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

Stokes Wave Calculators

Description : Stokes fifth order wave calculators.

Discussion : The Stokes fifth order wave theory models water waves as a fifth order harmonic, rather than the first order harmonic used by Airy wave theory. The wavelength L and the Stokes lamda parameter λ are calculated by solving

λ = π H / ( L ( ( B35 + B55 ) λ 4 + B33 λ 2 + 1 ) )
L = Lo tanh( 2 π d / L )( C2 λ 4 + C1 λ 2 + 1 )

where

λ = Stokes lamda Coefficient
L = Ocean Wave Length
Lo = Deep Water Wave Length = ( 9.80665 T 2 ) / ( 2 π )
B33, B35, B55, C1 and C2 are coefficients dependant on L

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.

Figures :

References :

Calculator Tools In This Module:

CALC : Ocean : Wave 101 : Stokes Fifth Order Wave Theory : Calculator
CALC : Ocean : Wave 102 : Stokes Wave Surface Profile And Wavelength Only : Calculator
CALC : Ocean : Wave 103 : Stokes Wave Horizontal Velocity And Acceleration Only : Calculator
CALC : Ocean : Wave 106 : Stokes Wave Check Calculations : Calculator


Link

Module List

CALC : Ocean : Wave 101 : Stokes Fifth Order Wave Theory : Calculator

Description : Calculate the Stokes fifth order wave parameters.

Discussion : Calculates all Stokes wave parameters : 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 has a value close to or equal to one. The working of the functinos is not shown in the check calculations. Use the Airy wave check calculator to check the function workings.

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
  • H* = Dimensionless Wave Height
  • 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
  • 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 )
list( y , w ) = StokesP( H , d , L , λ , Θ )
list( Up , dUp , c ) = StokesU( d , T , L , λ , Θ , z )
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 102 : Stokes Wave Surface Profile And Wavelength Only : Calculator

Description : Calculate the Stokes fifth order wave length and surface profile.

Discussion : Calculates only the wave length and surface profile. The surface profile height is measured from sea level. Check that the convergence check value is close to or equal to one. The function working is not included in the check calculations. Use the Stokes wave check calculator to check the function working.

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 ) = StokesTP( H , d , T , Θ )

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CALC : Ocean : Wave 103 : Stokes Wave Horizontal Velocity And Acceleration Only : Calculator

Description : Calculate Stokes fifth order wave horizontal water particle motion versus phase angle.

Discussion : Calculates only the horizontal water particle velocity and acceleration. Check that the convergence check is close to or equal to one. For brevity the function working is not shown in the check calculations. Use the Stokes wave check tool to check the function working.

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 ) = StokesTU( H , d , T , Θ , z )

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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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