1 | # |
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2 | # slide_tsunami function |
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3 | # |
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4 | |
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5 | """This function returns a callable object representing an initial water |
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6 | displacement generated by a submarine sediment slide. |
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7 | |
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8 | Using input parameters: |
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9 | |
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10 | Required |
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11 | length downslope slide length |
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12 | depth water depth to slide centre of mass |
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13 | slope bathymetric slope |
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14 | |
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15 | Optional |
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16 | x0 x origin (0) |
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17 | y0 y origin (0) |
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18 | alpha angular orientation of slide in xy plane (0) |
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19 | w slide width (0.25*length) |
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20 | T slide thickness (0.01*length) |
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21 | g acceleration due to gravity (9.8) |
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22 | gamma specific density of sediments (1.85) |
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23 | Cm added mass coefficient (1) |
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24 | Cd drag coefficient (1) |
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25 | Cn friction coefficient (0) |
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26 | psi (0) |
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27 | dx offset of second Gaussian (0.2*width of first Gaussian) |
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28 | kappa multiplier for sech^2 function (3.0) |
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29 | kappad multiplier for second Gaussian function (0.8) |
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30 | zsmall an amount near to zero (0.01) |
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31 | |
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32 | The following parameters are calculated: |
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33 | |
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34 | a0 initial acceleration |
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35 | ut theoretical terminal velocity |
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36 | s0 charactistic distance of motion |
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37 | t0 characteristic time of motion |
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38 | w initial wavelength of tsunami |
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39 | a2D 2D initial amplitude of tsunami |
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40 | a3D 3D initial amplitude of tsunami |
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41 | |
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42 | The returned object is a callable double Gaussian function that represents |
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43 | the initial water displacement generated by a submarine sediment slide. |
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44 | |
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45 | Adrian Hitchman |
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46 | Geoscience Australia, June 2005 |
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47 | """ |
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48 | |
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49 | def slide_tsunami(length, depth, slope, width=None, thickness=None, \ |
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50 | x0=0.0, y0=0.0, alpha=0.0, \ |
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51 | gravity=9.8, gamma=1.85, \ |
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52 | massco=1, dragco=1, frictionco=0, psi=0, \ |
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53 | dx=None, kappa=3.0, kappad=0.8, zsmall=0.01, |
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54 | verbose=False): |
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55 | |
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56 | from math import sin, tan, radians, pi, sqrt, exp |
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57 | |
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58 | #if width not provided, set to typical value |
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59 | if width is None: |
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60 | width = 0.25 * length |
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61 | |
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62 | #if thickness not provided, set to typical value |
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63 | if thickness is None: |
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64 | thickness = 0.01 * length |
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65 | |
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66 | #calculate some parameters of the slide |
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67 | |
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68 | sint = sin(radians(slope)) |
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69 | tant = tan(radians(slope)) |
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70 | tanp = tan(radians(psi)) |
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71 | |
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72 | a0 = gravity * sint * ((gamma-1)/(gamma+massco)) * (1-(tanp/tant)) |
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73 | ut = sqrt((gravity*depth) * (length*sint/depth) \ |
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74 | * (pi*(gamma-1)/(2*dragco)) * (1-(tanp/tant))) |
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75 | s0 = ut**2 / a0 |
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76 | t0 = ut / a0 |
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77 | |
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78 | #calculate some parameters of the water displacement produced by the slide |
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79 | |
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80 | w = t0 * sqrt(gravity*depth) |
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81 | a2D = s0 * (0.0574 - (0.0431*sint)) \ |
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82 | * (thickness/length) \ |
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83 | * ((length*sint/depth)**1.25) \ |
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84 | * (1 - exp(-2.2*(gamma-1))) |
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85 | a3D = a2D / (1 + (15.5*sqrt(depth/(length*sint)))) |
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86 | |
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87 | #a few temporary print statements |
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88 | if verbose is True: |
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89 | print '\nThe slide ...' |
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90 | print '\tLength: ', length |
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91 | print '\tDepth: ', depth |
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92 | print '\tSlope: ', slope |
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93 | print '\tWidth: ', width |
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94 | print '\tThickness: ', thickness |
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95 | print '\tx0: ', x0 |
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96 | print '\ty0: ', y0 |
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97 | print '\tAlpha: ', alpha |
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98 | print '\tAcceleration: ', a0 |
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99 | print '\tTerminal velocity: ', ut |
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100 | print '\tChar time: ', t0 |
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101 | print '\tChar distance: ', s0 |
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102 | print '\nThe tsunami ...' |
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103 | print '\tWavelength: ', w |
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104 | print '\t2D amplitude: ', a2D |
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105 | print '\t3D amplitude: ', a3D |
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106 | |
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107 | #keep an eye on some of the assumptions built into the maths |
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108 | |
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109 | if ((slope < 5) or (slope > 30)): |
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110 | if verbose is True: |
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111 | print 'WARNING: slope out of range (5 - 30 degrees) ', slope |
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112 | if ((depth/length < 0.06) or (depth/length > 1.5)): |
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113 | if verbose is True: |
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114 | print 'WARNING: d/b out of range (0.06 - 1.5) ', depth/length |
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115 | if ((thickness/length < 0.008) or (thickness/length > 0.2)): |
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116 | if verbose is True: |
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117 | print 'WARNING: T/b out of range (0.008 - 0.2) ', thickness/length |
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118 | if ((gamma < 1.46) or (gamma > 2.93)): |
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119 | if verbose is True: |
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120 | print 'WARNING: gamma out of range (1.46 - 2.93) ', gamma |
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121 | |
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122 | return Double_gaussian(a3D=a3D, wavelength=w, width=width, \ |
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123 | x0=x0, y0=y0, alpha=alpha, \ |
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124 | dx=dx, kappa=kappa, kappad=kappad, zsmall=zsmall) |
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125 | |
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126 | # |
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127 | # slump_tsunami function |
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128 | # |
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129 | |
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130 | """This function returns a callable object representing an initial water |
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131 | displacement generated by a submarine sediment slump. |
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132 | |
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133 | Using input parameters: |
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134 | |
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135 | Required |
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136 | length downslope slump length |
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137 | depth water depth to slump centre of mass |
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138 | slope bathymetric slope |
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139 | |
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140 | Optional |
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141 | x0 x origin (0) |
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142 | y0 y origin (0) |
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143 | alpha angular orientation of slide in xy plane (0) |
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144 | w slump width (1.0*length) |
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145 | T slump thickness (0.1*length) |
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146 | R slump radius of curvature (b^2/(8*T)) |
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147 | del_phi slump angular displacement (0.48) |
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148 | g acceleration due to gravity (9.8) |
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149 | gamma specific density of sediments (1.85) |
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150 | Cm added mass coefficient (1) |
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151 | Cd drag coefficient (1) |
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152 | Cn friction coefficient (0) |
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153 | dx offset of second Gaussian (0.2*width of first Gaussian) |
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154 | kappa multiplier for sech^2 function (3.0) |
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155 | kappad multiplier for second Gaussian function (0.8) |
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156 | zsmall an amount near to zero (0.01) |
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157 | |
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158 | The following parameters are calculated: |
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159 | |
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160 | a0 initial acceleration |
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161 | um maximum velocity |
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162 | s0 charactistic distance of motion |
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163 | t0 characteristic time of motion |
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164 | w initial wavelength of tsunami |
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165 | a2D 2D initial amplitude of tsunami |
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166 | a3D 3D initial amplitude of tsunami |
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167 | |
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168 | The returned object is a callable double Gaussian function that represents |
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169 | the initial water displacement generated by a submarine sediment slump. |
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170 | |
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171 | Adrian Hitchman |
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172 | Geoscience Australia, June 2005 |
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173 | """ |
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174 | |
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175 | def slump_tsunami(length, depth, slope, width=None, thickness=None, \ |
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176 | radius=None, dphi=0.48, x0=0.0, y0=0.0, alpha=0.0, \ |
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177 | gravity=9.8, gamma=1.85, \ |
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178 | massco=1, dragco=1, frictionco=0, \ |
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179 | dx=None, kappa=3.0, kappad=0.8, zsmall=0.01, |
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180 | verbose=False): |
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181 | |
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182 | from math import sin, radians, sqrt |
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183 | |
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184 | #if width not provided, set to typical value |
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185 | if width is None: |
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186 | width = length |
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187 | |
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188 | #if thickness not provided, set to typical value |
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189 | if thickness is None: |
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190 | thickness = 0.1 * length |
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191 | |
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192 | #if radius not provided, set to typical value |
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193 | if radius is None: |
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194 | radius = length**2 / (8.0 * thickness) |
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195 | |
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196 | #calculate some parameters of the slump |
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197 | |
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198 | sint = sin(radians(slope)) |
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199 | |
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200 | s0 = radius * dphi / 2 |
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201 | t0 = sqrt((radius*(gamma+massco)) / (gravity*(gamma-1))) |
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202 | a0 = s0 / t0**2 |
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203 | um = s0 / t0 |
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204 | |
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205 | #calculate some parameters of the water displacement produced by the slump |
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206 | |
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207 | w = t0 * sqrt(gravity*depth) |
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208 | a2D = s0 * (0.131/sint) \ |
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209 | * (thickness/length) \ |
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210 | * (length*sint/depth)**1.25 \ |
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211 | * (length/radius)**0.63 * dphi**0.39 \ |
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212 | * (1.47 - (0.35*(gamma-1))) * (gamma-1) |
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213 | a3D = a2D / (1 + (2.06*sqrt(depth/length))) |
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214 | |
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215 | #a few temporary print statements |
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216 | if verbose is True: |
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217 | print '\nThe slump ...' |
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218 | print '\tLength: ', length |
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219 | print '\tDepth: ', depth |
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220 | print '\tSlope: ', slope |
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221 | print '\tWidth: ', width |
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222 | print '\tThickness: ', thickness |
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223 | print '\tRadius: ', radius |
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224 | print '\tDphi: ', dphi |
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225 | print '\tx0: ', x0 |
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226 | print '\ty0: ', y0 |
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227 | print '\tAlpha: ', alpha |
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228 | print '\tAcceleration: ', a0 |
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229 | print '\tMaximum velocity: ', um |
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230 | print '\tChar time: ', t0 |
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231 | print '\tChar distance: ', s0 |
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232 | print '\nThe tsunami ...' |
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233 | print '\tWavelength: ', w |
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234 | print '\t2D amplitude: ', a2D |
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235 | print '\t3D amplitude: ', a3D |
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236 | |
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237 | #keep an eye on some of the assumptions built into the maths |
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238 | |
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239 | if ((slope < 10) or (slope > 30)): |
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240 | if verbose is True: |
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241 | print 'WARNING: slope out of range (10 - 30 degrees) ', slope |
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242 | if ((depth/length < 0.34) or (depth/length > 0.5)): |
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243 | if verbose is True: |
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244 | print 'WARNING: d/b out of range (0.34 - 0.5) ', depth/length |
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245 | if ((thickness/length < 0.10) or (thickness/length > 0.15)): |
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246 | if verbose is True: |
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247 | print 'WARNING: T/b out of range (0.10 - 0.15) ', thickness/length |
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248 | if ((radius/length < 1.0) or (radius/length > 2.0)): |
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249 | if verbose is True: |
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250 | print 'WARNING: R/b out of range (1 - 2) ', radius/length |
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251 | if ((dphi < 0.10) or (dphi > 0.52)): |
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252 | if verbose is True: |
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253 | print 'WARNING: del_phi out of range (0.10 - 0.52) ', dphi |
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254 | if ((gamma < 1.46) or (gamma > 2.93)): |
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255 | if verbose is True: |
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256 | print 'WARNING: gamma out of range (1.46 - 2.93) ', gamma |
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257 | |
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258 | return Double_gaussian(a3D=a3D, wavelength=w, width=width, \ |
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259 | x0=x0, y0=y0, alpha=alpha, \ |
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260 | dx=dx, kappa=kappa, kappad=kappad, zsmall=zsmall) |
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261 | |
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262 | # |
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263 | # Double_gaussian class |
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264 | # |
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265 | |
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266 | """This is a callable class representing the initial water displacment |
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267 | generated by a sediment slide or slump. |
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268 | |
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269 | Using input parameters: |
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270 | |
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271 | Required |
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272 | w initial wavelength of tsunami |
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273 | a3D 3D initial amplitude of tsunami |
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274 | width width of smf |
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275 | |
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276 | Optional |
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277 | x0 x origin of smf |
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278 | y0 y origin of smf |
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279 | alpha angular orientation of smf in xy plane (0) |
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280 | dx offset of second Gaussian (0.2*width of first Gaussian) |
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281 | kappa multiplier for sech^2 function (3.0) |
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282 | kappad multiplier for second Gaussian function (0.8) |
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283 | zsmall an amount near to zero (0.01) |
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284 | |
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285 | Adrian Hitchman |
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286 | Geoscience Australia, June 2005 |
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287 | """ |
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288 | |
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289 | class Double_gaussian: |
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290 | |
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291 | def __init__(self, a3D, wavelength, width, x0, y0, alpha, \ |
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292 | dx, kappa, kappad, zsmall): |
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293 | self.a3D = a3D |
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294 | self.wavelength = wavelength |
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295 | self.width = width |
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296 | self.x0 = x0 |
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297 | self.y0 = y0 |
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298 | self.alpha = alpha |
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299 | self.kappa = kappa |
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300 | self.kappad = kappad |
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301 | |
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302 | if dx is None: |
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303 | self.determineDX(zsmall=zsmall) |
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304 | else: |
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305 | self.dx = dx |
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306 | |
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307 | def __call__(self, x, y): |
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308 | """Make Double_gaussian a callable object. |
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309 | |
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310 | If called as a function, this object returns z values representing |
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311 | the initial 3D distribution of water heights at the points (x,y) |
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312 | produced by a submarine mass failure. |
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313 | """ |
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314 | |
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315 | from math import sin, cos, radians, exp, cosh |
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316 | from Numeric import zeros, Float |
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317 | |
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318 | #ensure vectors x and y have the same length |
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319 | N = len(x) |
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320 | assert N == len(y) |
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321 | |
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322 | am = self.a3D |
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323 | wa = self.wavelength |
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324 | wi = self.width |
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325 | x0 = self.x0 |
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326 | y0 = self.y0 |
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327 | alpha = self.alpha |
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328 | dx = self.dx |
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329 | kappa = self.kappa |
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330 | kappad = self.kappad |
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331 | |
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332 | #double Gaussian calculation assumes water displacement is oriented |
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333 | #E-W, so, for displacement at some angle alpha clockwise from the E-W |
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334 | #direction, rotate (x,y) coordinates anti-clockwise by alpha |
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335 | |
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336 | cosa = cos(radians(alpha)) |
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337 | sina = sin(radians(alpha)) |
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338 | |
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339 | xr = ((x-x0) * cosa - (y-y0) * sina) + x0 |
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340 | yr = ((x-x0) * sina + (y-y0) * cosa) + y0 |
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341 | |
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342 | z = zeros(N, Float) |
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343 | |
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344 | for i in range(N): |
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345 | try: |
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346 | z[i] = -am / ((cosh(kappa*(yr[i]-y0)/(wi+wa)))**2) \ |
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347 | * (exp(-((xr[i]-x0)/wa)**2) - \ |
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348 | kappad*exp(-((xr[i]-dx-x0)/wa)**2)) |
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349 | except OverflowError: |
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350 | pass |
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351 | |
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352 | return z |
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353 | |
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354 | def determineDX(self, zsmall): |
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355 | """Determine a suitable offset for the second Gaussian function. |
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356 | |
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357 | A suitable offset for the second Gaussian function is taken to |
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358 | be some fraction of the 'width' of the first Gaussian function. |
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359 | |
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360 | The 'width' of the first Gaussian is obtained from the range of |
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361 | the x coordinates over which the function takes values from |
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362 | 'near zero', through 1, and back to 'near zero'. |
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363 | |
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364 | The parameter zsmall passed to this function specifies how much |
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365 | 'near zero' is. |
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366 | """ |
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367 | |
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368 | from math import sqrt, log, e |
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369 | |
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370 | a = self.a3D |
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371 | c = self.wavelength |
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372 | |
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373 | self.dx = 2.0 * (c * sqrt(-log((zsmall/a),e))) / 5.0 |
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