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Pipeng Free Online Software : Cnoidal Wave Calculators
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Pipeng : Cnoidal Fifth Order Waves Calculation Module

Cnoidal Wave Calculators

Description : Cnoidal fifth order ocean wave calculators using Fentons method.

Discussion : The Cnoidal wave is defined by the elliptic modulus k, the wave trough depth w, and the wave alpha parameter α. The Cnoidal wave model is a truncated series and is only valid within certian ranges. The water wave theory selection figure shows the range of validity of the wave models based on the dimensionless water depth and the dimensionless wave height. It is advisable to also check the wave parameters for a range of elevation and phase angle. The Cnoidal wave theory is not recommended where the wavelength over water depth ratio (Lod) is less than 8.

Figures :

References :

Calculator Tools In This Module:

CALC : Ocean : Wave 201 : Cnoidal Fifth Order Wave Theory : Calculator
CALC : Ocean : Wave 202 : Cnoidal Wave Surface Profile And Wavelength Only : Calculator
CALC : Ocean : Wave 203 : Cnoidal Wave Horizontal Velocity And Acceleration Only : Calculator
CALC : Ocean : Wave 206 : Cnoidal Wave Check Calculations : Calculator


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Module List

CALC : Ocean : Wave 201 : Cnoidal Fifth Order Wave Theory : Calculator

Description : Calculate the Cnoidal fifth order wave parameters.

Discussion : Calculates all Cnoidal wave parameters : the wave k modulus, trough depth, wave height over trough depth ratio, celerity, alpha parameter, wave length, surface profile, horizontal velocity and acceleration, wave number, wave celerity, wave frequency, water depth over wave length ratio (dol), wave height over water depth ratio (hod), dimensionless wave height and dimensionless water depth. Check that the convergence check is close to or equal to one. For brevity the working of the functions is not shown in the check calculations. To check the working of the functions use the Cnoidal wave check tool.

Input Variables :

  • Θ = Wave Phase Angle
  • H = Ocean Wave Height
  • T = Ocean Wave Period
  • d = Water Depth
  • z = Height Above Seabed

Output Variables :

  • α = Wave Alpha Parameter
  • H* = Dimensionless Wave Height
  • HoW = Wave Height Over Trough Depth Ratio
  • L = Ocean Wave Length
  • Up = Horizontal Velocity Phase
  • c = Ocean Wave Celerity
  • cvg = Convergence Check
  • d* = Dimensionless Water Depth
  • dUp = Horizontal Acceleration Phase
  • f = Ocean Wave Frequency
  • hod = Wave Height Over Water Depth Ratio
  • k = Elliptical k Modulus
  • lod = Wave Length Over Water Depth Ratio
  • n = Ocean Wave Number
  • w = Wave Trough Depth
  • y = Ocean Wave Surface Profile

Calculation :

list( k , cvg ) = CnoidalK( H , d , T )
list( w , HoW ) = CnoidalW( H , d , k )
list( c , α , L ) = CnoidalL( k , w , HoW )
y = CnoidalP( d , k , w , HoW , Θ )
list( Up , dUp ) = CnoidalU( k , w , HoW , Θ , z , c , α )
n = ( 2 π ) / L
f = 1 / T
hod = H / d
lod = L / d
H* = H / ( 9.80665 T 2 )
d* = d / ( 9.80665 T 2 )

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CALC : Ocean : Wave 202 : Cnoidal Wave Surface Profile And Wavelength Only : Calculator

Description : Calculate the Cnoidal fifth order wave surface profile and wavelength.

Discussion : Calculates only the surface profile and wave length. The wave surface profile is the height of the wave surface above sea level. Check that the convergence check is close to or equal to one. For brevity the working of the functions is not shown in the check calculations. To check the working of the functions use the Cnoidal wave check calculator.

Input Variables :

  • Θ = Wave Phase Angle
  • H = Ocean Wave Height
  • T = Ocean Wave Period
  • d = Water Depth

Output Variables :

  • L = Ocean Wave Length
  • cvg = Convergence Check
  • y = Ocean Wave Surface Profile

Calculation :

list( L , y , cvg ) = CnoidalTP( H , d , T , Θ )

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CALC : Ocean : Wave 203 : Cnoidal Wave Horizontal Velocity And Acceleration Only : Calculator

Description : Calculate the Cnoidal fifth order wave horizontal velocity and acceleration versus phase angle.

Discussion : Calculates only the horizontal velcity and acceleration. Check that the convergence check is close to or equal to one. For brevity the working of the functions is not shown in the check calculations. To check the working of the functions use the Cnoidal wave check tool.

Input Variables :

  • Θ = Wave Phase Angle
  • H = Ocean Wave Height
  • T = Ocean Wave Period
  • d = Water Depth
  • z = Height Above Seabed

Output Variables :

  • Up = Horizontal Velocity Phase
  • cvg = Convergence Check
  • dUp = Horizontal Acceleration Phase

Calculation :

list( Up , dUp , cvg ) = CnoidalTU( H , d , T , Θ , z )

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CALC : Ocean : Wave 206 : Cnoidal Wave Check Calculations : Calculator

Description : Calculate the Cnoidal fifth order check values.

Discussion : Use this tool to check the Cnoidal wave calculations. Check that the convergence check is close to or equal to one and that the k modulus = the k check modulus.

Input Variables :

  • Θ = Wave Phase Angle
  • H = Ocean Wave Height
  • T = Ocean Wave Period
  • d = Water Depth
  • z = Height Above Seabed

Output Variables :

  • α = Wave Alpha Parameter
  • H* = Dimensionless Wave Height
  • HoW = Wave Height Over Trough Depth Ratio
  • L = Ocean Wave Length
  • Up = Horizontal Velocity Phase
  • c = Ocean Wave Celerity
  • chk = Check Numbers
  • d* = Dimensionless Water Depth
  • dUp = Horizontal Acceleration Phase
  • f = Ocean Wave Frequency
  • hod = Wave Height Over Water Depth Ratio
  • k = Elliptical k Modulus
  • kchk = Elliptical k Check
  • lod = Wave Length Over Water Depth Ratio
  • n = Ocean Wave Number
  • w = Wave Trough Depth
  • y = Ocean Wave Surface Profile

Calculation :

list( k , [cvg] ) = CnoidalK( H , d , T )
hod = H / d
KK = EllipticCK( k )
EE = EllipticCE( k )
M1 = k k
M2 = M1 M1
M3 = M1 M2
M4 = M1 M3
M5 = M1 M4
EOM1 = hod / M1
EOM2 = EOM1 EOM1
EOM3 = EOM1 EOM2
EOM4 = EOM1 EOM3
EOM5 = EOM1 EOM4
EK1 = EE / KK
EK2 = EK1 EK1
EK3 = EK1 EK2
EK4 = EK1 EK3
EK5 = EK1 EK4
SK1 = EOM1 ( 1 ( 1 0.25 ) + M1 ( 1 - 0.125 ) )
SK2 = EOM2 ( 1 ( 1 0.01485 + EK1 - 1.08333 + EK2 1 ) + M1 ( 1 - 0.01458 + EK1 0.54167 ) + M2 ( 1 - 0.07656 ) )
SK3 = EOM3 ( 1 ( 1 0.36121 + EK1 2.50417 + EK2 - 4.5 + EK3 2 ) + M1 ( 1 - 0.54182 + EK1 - 2.50417 + EK2 2.25 ) )
+ M2 ( 1 0.41216 + EK1 0.33229 ) + M3 ( 1 - 0.11578 )
SK4 = EOM4 ( 1 ( 1 - 1.86885 + EK1 - 4.22859 + EK2 15.19111 + EK3 - 13.66667 + EK4 4 ) )
+ M1 ( 1 3.73770 + EK1 6.34288 + EK2 - 15.19111 + EK3 6.83333 )
+ M2 ( 1 - 2.73031 + EK1 - 1.88433 + EK2 2.69111 ) + M3 ( 1 0.86147 + EK1 - 0.11498 ) + M4 ( 1 - 0.07582 )
kchk = T √( 9.80665 / d ) / ( ( 4 KK / √( 3 H / d ) )( 1 + SK1 + SK2 + SK3 + SK4 ) )
SW1 = EOM1 ( 1 ( 1 1 + EK1 - 1 ) + M1 ( 1 - 1 ) )
SW2 = EOM2 ( 1 ( 1 - 0.5 + EK1 0.5 ) + M1 ( 1 0.5 + EK1 - 0.25 ) )
SW3 = EOM3 ( 1 ( 1 0.665 + EK1 - 1.165 + EK2 0.5 ) + M1 ( 1 - 0.99750 + EK1 1.165 + EK2 - 0.25 ) + M2 ( 1 0.33250 + EK1 - 0.04 ) )
SW4 = EOM4 ( 1 ( 1 - 1.62667 + EK1 3.20667 + EK2 - 2.08 + EK3 0.5 ) + M1 ( 1 3.25333 + EK1 - 4.81 + EK2 2.08 + EK3 - 0.25 ) )
+ M2 ( 1 - 2.454 + EK1 2.17633 + EK2 - 0.1425 ) + M3 ( 1 0.82733 + EK1 - 0.2865 )
SW5 = EOM5 ( 1 ( 1 4.86659 + EK1 - 10.74409 + EK2 8.6225 + EK3 - 3.245 + EK4 0.5 ) )
+ M1 ( 1 - 12.16647 + EK1 21.48818 + EK2 - 12.93375 + EK3 3.245 + EK4 - 0.25 )
+ M2 ( 1 11.79929 + EK1 - 16.00776 + EK2 6.09025 + EK3 - 0.3075 )
+ M3 ( 1 - 5.53247 + EK1 5.26368 + EK2 - 0.8895 ) + M4 ( 1 1.03306 + EK1 - 0.20555 )
w = d ( 1 + SW1 + SW2 + SW3 + SW4 + SW5 )
HoW = H / w
EM1 = HoW / M1
EM2 = EM1 EM1
EM3 = EM1 EM2
EM4 = EM1 EM3
EM5 = EM1 EM4
SC1 = EM1 ( 1 ( 1 0.5 + EK1 - 1 ) )
SC2 = EM2 ( 1 ( 1 - 0.10833 + EK1 0.33333 ) + M1 ( 1 - 0.01667 + EK1 0.08333 ) + M2 ( 1 - 0.025 ) )
SC3 = EM3 ( 1 ( 1 - 0.1719 + EK1 0.09333 ) + M1 ( 1 0.33911 + EK1 - 0.34333 ) + M2 ( 1 - 0.16006 + EK1 0.21833 ) + M3 ( 1 0.04643 ) )
SC4 = EM4 ( 1 ( 1 0.02097 + EK1 0.3769 ) + M1 ( 1 0.17293 + EK1 - 0.68202 ) + M2 ( 1 - 0.56238 + EK1 1.04889 ) )
+ M3 ( 1 0.39861 + EK1 - 0.56668 ) + M4 ( 1 - 0.08531 )
SC5 = EM5 ( 1 ( 1 0.11046 + EK1 - 0.94038 ) + M1 ( 1 - 0.31285 + EK1 1.22117 ) + M2 ( 1 - 0.11262 + EK1 0.35314 ) )
+ M3 ( 1 0.91605 + EK1 - 1.75325 ) + M4 ( 1 - 0.73881 + EK1 1.00619 ) + M5 ( 1 0.15763 )
c = √( 9.80665 w ) ( 1 + SC1 + SC2 + SC3 + SC4 + SC5 )
SA1 = EM1 ( 1 0.25 + M1 - 0.875 )
SA2 = EM2 ( 1 0.03125 + M1 - 0.34375 + M2 0.86719 )
SA3 = EM3 ( 1 - 0.37743 + M1 0.51146 + M2 0.13743 + M3 - 0.8330 )
SA4 = EM4 ( 0.20322 + M1 0.44278 + M2 - 1.38945 + M3 0.54282 + M4 0.76773 )
α = √( 3 HoW / ( 4 k k ) ) ( 1 + SA1 + SA2 + SA3 + SA4 )
L = 2 KK w / α
Q = KK Θ / π
list( SN , CN , DN ) = EllipticFunc( Q , k )
CN1 = CN CN
CN2 = CN1 CN1
CN3 = CN2 CN1
CN4 = CN3 CN1
CN5 = CN4 CN1
SY1 = EM1 ( M1 ( CN1 1 ) )
SY2 = EM2 ( M2 ( CN1 - 0.75 + CN2 0.75 ) )
SY3 = EM3 ( M2 ( CN1 - 0.7625 + CN2 0.7625 ) + M3 ( CN1 1.3875 + CN2 - 2.65 + CN3 1.26250 ) )
SY4 = EM4 ( M2 ( CN1 - 0.80533 + CN2 0.80533 ) + M3 ( CN1 2.48904 + CN2 - 4.33146 + CN3 1.84242 ) )
+ M4 ( CN1 - 3.05188 + CN2 7.40646 + CN3 - 6.52546 + CN4 2.17088 )
SY5 = EM5 ( M2 ( CN1 0.43643 + CN2 - 0.43643 ) + M3 ( CN1 1.92280 + CN2 - 4.66167 + CN3 2.73888 ) )
+ M4 ( CN1 - 7.04588 + CN2 17.45561 + CN3 - 15.31697 + CN4 4.90723 )
+ M5 ( CN1 6.54722 + CN2 - 19.80887 + CN3 25.34187 + CN4 - 16.32709 + CN5 4.24687 )
y = w ( 1 + SY1 + SY2 + SY3 + SY4 + SY5 ) - d
ZH1 = ( z z ) / ( w w )
ZH2 = ZH1 ZH1
ZH3 = ZH2 ZH1
ZH4 = ZH3 ZH1
SU1 = EM1 ( 1 ( 1 ( - 0.5 1 ) + M1 ( 1 1 + - 1 CN1 ) ) )
SU2 = EM2 ( 1 ( 1 ( 0.225 1 ) + M1 ( - 0.6 1 + 1 CN1 ) + M2 ( 0.225 1 + - 1.25 CN1 + 1 CN2 ) ) )
+ ZH1 ( M1 ( 0.75 1 + - 1.5 CN1 ) + M2 ( - 0.75 1 + 3.0 CN1 + - 2.25 CN2 ) )
SU3 = EM3 ( 1 ( 1 ( - 0.07857 1 ) + M1 ( 0.14911 1 + - 0.4 CN1 ) + M2 ( 0.16161 1 + 1.35 CN1 + - 1.7 CN2 ) ) )
+ M3 ( - 0.17857 1 + - 0.475 CN1 + 1.9 CN2 + - 1.2 CN3 ) + ZH1 ( M1 ( - 0.375 1 + 0.75 CN1 ) )
+ M2 ( 0 1 + - 5.25 CN1 + 7.125 CN2 ) + M3 ( 0.375 1 + 3 CN1 + - 10.875 CN2 + 7.5 CN3 )
+ ZH2 ( M1 ( 0.1875 1 + - 0.375 CN1 ) + M2 ( - 0.5625 1 + 3.1875 CN1 + - 2.8125 CN2 ) )
+ M3 ( 0.375 1 + - 3.1875 CN1 + 5.625 CN2 + - 2.8125 CN3 )
SU4 = EM4 ( 1 ( 1 ( 0.39788 1 ) + M1 ( - 0.74576 1 + 0.09643 CN1 ) + M2 ( 0.73477 1 + - 0.46431 CN1 + 0.81967 CN2 ) ) )
+ M3 ( - 0.67908 1 + 0.55694 CN1 + - 3.04667 CN2 + 2.90133 CN3 ) + M4 ( 0.23701 1 + 0.17018 CN1 + 1.03417 CN2 + - 3.10267 CN3 + 1.576 CN4 )
+ ZH1 ( M1 ( 0.01875 1 + - 0.0375 CN1 ) + M2 ( - 0.09375 1 + 5.625 CN1 + - 7.25625 CN2 ) )
+ M3 ( 0.66562 1 + - 7.55625 CN1 + 29.025 CN2 + - 25.2 CN3 ) + M4 ( - 0.59063 1 + 3.0375 CN1 + - 17.07188 CN2 + 33.525 CN3 + - 18.9 CN4 )
+ ZH2 ( M1 ( 0 1 + 0 CN1 ) + M2 ( 0.14063 1 + - 4.78125 CN1 + 6.0 CN2 ) + M3 ( 0.14063 1 + 9.70312 CN1 + - 33.42187 CN2 + 24.375 CN3 ) )
+ M4 ( - 0.28125 1 + - 4.07812 CN1 + 27.09375 CN2 + - 42.42187 CN3 + 19.6875 CN4 )
+ ZH3 ( M1 ( 0.01875 1 + - 0.0375 CN1 ) + M2 ( - 0.17813 1 + 1.2375 CN1 + - 1.18125 CN2 ) )
+ M3 ( 0.31875 1 + - 3.4875 CN1 + 7.0875 CN2 + - 3.9375 CN3 ) + M4 ( - 0.15938 1 + 2.325 CN1 + - 7.0875 CN2 + 7.875 CN3 + - 2.95313 CN4 )
SU5 = EM5 ( 1 ( 1 ( - 0.82992 1 ) + M1 ( 1.76991 1 + - 0.78281 CN1 ) + M2 ( - 0.99629 1 + 0.89794 CN1 + 0.49893 CN2 ) ) )
+ M3 ( - 0.41272 1 + - 0.66324 CN1 + 1.20740 CN2 + - 2.26014 CN3 )
+ M4 ( 0.83138 1 + 0.70429 CN1 + - 2.71917 CN2 + 7.12267 CN3 + - 4.96993 CN4 )
+ M5 ( - 0.3425 1 + - 0.26144 CN1 + 0.43733 CN2 + - 2.69367 CN3 + 5.21186 CN4 + - 2.19396 CN5 )
+ ZH1 ( M1 ( - 0.5705 1 + 1.141 CN1 ) + M2 ( 1.44569 1 + - 4.45837 CN1 + 2.2795 CN2 ) )
+ M3 ( - 1.29006 1 + 5.8665 CN1 + - 31.13156 CN2 + 31.778 CN3 )
+ M4 ( - 0.38012 1 + - 2.12075 CN1 + 42.62962 CN2 + - 110.618 CN3 + 74.07 CN4 )
+ M5 ( 0.795 1 + - 2.23875 CN1 + - 11.405 CN2 + 61.73675 CN3 + - 91.44 CN4 + 42.552 CN5 )
+ ZH2 ( M1 ( - 0.01875 1 + 0.0375 CN1 ) + M2 ( - 0.45 1 + 3.84375 CN1 + - 3.91875 CN2 ) )
+ M3 ( 0.96797 1 + - 16.09219 CN1 + 63.225 CN2 + - 53.23125 CN3 )
+ M4 ( - 1.16016 1 + 16.30547 CN1 + - 110.93203 CN2 + 225.6375 CN3 + - 131.90625 CN4 )
+ M5 ( 0.66094 1 + - 5.39297 CN1 + 47.56406 CN2 + - 162.53203 CN3 + 204.75 CN4 + - 85.05 CN5 )
+ ZH3 ( M1 ( 0.00937 1 + - 0.01875 CN1 ) + M2 ( 0.11719 1 + - 1.59375 CN1 + 1.80938 CN2 ) )
+ M3 ( - 0.25313 1 + 9.84375 CN1 + - 31.55625 CN2 + 22.96875 CN3 )
+ M4 ( 0.05156 1 + - 13.2375 CN1 + 73.18125 CN2 + - 115.5 CN3 + 55.61719 CN4 )
+ M5 ( 0.075 1 + 4.875 CN1 + - 41.7 CN2 + 103.6875 CN3 + - 102.375 CN4 + 35.4375 CN5 )
+ ZH4 ( M1 ( 0.001 1 + - 0.00201 CN1 ) + M2 ( - 0.03415 1 + 0.26016 CN1 + - 0.25614 CN2 ) )
+ M3 ( 0.12656 1 + - 1.64833 CN1 + 3.70647 CN2 + - 2.21484 CN3 )
+ M4 ( - 0.15569 1 + 2.77634 CN1 + - 9.58259 CN2 + 11.70703 CN3 + - 4.74609 CN4 )
+ M5 ( 0.06228 1 + - 1.38817 CN1 + 6.38839 CN2 + - 11.70703 CN3 + 9.49219 CN4 + - 2.84766 CN5 )
Up = c - √( 9.80665 w ) ( 1 + SU1 + SU2 + SU3 + SU4 + SU5 )
SUA1 = EM1 ( 1 ( M1 ( - 1 1 ) ) )
SUA2 = EM2 ( 1 ( M1 ( 1 1 ) + M2 ( - 1.25 1 + 1 2 CN1 ) ) + ZH1 ( M1 ( - 1.5 1 ) + M2 ( 3.0 1 + - 2.25 2 CN1 ) ) )
SUA3 = EM3 ( 1 ( M1 ( - 0.4 1 ) + M2 ( 1.35 1 + - 1.7 2 CN1 ) + M3 ( - 0.475 1 + 1.9 2 CN1 + - 1.2 3 CN2 ) ) )
+ ZH1 ( M1 ( 0.75 1 ) + M2 ( - 5.25 1 + 7.125 2 CN1 ) + M3 ( 3 1 + - 10.875 2 CN1 + 7.5 3 CN2 ) )
+ ZH2 ( M1 ( - 0.375 1 ) + M2 ( 3.1875 1 + - 2.8125 2 CN1 ) + M3 ( - 3.1875 1 + 5.625 2 CN1 + - 2.8125 3 CN2 ) )
SUA4 = EM4 ( 1 ( M1 ( 0.09643 1 ) + M2 ( - 0.46431 1 + 0.81967 2 CN1 ) + M3 ( 0.55694 1 + - 3.04667 2 CN1 + 2.90133 3 CN2 ) ) )
+ M4 ( 0.17018 1 + 1.03417 2 CN1 + - 3.10267 3 CN2 + 1.576 4 CN3 )
+ ZH1 ( M1 ( - 0.0375 1 ) + M2 ( 5.625 1 + - 7.25625 2 CN1 ) + M3 ( - 7.55625 1 + 29.025 2 CN1 + - 25.2 3 CN2 ) )
+ M4 ( 3.0375 1 + - 17.07188 2 CN1 + 33.525 3 CN2 + - 18.9 4 CN3 )
+ ZH2 ( M1 ( 0 1 ) + M2 ( - 4.78125 1 + 6.0 2 CN1 ) + M3 ( 9.70312 1 + - 33.42187 2 CN1 + 24.375 3 CN2 ) )
+ M4 ( - 4.07812 1 + 27.09375 2 CN1 + - 42.42187 3 CN2 + 19.6875 4 CN3 )
+ ZH3 ( M1 ( - 0.0375 1 ) + M2 ( 1.2375 1 + - 1.18125 2 CN1 ) + M3 ( - 3.4875 1 + 7.0875 2 CN1 + - 3.9375 3 CN2 ) )
+ M4 ( 2.325 1 + - 7.0875 2 CN1 + 7.875 3 CN2 + - 2.95313 4 CN3 )
SUA5 = EM5 ( 1 ( M1 ( - 0.78281 1 ) + M2 ( 0.89794 1 + 0.49893 2 CN1 ) + M3 ( - 0.66324 1 + 1.20740 2 CN1 + - 2.26014 3 CN2 ) ) )
+ M4 ( 0.70429 1 + - 2.71917 2 CN1 + 7.12267 3 CN2 + - 4.96993 4 CN3 )
+ M5 ( - 0.26144 1 + 0.43733 2 CN1 + - 2.69367 3 CN2 + 5.21186 4 CN3 + - 2.19396 5 CN4 )
+ ZH1 ( M1 ( 1.141 1 ) + M2 ( - 4.45837 1 + 2.2795 2 CN1 ) + M3 ( 5.8665 1 + - 31.13156 2 CN1 + 31.778 3 CN2 ) )
+ M4 ( - 2.12075 1 + 42.62962 2 CN1 + - 110.618 3 CN2 + 74.07 4 CN3 )
+ M5 ( - 2.23875 1 + - 11.405 2 CN1 + 61.73675 3 CN2 + - 91.44 4 CN3 + 42.552 5 CN4 )
+ ZH2 ( M1 ( 0.0375 1 ) + M2 ( 3.84375 1 + - 3.91875 2 CN1 ) + M3 ( - 16.09219 1 + 63.225 2 CN1 + - 53.23125 3 CN2 ) )
+ M4 ( 16.30547 1 + - 110.93203 2 CN1 + 225.6375 3 CN2 + - 131.90625 4 CN3 )
+ M5 ( - 5.39297 1 + 47.56406 2 CN1 + - 162.53203 3 CN2 + 204.75 4 CN3 + - 85.05 5 CN4 )
+ ZH3 ( M1 ( - 0.01875 1 ) + M2 ( - 1.59375 1 + 1.80938 2 CN1 ) + M3 ( 9.84375 1 + - 31.55625 2 CN1 + 22.96875 3 CN2 ) )
+ M4 ( - 13.2375 1 + 73.18125 2 CN1 + - 115.5 3 CN2 + 55.61719 4 CN3 )
+ M5 ( - 41.7 2 CN1 + 103.6875 3 CN2 + - 102.375 4 CN3 + 35.4375 5 CN4 )
+ ZH4 ( M1 ( - 0.00201 1 ) + M2 ( 0.26016 1 + - 0.25614 2 CN1 ) + M3 ( - 1.64833 1 + 3.70647 2 CN1 + - 2.21484 3 CN2 ) )
+ M4 ( 2.77634 1 + - 9.58259 2 CN1 + 11.70703 3 CN2 + - 4.74609 4 CN3 )
+ M5 ( - 1.38817 1 + 6.38839 2 CN1 + - 11.70703 3 CN2 + 9.49219 4 CN3 + - 2.84766 5 CN4 )
dUp = 2 c α √( 9.80665 / w ) CN SN DN ( SUA1 + SUA2 + SUA3 + SUA4 + SUA5 )
n = ( 2 π ) / L
f = 1 / T
lod = L / d
H* = H / ( 9.80665 T 2 )
d* = d / ( 9.80665 T 2 )

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