Cnoidal Fifth Order Wave

Calculate Cnoidal wave velocity, acceleration and surface profile using Fentons 1999 fifth order wave method.

The Cnoidal wave is defined by the elliptic modulus m, the wave trough depth w, and the wave alpha parameter α. The Cnoidal wave model is a truncated series and is only valid within certain ranges. The Cnoidal wave theory is not recommended where the wavelength over water depth ratio (Lod) is less than 8. The recommended wave type is displayed below the calc bar.

Note : The cnoidal wave theory uses a truncated infinite series. The truncated series is only valid for conditions where the series converges (m > 0.8). For deep water waves with small m, the series does not converge (use the Stokes wave instead).

Check that the convergence is close to or equal to one. The wave period should be measured at zero current velocity to avoid Doppler effects.

Reference : J D Fenton, The Cnoidal Theory Of Water Waves, Developments in Offshore Engineering, Gulf, Houston, chapter 2, 1999

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CALCULATOR : Cnoidal Wave Velocity And Acceleration Using Fenton's 1990 Method [FREE] ±

Calculate cnoidal wave velocity and acceleration.

The wave can be defined by the wave height and water depth, and either the wave length, or the wave period. The wave velocity and acceleration can be calculated at either the wave surface (varies with phase angle), the wave trough height, the seabed, or at a user defined height above the seabed.

Use the Result Plot option to plot wave velocity and acceleration versus phase angle.

Tool Input

lttype : Wave Parameter Type
- Tu : User Defined Wave Period
- Lu : User Defined Wave Length
ztype : Height Above Seabed Type
- zhu : User Defined Height
d : Water Depth
H : Wave Height
Θ : Wave Phase Angle

Tool Output

α : Cnoidal Alpha Parameter
δ : Cnoidal Delta Parameter
η : Wave Surface Relative To Mean Sea Level
Ah : Wave Horizontal Acceleration
Am : Wave Acceleration Magnitude
Av : Wave Vertical Acceleration
L : Wave Length
Lo : Deep Water Wave Length
T : Wave Period
Vh : Wave Horizontal Velocity
Vm : Wave Velocity Magnitude
Vv : Wave Vertical Velocity
c : Wave Celerity
cvg : Convergence Check
m : Cnoidal Elliptic Parameter
zh : Height Above Seabed
zs : Wave Surface Height Above Seabed
zw : Wave Trough Height Above Seabed

CALCULATOR : Cnoidal Wave Elliptic Integral And Elliptic Function [FREE] ±

Calculate cnoidal wave elliptic integrals, and elliptic functions using Fentons approximate method.

The elliptic integrals are calculated from the elliptic parameter m. The elliptic functions are calculated from the wave phase angle and the elliptic parameter. Fentons approximate method is valid for m > 0.1 (less than 1% error). Use the Result Plot option to plot the elliptic integrals versus elliptic parameter, or the elliptic functions versus phase angle.

Note : The wave phase angle is not equal to the elliptic angle Φ. The cnoidal wave is periodic on an elliptic angle of π

Tool Input

m : Wave Elliptic Parameter
Θ : Wave Phase Angle

Tool Output

E : Complete Elliptic Integral Of The Second Kind
K : Complete Elliptic Integral Of The First Kind
K' : Complemetary Elliptic Integral Of The First Kind
cn : Elliptic cn
dn : Elliptic dn
e(m) : Elliptic e(m)
sn : Elliptic sn

CALCULATOR : Airy Stokes And Cnoidal Wave Selection [FREE] ±

Select the most suitable wave theory (Airy, Stokes or cnoidal) from the dimensionless wave height, water depth and wave length (combined ISO method and Fenton method).

Use the Result Plot option to plot either dimensionless wave height versus water depth (ISO method), or dimensionless wave height versus wave length (Fenton method). The observed breaking wave height is lower than theoretical wave height.

Tool Input

wavetype : Wave Type
lttype : Wave Parameter Type
- Tu : User Defined Wave Period
- Lu : User Defined Wave Length
d : Water Depth
H : Wave Height

Tool Output

H* : Dimensionless Wave Height Number
L : Wave Length
T : Wave Period
Ur : Dimensionless Ursell Number
c : Wave Celerity
cvg : Convergence Check
d* : Dimensionless Water Depth Number
f : Wave Frequency
hod : Wave Height Over Water Depth Ratio
k : Wave Number
lod : Wave Length Over Water Depth Ratio
w : Wave Trough Height Above Seabed

CALCULATOR : Airy Stokes And Cnoidal Wave Velocity And Acceleration [FREE] ±

Calculate cnoidal wave reduced velocity and acceleration.

The wave parameters can be defined by the wave height and water depth, and either the wave length, or the wave period. The wave velocity and acceleration can be calculated at either the wave surface (varies with phase angle), the wave trough height, the seabed, or at a user defined height above the seabed. Wave phase angle can be defined from angle Θ, displacement x or ± time t (Θ = 360 (x - c t) / L).

Use the Result Plot option to plot either wave velocity and acceleration versus phase angle, wave profile versus wave type, or wave acceleration or velocity versus wave type.

Tool Input

wavetype : Wave Type
lttype : Wave Parameter Type
- Tu : User Defined Wave Period
- Lu : User Defined Wave Length
phasetype : Phase Angle Type
- Θu : User Defined Phase Angle
- xu : User Defined Displacement
- tu : User Defined Time
ztype : Height Above Seabed Type
- zhu : User Defined Height
d : Water Depth
H : Wave Height

Tool Output

Θ : Wave Phase Angle
η : Wave Surface Relative To Mean Sea Level
Ah : Wave Horizontal Acceleration
Am : Wave Acceleration Magnitude
Av : Wave Vertical Acceleration
L : Wave Length
T : Wave Period
Vh : Wave Horizontal Velocity
Vm : Wave Velocity Magnitude
Vv : Wave Vertical Velocity
c : Wave Celerity
cvg : Convergence Check
zh : Height Above Seabed
zs : Wave Surface Height Above Seabed
zw : Wave Trough Height Above Seabed

CALCULATOR : Airy Stokes And Cnoidal Wave Velocity And Acceleration From Return Period Data [PLUS] ±

Calculate cnoidal wave reduced velocity and acceleration from return period data.

Wave height and period can be calculated using linear regression on either a Weibull, Gumbel, or Frechet distribution. The three parameter option can be used with the Weibull and Frechet distributions to account for the minimum amplitude. Enter data points as sets of three data points separated by a comma or tab (R, H, T), with each set on a new line. Data can also be copied and pasted from a spreadsheet.

A relative heading of zero degrees is parallel to the structure, or ninety degrees is perpendicular to the structure. Wave phase angle can be defined from angle Θ, displacement x or ± time t (Θ = 360 (x - c t) / L).

Use the Result Plot option to plot either wave velocity and acceleration versus phase angle, wave profile versus wave type, or wave acceleration or velocity versus wave type.

Tool Input

sptype : Wave Sample Period
- Su : User Defined Wave Sample Period
offtype : Probability Function Parameter Type
- Hou : User Defined Minimum Wave Height
- Tou : User Defined Minimum Wave Period
rtype : Linear Regression Type
wavetype : Wave Type
phasetype : Phase Angle Type
- Θu : User Defined Phase Angle
- x : User Defined Displacement
- t : User Defined Time
ztype : Design Elevation Above Seabed Type
- Zu : User Defined Elevation
Rdata : Return Period Data
Hdata : Wave Height Data
Tdata : Wave Period Data
R : Design Return Period
d : Water Depth

Tool Output

Θ : Wave Phase Angle
η : Wave Surface Relative To Mean Sea Level
Ah : Wave Horizontal Acceleration
Am : Wave Acceleration Magnitude
Av : Wave Vertical Acceleration
Ho : Minimum Wave Height
Hsm : Mean Wave Height
Hw : Wave Height
L : Wave Length
S : Sample Period
To : Minimum Wave Period
Tpm : Mean Wave Period
Tw : Wave Period
Vh : Wave Horizontal Velocity
Vm : Wave Velocity Magnitude
Vv : Wave Vertical Velocity
Z : Design Elevation Above Seabed
Zs : Wave Surface Height Above Seabed
Zw : Wave Trough Height Above Seabed
c : Wave Celerity
cvg : Convergence Factor

CALCULATOR : Cnoidal Wave Spreading And Velocity Reduction Factor [FREE] ±

Calculate cnoidal wave velocity reduction factor from relative heading and spreading factor.

The spreading factor accounts for wave 'choppiness', the tendency for intersecting waves from multiple directions. Smaller spreading factor indicates more choppiness. A relative heading of zero degrees is parallel to the pipeline or structure, or ninety degrees is perpendicular to the pipeline or structure.

Use the Result Plot option to plot reduction factor versus relative heading and spreading factor.

Tool Input

sfactype : Shape Factor Type
- su : User Defined Spreading Factor
rdtype : Velocity Reduction Factor Type
Φ : Relative Wave Heading

Tool Output

Rdw : Wave Velocity Reduction Factor
kw : Wave Shape kw Factor
sf : Wave Shape Factor