
propagation.
Now, we will obtain the solution of one dimensional wave equation in
free space. For this, consider the Cauchy problem of an infinite string
with the initial conditions

2
0 , , 0
tt xx
u c u x R t ? ? ? ?       (3.1)
? ? ? ? ,0 , u x f x x R ??        (3.2)
? ? ? ? ,0 ,
t
u x g x x R ??       (3.3)
To solve equation (3.1), we first reduce it into canonical form. The two
characteristic coordinates
,, x ct x ct ?? ? ? ? ?
transforms equation (3.1) into 0. u
??
?
After performing two straightforward integration, we get
? ? ? ? ? ? , u ? ? ? ? ? ? ?? ,
where ? and ? are arbitrary functions to be determined, (provided
they are differentiable twice). Thus, the general solution of wave
equation in terms of original variables x and t is
? ? ? ? ? ? , u x t x ct x ct ?? ? ? ? ? ,      (3.4)
? ? ? ? ? ? ? ? ,0 u x x x f x ?? ? ? ? ,      (3.5)
Page 5

Wave Equations
Institute of Lifelong Learning, University of Delhi                                                                                         pg. 1

Subject: Math
Lesson: Wave Equations
Course Developer: Dr. Preeti Jain
College/Department: A.R.S.D. College, University of Delhi

Wave Equations
Institute of Lifelong Learning, University of Delhi                                                                                         pg. 2

Chapter : Wave Equations
? 1: Learning Outcomes
? 2: Introduction
? 3: Homogeneous Wave Equation
? 4: Initial Boundary Value Problem
? 5: Non- Homogeneous Boundary Conditions
? 6: Vibration of Finite Strings with Fixed Ends
? 7: Non- homogeneous Wave Equations
? 8: Riemann Method
? 9: Goursat Problem
? 10: Spherical Wave Equation
? Summary
? Exercises
? Glossary

Wave Equations
Institute of Lifelong Learning, University of Delhi                                                                                         pg. 3

1. Learning Outcomes:
After you have read this lesson , you will be able to understand, define
and simplify the homogeneous wave equation, non- homogeneous
wave equation, non- homogeneous boundary conditions, initial
boundary value problem, finite string problem with fixed ends,
Riemann problem, Goursat problem and spherical wave equation. You
should be able to differentiate between homogenous wave equations
and non- homogeneous wave equations. You will also be able to
understand why it is said that the solution of finite vibrating string
problem with fixed ends is more complicated then the problem of
infinite vibrating string. Moreover, you should be able to apply the
knowledge to solve various initial boundary problems which arises in
many practical problems.

2. Introduction:
Differential equations enables us to solve various problems which we
come across in many mathematical problems. Our main concern here
is on boundary value problems. Boundary value problems are
problems in differential equations in which certain conditions (which
we can also call as constraints) are imposed on the equations. After
finding the general solution of these problems, we apply these
additional condition to get a solution which also satisfies the boundary
conditions. We also come across the boundary value problems in many
physical quantities. Generally, these conditions are specified at the
extremes. Here, when we are dealing with wave equations, we will be
considering both the cases homogeneous wave equation and non-
homogeneous wave equation. If we define (initialize) the independent
variable at the lower boundary of the domain, we can often term
homogeneous wave equations as initial boundary value problems.
Wave Equations
Institute of Lifelong Learning, University of Delhi                                                                                         pg. 4

3. Homogeneous Wave Equation:
The equation
2
0
tt xx
u c u ?? is the standard example of hyperbolic
equation. There are two real characteristic slopes at each point ? ? , xt .
This equation is popularly known as wave equation in one dimension
and describes the propagation (bi-directional) of waves with finite
speed c ? . The characteristics are therefore, curves in the real domain
of the problem. Oscillatory (not always periodic) behaviour in time.
Here time reversible is permissible as it reverses the direction of wave
propagation.
Now, we will obtain the solution of one dimensional wave equation in
free space. For this, consider the Cauchy problem of an infinite string
with the initial conditions

2
0 , , 0
tt xx
u c u x R t ? ? ? ?       (3.1)
? ? ? ? ,0 , u x f x x R ??        (3.2)
? ? ? ? ,0 ,
t
u x g x x R ??       (3.3)
To solve equation (3.1), we first reduce it into canonical form. The two
characteristic coordinates
,, x ct x ct ?? ? ? ? ?
transforms equation (3.1) into 0. u
??
?
After performing two straightforward integration, we get
? ? ? ? ? ? , u ? ? ? ? ? ? ?? ,
where ? and ? are arbitrary functions to be determined, (provided
they are differentiable twice). Thus, the general solution of wave
equation in terms of original variables x and t is
? ? ? ? ? ? , u x t x ct x ct ?? ? ? ? ? ,      (3.4)
? ? ? ? ? ? ? ? ,0 u x x x f x ?? ? ? ? ,      (3.5)
Wave Equations
Institute of Lifelong Learning, University of Delhi                                                                                         pg. 5

? ? ? ? ? ? ? ? ,0
t
u x c x x g x ?? ?? ? ? ? .      (3.6)
Integrating the last equation and then simplifying for ? and ? , we get
? ? ? ? ? ?
0
11
2 2 2
x
x
K
x f x g d
c
? ? ? ? ? ?
?
,      (3.7)
? ? ? ? ? ?
0
11
2 2 2
x
x
K
x f x g d
c
? ? ? ? ? ?
?
,     (3.8)
0
x and K called the arbitrary constants.
The solution of wave equation is, therefore, given by
? ? ? ? ? ? ? ?
11
,
22
x ct
x ct
u x t f x ct f x ct g d
c
??
?
?
? ? ? ? ? ??
?? ?
.   (3.9)
Solution shown in equation (3.9) is known as D’Alembert solution of
the Cauchy problem for one dimensional wave equation.

if double derivative of f and derivative of g exist then by direct
substitution it is evident that ? ? , u x t satisfies  the equation (3.1). The
D’Alembert solution describes two distinct waves- one moves to right
direction and other on the left both with speed c. Physically,
? ? x ct ? ? represents a propagating wave progressing in the negative x-
x
y
? ?
00
, xt
R
O
Figure 1
Range of influence
Domain of dependence
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