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1 | (14) |
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1.1. Elements of an Ultrasonic NDE System |
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1 | (2) |
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3 | (2) |
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1.3. Ultrasonic Transducers |
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5 | (4) |
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1.4. Ultrasonic Digitizers |
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9 | (1) |
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1.5. Ultrasonic Terminology |
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10 | (2) |
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1.6. About the Literature |
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12 | (1) |
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12 | (1) |
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13 | (2) |
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2. Linear Systems and the Fourier Transform |
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15 | (14) |
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2.1. Linear Time-Shift Invariant Systems |
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15 | (1) |
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16 | (3) |
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2.3. LTI Systems and the Impulse Response Function |
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19 | (2) |
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2.4. An Ultrasonic NDE Measurement System as an LTI System |
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21 | (2) |
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2.5. About the Literature |
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23 | (1) |
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24 | (4) |
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28 | (1) |
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29 | (20) |
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3.1. Governing Equations for a Fluid |
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29 | (4) |
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3.1.1. Equations of Motion |
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29 | (1) |
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3.1.2. Constitutive Equations |
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30 | (1) |
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31 | (1) |
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3.1.4. Interface/Boundary Conditions |
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32 | (1) |
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3.2. Governing Equations for an Elastic Solid |
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33 | (12) |
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3.2.1. Equations of Motion |
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34 | (1) |
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3.2.2. Constitutive Equations |
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35 | (1) |
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3.2.3. Navier's Equations |
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36 | (1) |
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3.2.4. Interface/Boundary Conditions |
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37 | (2) |
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3.2.5. Wave Equations for Potentials |
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39 | (1) |
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3.2.6. Dilatation and Rotation |
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40 | (1) |
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3.2.7. Governing Equations in Cartesian Coordinates |
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40 | (5) |
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3.3. About the Literature |
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45 | (1) |
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45 | (3) |
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48 | (1) |
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4. Propagation of Bulk Waves |
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49 | (20) |
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4.1. Plane Waves in a Fluid |
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49 | (4) |
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4.1.1. One-Dimensional Waves |
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49 | (1) |
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4.1.2. Fourier Transform Relations |
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50 | (1) |
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51 | (1) |
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4.1.4. Three-Dimensional Waves |
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52 | (1) |
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4.2. Plane Waves in an Elastic Solid |
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53 | (4) |
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4.2.1. One-Dimensional Solutions to Navier's Equations |
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53 | (1) |
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4.2.2. Three-Dimensional Solutions to Navier's Equations |
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53 | (4) |
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4.3. Spherical Waves in a Fluid |
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57 | (5) |
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4.3.1. Fundamental Solution |
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57 | (2) |
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4.3.2. Integral Forms of the Fundamental Solution |
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59 | (2) |
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4.3.3. Far-Field Form of G and Its Derivatives |
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61 | (1) |
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4.4. Spherical Waves in an Elastic Solid |
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62 | (4) |
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4.4.1. Fundamental Solution |
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62 | (4) |
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4.4.2. Far-Field Form of G(ji) and Its Derivatives |
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66 | (1) |
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4.5. About the Literature |
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66 | (1) |
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67 | (1) |
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68 | (1) |
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5. Reciprocal Theorem and Other Integral Relations |
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69 | (22) |
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5.1. Reciprocal Theorem for a Fluid |
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69 | (8) |
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5.1.1. Integral Representation Theorem |
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70 | (2) |
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5.1.2. Sommerfeld Radiation Conditions |
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72 | (3) |
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5.1.3. Integral Equations for Scattering Problems |
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75 | (2) |
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5.2. Reciprocal Theorem for an Elastic Solid |
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77 | (6) |
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5.2.1. Integral Representation Theorem |
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78 | (1) |
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5.2.2. Radiation Conditions |
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79 | (2) |
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5.2.3. Integral Equations for Scattering Problems |
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81 | (2) |
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5.3. An Electromechanical Reciprocal Theorem |
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83 | (3) |
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5.3.1. Governing Equations |
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83 | (1) |
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5.3.2. Reciprocal Theorem for a Piezoelectric Medium |
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84 | (2) |
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5.4. About the Literature |
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86 | (1) |
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86 | (2) |
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88 | (3) |
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6. Reflection and Refraction of Bulk Waves |
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91 | (50) |
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6.1. Reflection and Refraction at a Fluid-Fluid Interface (Normal Incidence) |
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91 | (6) |
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6.1.1. Reflection and Transmission Coefficients |
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91 | (3) |
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6.1.2. Acoustic Intensity of a Plane Wave |
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94 | (3) |
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6.2. Reflection and Refraction of a Plane Wave at a Fluid-Fluid Interface (Oblique Incidence) |
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97 | (18) |
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6.2.1. Reflection and Transmission Coefficients |
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97 | (1) |
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6.2.2. Critical Angles and Inhomogeneous Waves |
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98 | (2) |
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6.2.3. Energy Reflection and Transmission below the Critical Angle |
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100 | (1) |
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6.2.4. Energy Reflection and Transmission above the Critical Angle |
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101 | (1) |
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102 | (4) |
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106 | (2) |
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6.2.7. Reflection and Refraction at a Fluid-Fluid Interface in Three Dimensions |
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108 | (4) |
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6.2.8. Snell's Law and Stationary Phase |
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112 | (3) |
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6.3. Reflection and Refraction at a Fluid-Solid Interface at Oblique Incidence |
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115 | (8) |
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6.3.1. Reflection and Transmission Coefficients |
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115 | (4) |
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6.3.2. Energy Flux and Intensity for Elastic Waves |
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119 | (3) |
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6.3.3. Stokes' Relations (Fluid-Solid Interface) |
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122 | (1) |
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6.4. Reflection and Refraction at a Solid-Solid Interface (Smooth Contact) |
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123 | (4) |
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6.5. Reflection and Refraction at a Solid-Solid Interface (Welded Contact) |
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127 | (6) |
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6.5.1. Incident P- and SV-Waves |
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127 | (4) |
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131 | (2) |
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6.6. Reflection at a Stress-Free Surface |
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133 | (1) |
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6.7. About the Literature |
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134 | (1) |
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135 | (5) |
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140 | (1) |
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7. Propagation of Surface and Plate Waves |
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141 | (16) |
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7.1. Rayleigh Surface Waves |
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141 | (4) |
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7.2. Plate Waves-Horizontal Shearing Motions |
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145 | (4) |
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149 | (4) |
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149 | (3) |
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152 | (1) |
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7.4. Other Waves in Bounded Media |
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153 | (1) |
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7.5. About the Literature |
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153 | (1) |
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154 | (1) |
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155 | (2) |
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8. Ultrasonic Transducer Radiation |
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157 | (126) |
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8.1. Planar Piston Transducer in a Fluid |
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157 | (24) |
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8.1.1. Rayleigh-Sommerfeld Theory |
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158 | (2) |
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160 | (5) |
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165 | (13) |
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8.1.4. Angular Spectrum of Plane Waves and Boundary Diffraction Wave Theory |
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178 | (3) |
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8.2. Spherically Focused Piston Transducer in a Fluid |
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181 | (18) |
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8.2.1. O'Neil Model and Others |
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181 | (2) |
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183 | (6) |
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189 | (8) |
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8.2.4. Focusing by an Acoustic Lens |
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197 | (2) |
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8.3. Beam Propagation through a Planar Interface-Planar Probe |
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199 | (16) |
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8.3.1. Fluid-Fluid Interface-Normal Incidence |
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199 | (7) |
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8.3.2. Fluid-Solid Interface-Normal Incidence |
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206 | (3) |
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8.3.3. Fluid-Fluid Interface-Oblique Incidence |
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209 | (5) |
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8.3.4. Fluid-Solid Interface-Oblique Incidence |
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214 | (1) |
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8.4. Beam Propagation through a Planar Interface-Focused Probe |
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215 | (6) |
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8.4.1. Fluid-Fluid Interface |
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215 | (4) |
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8.4.2. Fluid-Solid Interface |
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219 | (2) |
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8.5. Beam Propagation through a Curved Interface |
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221 | (23) |
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8.5.1. Fluid-Fluid Interface |
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221 | (17) |
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8.5.2. Fluid-Solid Interface |
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238 | (6) |
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8.6. Numerical Evaluation of Beam Models |
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244 | (16) |
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246 | (10) |
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8.6.2. Curved Interface Problems with Edge Elements |
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256 | (4) |
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260 | (7) |
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8.8. Angle Beam Shear Wave Transducer |
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267 | (9) |
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8.8.1. Angle Beam Transducer Model |
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267 | (5) |
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272 | (4) |
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8.9. About the Literature |
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276 | (1) |
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276 | (5) |
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281 | (2) |
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9. Material Attenuation and Efficiency Factors |
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283 | (22) |
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9.1. Sources of Attenuation |
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283 | (4) |
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9.2. General Model for Measuring Material Attenuation and the System Efficiency Factor |
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287 | (14) |
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9.2.1. Diffraction Correction Integral |
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289 | (6) |
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9.2.2. Attenuation Measurement by a Deconvolution Model and the Wiener Filter |
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295 | (3) |
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9.2.3. Efficiency Factor Measurement by a Deconvolution Model and the Wiener Filter |
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298 | (3) |
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9.3. About the Literature |
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301 | (1) |
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301 | (3) |
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304 | (1) |
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305 | (80) |
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10.1. Far-Field Scattering Amplitude in a Fluid |
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305 | (2) |
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305 | (2) |
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307 | (1) |
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10.2. Far-Field Scattering Amplitude in an Elastic Solid |
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307 | (4) |
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307 | (3) |
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310 | (1) |
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10.3. Approximate Scattering Solutions-Fluid Model |
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311 | (25) |
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10.3.1. Kirchhoff Approximation-Volumetric Flaws |
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312 | (10) |
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10.3.2. Kirchhoff Approximation-Cracks |
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322 | (6) |
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10.3.3. Born Approximation |
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328 | (8) |
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10.4. Approximate Scattering Solutions-Elastic Solid Model |
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336 | (21) |
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10.4.1. Kirchhoff Approximation-Volumetric Flaws |
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336 | (8) |
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10.4.2. Kirchhoff Approximation-Cracks |
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344 | (8) |
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10.4.3. Born Approximation |
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352 | (5) |
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10.5. Far-Field Scattering Amplitude and Reciprocity |
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357 | (5) |
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10.5.1. Scattering Amplitude in a Fluid |
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357 | (3) |
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10.5.2. Scattering Amplitude in an Elastic Solid |
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360 | (2) |
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10.6. Scattering by a Sphere-Separation of Variables |
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362 | (16) |
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10.6.1. Sphere in a Fluid |
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362 | (9) |
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10.6.2. Sphere in an Elastic Solid |
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371 | (7) |
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10.7. About the Literature |
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378 | (1) |
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379 | (4) |
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383 | (2) |
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11. Transducer Reception Process |
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385 | (14) |
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11.1. Reception in a Single-Fluid Medium |
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385 | (2) |
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11.2. Reception Across a Plane Fluid-Fluid Interface |
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387 | (4) |
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11.3. Reception Across a Plane Fluid-Solid Interface |
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391 | (4) |
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11.4. About the Literature |
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395 | (1) |
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395 | (1) |
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396 | (3) |
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12. Ultrasonic Measurement Models |
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399 | (36) |
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12.1. LTI Model for a Single-Fluid Medium |
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399 | (4) |
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12.2. LTI Model for Immersion Testing |
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403 | (4) |
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12.2.1. Fluid-Fluid Model |
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404 | (1) |
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12.2.2. Fluid-Solid Model |
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405 | (2) |
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12.3. Reciprocity-Based Model for Immersion Testing |
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407 | (12) |
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407 | (8) |
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12.3.2. Reduction to an LTI Model |
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415 | (4) |
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12.4. Reciprocity-Based Model for Angle Beam Shear Wave Testing |
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419 | (5) |
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12.5. Electromechanical Reciprocity-Based Measurement Model |
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424 | (3) |
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12.6. Measurement Models and Their Limitations |
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427 | (2) |
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12.7. About the Literature |
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429 | (1) |
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430 | (3) |
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433 | (2) |
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13. Near-Field Measurement Models |
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435 | (22) |
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13.1. Model for a Single-Fluid Medium |
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435 | (11) |
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13.1.1. On-Axis Response to a Circular Transducer |
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440 | (1) |
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13.1.2. Scattering from a Sphere |
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441 | (2) |
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13.1.3. Scattering from the Flat End of a Cylinder |
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443 | (3) |
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13.1.4. Paraxial Approximation Limit |
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446 | (1) |
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13.2. Other Models for a Single-Fluid Medium |
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446 | (5) |
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13.3. Model for a Fluid-Solid Interface (Normal Incidence) |
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451 | (3) |
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13.4. About the Literature |
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454 | (1) |
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454 | (2) |
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456 | (1) |
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14. Quantitative Ultrasonic NDE with Models |
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457 | (34) |
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14.1. Transducer/System Characterization |
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458 | (11) |
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14.1.1. Effective Radius-Planar Transducer |
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458 | (2) |
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14.1.2. Effective Parameters-Spherically Focused Transducer |
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460 | (2) |
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14.1.3. System Efficiency Factor |
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462 | (1) |
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14.1.4. Experimental Results |
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463 | (6) |
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14.2. Flat-Bottom Hole Models and DGS Diagrams |
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469 | (12) |
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14.2.1. Fluid-Fluid Model |
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476 | (1) |
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477 | (1) |
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478 | (3) |
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14.3. Deconvolution and Far-Field Scattering Amplitudes |
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481 | (3) |
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14.4. Model-Based Ultrasonic Simulation |
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484 | (2) |
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14.5. About the Literature |
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486 | (1) |
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487 | (1) |
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488 | (3) |
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15. Model-Based Flaw Sizing |
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491 | (26) |
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15.1. Concept of Equivalent Flaw Sizing |
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491 | (1) |
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15.2. Kirchhoff Sizing for Cracks |
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491 | (7) |
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15.2.1. Nonlinear Least Squares Sizing Method |
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493 | (1) |
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15.2.2. Linear Least Squares/Eigenvalue Sizing Method |
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494 | (4) |
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15.3. Born Sizing for Volumetric Flaws |
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498 | (7) |
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505 | (3) |
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15.5. Other Sizing Methods |
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508 | (1) |
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15.6. About the Literature |
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509 | (1) |
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509 | (5) |
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514 | (3) |
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Appendix A. Fourier Transform |
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517 | (10) |
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A.1. Properties of the Fourier Transform |
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517 | (2) |
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A.2. Some Fourier Transform Pairs |
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519 | (1) |
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A.3. Discrete Fourier Transform |
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520 | (4) |
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A.4. Fast Fourier Transform |
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524 | (1) |
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525 | (1) |
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525 | (2) |
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Appendix B. Dirac Delta Function |
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527 | (2) |
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B.1. Properties of the Delta Function |
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527 | (1) |
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528 | (1) |
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Appendix C. Basic Notations and Concepts |
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529 | (12) |
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529 | (3) |
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532 | (1) |
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532 | (1) |
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533 | (1) |
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C.3. Strain and Deformation |
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533 | (3) |
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C.4. Conservation of Mass |
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536 | (1) |
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536 | (3) |
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536 | (1) |
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537 | (1) |
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C.5.3. Tractions and Stresses |
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538 | (1) |
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539 | (2) |
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Appendix D. Hilbert Transform |
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541 | (2) |
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D.1. Properties of the Hilbert Transform |
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541 | (1) |
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542 | (1) |
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Appendix E. Stationary Phase Method |
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543 | (8) |
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E.1. Single-Integral Forms |
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543 | (3) |
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E.2. Double-Integral Forms |
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546 | (1) |
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E.3. Curved-Surface Integral |
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547 | (2) |
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549 | (2) |
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Appendix F. Properties of Ellipsoids |
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551 | (4) |
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F.1. Geometry of an Ellipsoid |
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551 | (3) |
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554 | (1) |
| Index |
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555 | |