### 1. Introduction

^{th}century by Airy that efforts to provide a mathematical description of wave motion began in earnest (Henry, 2008). Since the early development of the theory describing the waves using the perturbation method by Stokes (1847), many nonlinear solutions were obtained, both analytically and numerically.

*N*= 1, the stream function theory reduces to the Airy wave theory. As the breaking wave height is approached, more terms are required to accurately represent the wave. Det Norske Veritas (DNV, 2010) described that stream function wave theory has an error of more than 1% in deep water waves whose height is over 90% of the breaking limit even though the required order is increased (See Figs. 3–4 in DNV (2010)). Chaplin (1979) applied the Schmidt orthogonalization process as an alternative to Dean’s method and obtained improved results (Tao et al., 2007).

*N+2*unknown constants, which are N Fourier coefficients, the wave steepness, and the reference depth parameter, while the stream function contains

*2N+6*unknown constants (Rienecker and Fenton, 1981; Fenton, 1988). The wavelength and the reference depth parameter are not coupled to the Fourier coefficients because the dimensionless coordinate system was defined with the phase and the dimensionless elevation. Therefore, the wavelength and the reference depth parameter were determined independently. As a result, the required order was reduced in Shin (2016) and Shin (2021)

*N ≤*12. This is why the other approximations are unsuitable for deep water waves.

### 2. Complete Solution

*t*,

*x, y*) shown in Fig. 1. The origin is located on the still water line. The

*x*-axis is in the direction of the wave propagation, the

*y*-axis points upwards, and

*t*is time. The fluid domain is bounded by the free surface y =

*η*(

*t,x*)

*.*

*β, α*) is the dimensionless stationary frame shown in Fig. 2. The origin is located at the point under the crest on the reference line, which is the horizontal line passing through two points at

*η*(±90°) on the free surface. Therefore, the wave profile is a fixed, periodic, and even function in the system. The horizontal axis is the phase,

_{o}= η*β = kx*−

*ω*t, whose domain is −

*π*≤

*β*≤

*π,*where

*k*is the wave number defined by

*ω*is the angular frequency defined by

*.*The vertical axis is the dimensionless elevation from the reference line,

*α = k*(

*y*−

*η*) in

_{0}*α*≤

*ζ,*where

*ζ*=

*k*(

*η*−

*η*) is the dimensionless free surface elevation from the reference line.

_{0}*a*is a Fourier coefficient, which is dimensionless. The horizontal water-particle velocity is where

_{n}*c = ω/k*is the wave celerity. The vertical water-particle velocity is

*ζ*(

*±π*/2)

*=*0. Eq. (6) contains only

*N*coefficients, while Rienecker et al. (1981) contains

*N+5*unknown constants, which are

*N+1*coefficients, the wave number, celerity, volume rate, and Bernoulli’s constant. Compared to Rienecker et al. (1981), Eq. (6) is simple. Unknown constants are reduced dramatically by introducing the coordinate system in Fig. 2. Bernoulli’s equation is represented in the dimensionless coordinate system as follows: where

*ρ*is the water density, and

*U*and

*V*are dimensionless horizontal and vertical velocities defined by

*p*(

*±π*/2,0) = 0

*. U*(

_{0}= U*±π*/2,0) and

*V*(

_{0}= V*±π*/2,0) are velocities at the phase of

*β = ±π*/2 on the free surface. The linear steepness is defined with

*g*is gravity and

*H*is the wave height. The linear steepness is a constant for a particular wave. The other wave profile is calculated by applying the DFSBC (

*p*(

*β, ζ =*0) to Eq. (7), as follows:

*N*coefficients and wave steepness. In addition, Eq. (8) also satisfies that

*ζ*(

*±π/*2)

*=*0. Dimensionless wave height (steepness) is defined by

*k = S/H*. The reference line is determined by the water depth condition presented as follows:

*N*→ ∞, Eqs. (1)—(10) provide the complete solution to irrotational deep water waves. The solution contains

*N+2*unknown constants, i.e., the Fourier coefficient,

*a*

_{n}, wave steepness,

*S*, and reference depth parameter,

*kη*

_{0}.

*N*phases. In addition, there are two equations, i.e., Eqs. (9) and (10). As a result, there are

*N+2*equations to determine the unknown constants.

*N*= 1 and for

*N*= 2 is presented in Appendix B and C, and Shin (2016) and Shin (2021) presented an example for

*N*= 3. This method is further simplified and generalized by tensor analysis in the next chapter.

### 3. The Fourier Coefficients and the Steepness

*N*.

*η*is directly calculated instead of

*ζ*. Therefore, Eq. (9) must be simultaneously calculated with the coefficients. In this study, however, the profile

*η*was calculated with

*ζ*. Therefore, Eq. (9) was not coupled with the coefficients in this study and was calculated after they were determined. Hence, Eq. (9) is not considered in this chapter.

*ζ*is prescribed in advance, it is possible to convert Eq. (6) to a set of linear equations to determine the coefficients. Because the wave profile is an even function, the phases β

*for*

_{m}*m =*1,2,

*...,N*are considered in the range, 0 ≤

*β*≤

_{m}*π*and

*β*≠

_{m}*π*/2 because Eqs. (6) and (8) already satisfy that

*ζ*(

*±π*/2) = 0. β

_{1}= 0 and β

*=*

_{N}*π.*Letting

*X*ζ(

_{m}=*β*)

_{m}*,*Eq. (6) is represented at the phase β

*as follows:*

_{m}*“n”*is a dummy subscript.

*K*is a second-order tensor,

_{mn}*a*and

_{n}*X*are vectors in N-dimensional space. The component of the tensor

_{m}*K*is presented as follows:

_{mn}*R*of the tensor

_{mn}*K*the solution to Eq. (11) is determined easily as follows:

_{mn},*E*in the DFSBC is defined as where

_{m}*S = X*from the wave height condition in Eq. (10). Therefore, the wave height condition is automatically integrated in Eq. (14). The components of the second-order tensors in Eq. (14) are represented as follows:

_{1}— X_{N}*N*nonlinear equations represented by

*E*0 for

_{m}=*X*are obtained by substituting Eq. (13) into Eq. (14). The set of equations is solved with Newton’ s method. A set of linear equations can be derived by denoting partial derivatives of a tensor using commas and indices as

_{m}*r*] with [ ] means the step of Newton’s method. All the steps are r

^{th}in all equations except Eq. (21) in this chapter. Therefore, for the simplification of equations, the superscript [

*r*] was omitted in all the equations except Eq. (21). Differentiating Eq. (14) with respect to

*X*the partial derivative

_{i},*E*of the error vector is

_{m,i}*X*, we have where

_{p}*X*and

_{m p}= δ_{mp}*δ*is the second-order isotropic tensor. Multiplying Eq. (26) by the tensor

_{mp}*R*, the partial derivative of the coefficient is easily determined as follows: where

_{im}*R*and

_{im}K_{mn}= δ_{in}*δ*Differentiating Eq. (12) with respect to

_{in}a_{n,p}= a_{i,p}.*X*

_{p},*E*by

_{m}*N*, the RMSE in the DFSBC (Dean, 1965) is defined as follows:

^{−13}% within three steps) when the result of Shin (2021) was used as the first step solution. There are some differences in the above approach compared to Fenton’s method (Rienecker et al., 1981; Fenton, 1988).

All the partial derivatives with respect to wavelength are not required.

All equations can be formulated with tensor analysis.

*X*are merely parameters for calculating the coefficients and the steepness in this study. After determining the coefficients,_{i}*X*are no longer used. Hence, the wave elevation is denoted with_{t}*X*instead of_{i}*ζ*in Eqs. (11)—(28)._{i}After determining the coefficients and the steepness, the wave profile and water depth condition were calculated separately, as presented in the next chapter.

### 4. Wave Profile and Reference Depth Parameter

*F*(

*β, ζ*) = 0. The power series of

*F*(

*β, ζ*) is represented as follows: where

*F*(

*β ζ*) = 0, Eq. (31) provides a polynomial equation with regard to

*ζ*. For

*q*= 1, Eq. (31) provides a linear equation. Eq. (31) provides a quadratic equation for

*q*= 2, and Eq. (31) provides a cubic equation for

*q*= 3. The equations provide good approximations to the wave profile because |

*ζ*| < 1 for all cases. For q=2, the following approximation is obtained:

*ζ*| < 1. The approximation is used as the first step in the Newton’s method defined as follows. The superscript [

*n*] means the

*n*’

^{th}step of Newton’s method (where n is a natural number), while the superscript (

*n*) means the

*n*’

^{th}order partial derivative with respect to

*ζ*. The wave profile is calculated as follows by applying Newton’s method: where

*F*‘(

*β*,

*ζ*)| ≥ |

*F*’(0,

*ζ*)| for all waves and |

_{c}*F*‘(0, ζ

*)| = 0 is the breaking condition proposed by Stokes (Chakrabarti, 1987). Therefore, Eq. (34) is valid for all waves except breaking waves. The Newton method rapidly converges to the complete solution. The wave elevations*

_{c}*ζ*

_{i}=

*ζ*(

*β*) were calculated using Eq. (33). Note that

_{i}*ζ*

_{i}stands for the free surface elevation at phase

*β*. The integral is numerically calculated by substituting the results in water depth conditions as follows:

_{i}*ζ*is an even function,

*β*

_{1}= 0,

*β*= (

_{i}*i*− 1)

*π*/

*M*and

*β*

_{M+1}=

*π*. Note that M is independent of

*N*. When M is increased, Eq. (35) can be calculated more accurately rather than Rienecker et al. (1981) and Fenton (1988).

### 5. Numerical Analysis Procedure

^{−13}%),

*F*(

*β*,

_{i}*ζ*)| ≤ 1 × 10

_{i}^{−17}) . Substituting

*ζ*into Eq. (35), the reference depth parameter

_{i}*kη*was calculated. With the results, the wave profile was calculated as follows:

_{o}### 6. Results

*a*the steepness,

_{n},*S*, and the reference depth parameter,

*η*

_{0}are functions of one variable whose independent variable is the linear steepness,

*θ*. Consequently, Shin (2019) numerically calculated some data for the coefficients, steepness, and reference depth parameters in the range, 0 <

*θ*< 1. By curve fitting the data, the Fourier coefficients, steepness, and reference depth parameter are represented by Newton’s polynomials in Shin (2021), which give closed-form solutions. In this calculation,

*N*= 3 was considered. The results satisfied the Laplace equation, the BBC, and the KFSBC. The RMSE in the DFSBC was less than 1% in the range,

*H/L*≤ 0.142 where

_{o}*L*is the linear wavelength defined as follows:

_{o}*θ*=

*2πH/L*= 0.142 corresponds to

_{o}, H/L_{o}*θ*= 0.892. As a result, Shin (2016, 2021) reported a good approximation whose error was less than 1% in the range, 0 ≤

*θ*≤ 0.892. Although the error increases, Shin (2021) still applies to the waves in the range, 0.892 ≤

*θ*< 0.999. When

*θ >*0.999, wave profile by Shin (2021) does not converge. Each researcher has a different criterion regarding deep water breaking limitations. Chakarabarti (1987) proposes

*H/L*0.142. According to Dean and Dalrymple, (1984), Michell theory is

_{o}=*H/L*0.17, which corresponds to

_{o}=*H/L =*0.142. According to Stokes, the breaking criterion is

*u*= c at the crest. In this study, the limitation was checked and the errors according to the required order were also checked. It is also discussed why the other Fourier approximations are unsuitable for deep water waves. For the verification, three waves with a period of 6 s are considered, which were tabulated in Table 1. The following series order was considered:

*N*= 1;

*N*= 3, 6, 10, 12, 13, and 35;

*M*= 180. The RMSE in the DFSBC was calculated and tabulated in Tables 2 and 3. As the required order is increased, the error is decreased. When

*N*≥13, Newton’s method in Ch. 3 did not converge because

*e*is very large and

^{Nζ}*a*is very small. As a result, this study is available for

_{N}*N*≤12.

*N = M*and Eq. (35) is calculated with

*M*should be increased to reduce it. Fenton’s method,

*M = N =*64, is greater than

*N*=13. Therefore, Fenton’s method is unsuitable for deep water waves.

^{th}-order Stokes wave. When

*θ*> 1.028, this study does not converge. More precisely, Newton’s method does not converge because the wave profile at the crest is sharp when

*θ*> 1.028. Therefore,

*θ*= 1.028 is the limitation of this study, which corresponds

*H/L =*0.137 and is slightly less than 0.142 according to Dean et al., (1984).

*θ*≤ 0.892, Shin (2021) is acceptable as the first step solution in Newton’s method. For

*θ*> 0.892, the method reported by Shin (2021) is unsuitable for the first step solution because the Newton method does not converge. This problem can be avoided using the sequence of height steps. For waves in the range 0.892 <

*θ*≤ 0.999, the profile for

*θ*= 0.892 is used as the first step solution, and for waves in the range 0.999 <

*θ*≤ 1.028, the profile for

*θ*= 0.999 is used as the first step solution. The profiles are shown in Fig. 4; the horizontal velocities are shown in Fig. 5; the vertical velocities are shown in Fig. 6. The relative horizontal velocity at the crest is 0.782 for wave (c), which is less than Stokes’ criteria, 1. The Bernoulli’s constant in Fig. 4 is calculated for wave (c).

### 7. Conclusions

Although the moving coordinate system by Dean was adopted in Fenton’s method, a dimensionless coordinated system was adopted in this study. As a result, all the partial derivatives with respect to wavelength are not required. Some parameters were eliminated.

All equations were formulated by tensor analysis. Therefore, numerical equations were much more simplified and had no errors.

In the other Fourier approximation, the required order was determined to reduce the error of the numerical integration in the water depth condition because it was solved simultaneously with the Fourier coefficients. On the other hand, the water depth condition was calculated independently. Therefore, the required order was reduced dramatically.

*N*≥ 13. Deep water breaking limitation was checked. This study is valid for waves in the range, 0 <

*θ*≤ 1.028. The limitation 1.028 correspond to

*H/L*= 0.137, which is slightly lower than the Michell theory.