So shall my word be that goes forth from my mouth. It shall not return to me void, but it shall accomplish what I please, and it shall prosper in the thing for which I sent it. Isiah 55:11
Pipeng Free Online Software : Wave Check Calculators
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Pipeng : Wave Check Calculations : Cnoidal Stokes And Airy Wave Calculation Module

Wave Check Calculators

Description : Water wave check calculations.

Discussion : Airy Stokes and Cnoidal wave models with check calculations. Use the check tools to validate wave results used in other tools.

Figures :

Calculator Tools In This Module:

CALC : Ocean : Wave 026 : Airy Wave Check Calculations : Calculator
CALC : Ocean : Wave 106 : Stokes Wave Check Calculations : Calculator
CALC : Ocean : Wave 206 : Cnoidal Wave Check Calculations : Calculator


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Module List

CALC : Ocean : Wave 026 : Airy Wave Check Calculations : Calculator

Description : Calculate Airy linear gravity wave check values.

Discussion : Use this tool to check the Airy wave calculations. Check that the convergence check is close to or equal to one, and that the check wavelength is equal to the Airy wave length.

Input Variables :

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

Output Variables :

  • H* = Dimensionless Wave Height
  • L = Ocean Wave Length
  • Lchk = Ocean Wave Length Check
  • Ldw = Deep Water Wave Length
  • Ua = Horizontal Velocity Amplitude
  • Up = Horizontal Velocity Phase
  • Va = Vertical Velocity Amplitude
  • Vp = Vertical Velocity Phase
  • c = Ocean Wave Celerity
  • cvg = Convergence Check
  • d* = Dimensionless Water Depth
  • dUa = Horizontal Acceleration Amplitude
  • dUp = Horizontal Acceleration Phase
  • dVa = Vertical Acceleration Amplitude
  • dVp = Vertical Acceleration Phase
  • f = Ocean Wave Frequency
  • hod = Wave Height Over Water Depth Ratio
  • lod = Wave Length Over Water Depth Ratio
  • n = Ocean Wave Number
  • w = Wave Trough Depth
  • y = Ocean Wave Surface Profile

Calculation :

list( L , cvg ) = AiryL( d , T )
Ldw = ( 9.80665 T 2 ) / ( 2 π )
Lchk = Ldw tanh( 2 π d / L )
y = H / 2 cos( Θ )
Ua = ( ( π H ) / T )( cosh( 2 π z / L ) / sinh( 2 π d / L ) )
dUa = ( 2 π 2 H / T 2 )( cosh( 2 π z / L ) / sinh( 2 π d / L ) )
Va = ( ( π H ) / T )( sinh( 2 π z / L ) / sinh( 2 π d / L ) )
dVa = ( 2 π 2 H / T 2 )( sinh( 2 π z / L ) / sinh( 2 π d / L ) )
Up = ( ( π H ) / T )( cosh( 2 π z / L ) / sinh( 2 π d / L ) )cos( Θ )
dUp = ( 2 π 2 H / T 2 )( cosh( 2 π z / L ) / sinh( 2 π d / L ) )sin( Θ )
Vp = ( ( π H ) / T )( sinh( 2 π z / L ) / sinh( 2 π d / L ) )sin( Θ )
dVp = - ( 2 π 2 H / T 2 )( sinh( 2 π z / L ) / sinh( 2 π d / L ) )cos( Θ )
c = L / T
w = d - H / 2
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 106 : Stokes Wave Check Calculations : Calculator

Description : Calculate Stokes fifth order wave check values.

Discussion : Use this tool to check the Stokes wave calculations. Check that the convergence check is close to or equal to one, and that the check wavelength and lamda values are equal to the Stokes wave length and lamda values.

Input Variables :

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

Output Variables :

  • λ = Stokes Lamda Coefficient
  • λk = Stokes Lamda Coefficient Check
  • H* = Dimensionless Wave Height
  • L = Ocean Wave Length
  • Lchk = Ocean Wave Length Check
  • Ldw = Deep Water Wave Length
  • Up = Horizontal Velocity Phase
  • c = Ocean Wave Celerity
  • chk = Check Numbers
  • cvg = Convergence Check
  • d* = Dimensionless Water Depth
  • dUp = Horizontal Acceleration Phase
  • f = Ocean Wave Frequency
  • hod = Wave Height Over Water Depth Ratio
  • lod = Wave Length Over Water Depth Ratio
  • n = Ocean Wave Number
  • w = Wave Trough Depth
  • y = Ocean Wave Surface Profile

Calculation :

list( L , λ , cvg ) = StokesL( d , T , H )
CH = cosh( 2 π d / L )
SH = sinh( 2 π d / L )
B33 = 3 ( 8 CH 6 + 1 ) / ( 64 SH 6 )
B35 = ( 88128 CH 14 - 208224 CH 12 + 70848 CH 10 + 54000 CH 8 - 21816 CH 6 + 6264 CH 4 )
- 54 CH 2 - 81 / ( 12288 SH 12 ( 6 CH 2 - 1 ) )
B55 = ( 192000 CH 16 - 262720 CH 14 + 83680 CH 12 + 20160 CH 10 - 7280 CH 8 + 7160 CH 6 )
- 1800 CH 4 - 1050 CH 2 + 225 / ( 12288 SH 10 ( 6 CH 2 - 1 )( 8 CH 4 - 11 CH 2 + 3 ) )
C1 = ( 8 CH 4 - 8 CH 2 + 9 ) / ( 8 SH 4 )
C2 = ( 3840 CH 12 - 4096 CH 10 + 2592 CH 8 - 1008 CH 6 + 5944 CH 4 - 1830 CH 2 + 147 )
1 / ( 512 SH 10 ( 6 CH 2 - 1 ) )
Ldw = ( 9.80665 T 2 ) / ( 2 π )
λk = π H / ( L ( ( B35 + B55 ) λ 4 + B33 λ 2 + 1 ) )
Lchk = Ldw tanh( 2 π d / L )( C2 λ 4 + C1 λ 2 + 1 )
B22 = ( ( 2 CH 2 + 1 ) CH ) / ( 4 SH 3 )
B24 = CH ( 272 CH 8 - 504 CH 6 - 192 CH 4 + 322 CH 2 + 21 ) / ( 384 SH 9 )
B44 = CH ( 768 CH 10 - 448 CH 8 - 48 CH 6 + 48 CH 4 + 106 CH 2 - 21 ) / ( 384 SH 9 ( 6 CH 2 - 1 ) )
y = ( λ cos( Θ ) + ( λ 2 B22 + λ 4 B24 )cos( 2 Θ ) + ( λ 3 B33 + λ 5 B35 )cos( 3 Θ ) )
+ λ 4 B44 cos( 4 Θ ) + λ 5 B55 cos( 5 Θ ) L / ( 2 π )
w = d - H + ( λ + ( λ 2 B22 + λ 4 B24 ) + ( λ 3 B33 + λ 5 B35 ) )
+ λ 4 B44 + λ 5 B55 L / ( 2 π )
A11 = 1 / SH
A13 = ( - CH 2 ( 5 CH 2 + 1 ) ) / ( 8 SH 5 )
A15 = - ( 1184 CH 10 - 1440 CH 8 - 1992 CH 6 + 2641 CH 4 - 249 CH 2 + 18 ) / ( 1536 SH 11 )
A22 = 3 / ( 8 SH 4 )
A24 = ( 192 CH 8 - 424 CH 6 - 312 CH 4 + 480 CH 2 - 17 ) / ( 768 SH 10 )
A33 = ( 13 - 4 CH 2 ) / ( 64 SH 7 )
A35 = ( 512 CH 12 + 4224 CH 10 - 6800 CH 8 - 12808 CH 6 + 16704 CH 4 - 3154 CH 2 + 107 )
1 / ( ( 4096 SH 13 )( 6 CH 2 - 1 ) )
A44 = ( 80 CH 6 - 816 CH 4 + 1338 CH 2 - 197 ) / ( 1536 SH 10 ( 6 CH 2 - 1 ) )
A55 = - ( 2880 CH 10 - 72480 CH 8 + 324000 CH 6 - 432000 CH 4 + 163470 CH 2 - 16245 )
1 / ( 61440 SH 11 ( 6 CH 2 - 1 )( 8 CH 4 - 11 CH 2 + 3 ) )
U1 = ( λ A11 ) + ( λ 3 A13 ) + ( λ 5 A15 )
U2 = ( λ 2 A22 ) + ( λ 4 A24 )
U3 = ( λ 3 A33 ) + ( λ 5 A35 )
U4 = ( λ 4 A44 )
U5 = ( λ 5 A55 )
ZOL = 2 π z / L
c = L / T
Up = c U1 cosh( ZOL )cos( Θ ) + 2 c U2 cosh( 2 ZOL )cos( 2 Θ ) + 3 c U3 cosh( 3 ZOL )cos( 3 Θ )
+ 4 c U4 cosh( 4 ZOL )cos( 4 Θ ) + 5 c U5 cosh( 5 ZOL )cos( 5 Θ )
dUp = ( 2 π ) / L ( c 2 U1 cosh( ZOL )sin( Θ ) + ( 2 c ) 2 U2 cosh( 2 ZOL )sin( 2 Θ ) )
+ ( 3 c ) 2 U3 cosh( 3 ZOL )sin( 3 Θ ) + ( 4 c ) 2 U4 cosh( 4 ZOL )sin( 4 Θ )
+ ( 5 c ) 2 U5 cosh( 5 ZOL )sin( 5 Θ )
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 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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