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16.3 Wave Motion in an Infinite Medium 579
16.4 Wave Motion in a Semi-Infinite Medium 585
16.4.1 Solution by Fourier Sine or Cosine Transform 586
16.5 Laplace Transform Techniques 587
16.6 Characteristics and d’Alembert’s Solution 594
16.6.1 Forward and Backward Waves 596
16.6.2 Forced Wave Motion 599
16.7 Vibrations in a Circular Membrane I 602
16.7.1 Normal Modes of Vibration 604
16.8 Vibrations in a Circular Membrane II 605
16.9 Vibrations in a Rectangular Membrane 608
CHAPTER 17 The Heat Equation 611
17.1 Initial and Boundary Conditions 611
17.2 The Heat Equation on [0, L] 612
17.2.1 Ends Kept at Temperature Zero 612
17.2.2 Insulated Ends 614
17.2.3 Radiating End 615
17.2.4 Transformation of Problems 618
17.2.5 The Heat Equation with a Source Term 619
17.2.6 Effects of Boundary Conditions and Constants 622
17.3 Solutions in an Infinite Medium 626
17.3.1 Problems on the Real Line 626
17.3.2 Solution by Fourier Transform 627
17.3.3 Problems on the Half-Line 629
17.3.4 Solution by Fourier Sine Transform 630
17.4 Laplace Transform Techniques 631
17.5 Heat Conduction in an Infinite Cylinder 636
17.6 Heat Conduction in a Rectangular Plate 638
CHAPTER 18 The Potential Equation 641
18.1 Laplace’s Equation 641
18.2 Dirichlet Problem for a Rectangle 642
18.3 Dirichlet Problem for a Disk 645
18.4 Poisson’s Integral Formula 648
18.5 Dirichlet Problem for Unbounded Regions 649
18.5.1 The Upper Half-Plane 650
18.5.2 The Right Quarter-Plane 652
18.6 A Dirichlet Problem for a Cube 654
18.7 Steady-State Equation for a Sphere 655
18.8 The Neumann Problem 659
18.8.1 A Neumann Problem for a Rectangle 660
18.8.2 A Neumann Problem for a Disk 662
18.8.3 A Neumann Problem for the Upper Half-Plane 664
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October 15, 2010 17:43 THM/NEIL Page-x 27410_00_fm_pi-xiv
