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 dependenceRead More

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