Lecture 10 - Wave Equations | Differential Equation and Mathematical Modeling-II - Engineering Mathematics PDF Download

 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 
 
Table of Contents: 
 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 
? References/ Further Reading 
 
 
 
 
 
 
 
 
 
 
 
 
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 
 
Table of Contents: 
 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 
? References/ Further Reading 
 
 
 
 
 
 
 
 
 
 
 
 
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 
 
Table of Contents: 
 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 
? References/ Further Reading 
 
 
 
 
 
 
 
 
 
 
 
 
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 
 
Table of Contents: 
 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 
? References/ Further Reading 
 
 
 
 
 
 
 
 
 
 
 
 
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