### 1. Introduction

Water waves are considered one of the most fascinating natural phenomena in the marine environment. Waves in the open ocean are generally generated by winds. The transmission of energy from the wind to the water is facilitated by friction between these two fluid media. This energy is transmitted as a wave across the water surface. Consequently, friction plays a pivotal role in the circulation of fluids and is a crucial factor in wave generation. The energy transferred from the wind to waves diminishes as the wind intensity decreases, leading to the transformation of waves into swells that eventually dissipate. Therefore, friction also contributes to the dissipation of waves. Nevertheless, the mathematical description of water waves has been conducted under the assumption of irrotational, inviscid flow (Chaplin, 1980; Chappeleakimr, 1961; De, 1955; Dean, 1965; Fenton, 1988; Kishida and Sobey, 1988; Korteweg and de Vries, 1895; Rienecker and Fenton, 1981; Shin, 2016, 2019; Stokes, 1847; Stoke, 1880; Skjelbreia and Hendrickson, 1960; Vanden-Broeck and Schwartz., 1979) or under the assumption of rotational, inviscid flow (Rankine, 1863; Chen and Zou, 2019; Constantin, 2005; Dalrymple, 1974; Henry; 2008; Nwogu, 2009; Shin, 2022, 2023). Hence, these studies cannot account for the natural dissipation of waves. In a few exceptional cases, viscosity is considered in the linearized Navier-Stokes equations, e.g., water waves over a viscous mud bottom and seiche phenomena in coastal regions. The equation (Dean and Dalrymple, 1984) for long waves with bottom friction is analogous to that of a damped harmonic oscillator. The amplitude decreases exponentially with time, and the angular frequency also changes, providing an important clue for solving the Navier–Stokes (NS) equations. NS equations are a set of nonlinear partial differential equations that govern the flow of Newtonian fluids. These equations are valuable because of their ability to elucidate numerous physical phenomena. The structural design process commences by evaluating the external appearance of the design object and the loads on the surface generated by fluid flow. The NS equations are used for these determinations. Despite the wide range of practical applications, the existence of smooth solutions in three dimensions has not been proven definitively. Theoretical solutions remain limited for most cases, with only a few exceptions. Analytical approaches are deemed almost impossible for solving these problems. Thus, only numerical and experimental methods are allowed. This study applied a theoretical approach with tensor calculus to address these limitations. General solutions were introduced without considering the boundary conditions and initial conditions

**.**NS equations are the Euler equations with the addition of the gradient of the viscous stress tensor. The viscous stress tensor is proportional to the strain rate, making the gradient of viscous stress a linear function of the velocity field. The convective acceleration is the only nonlinear term included in the Euler equation. Shin (2022, 2023) reported the complete solutions to the Euler equations. They transformed the Euler equations into Bernoulli’s principle when the velocity field was described by the solutions of the Helmholtz equation. Using the methodology used in the studies conducted by Shin (2022, 2023) with the solution for a damped harmonic oscillator, the findings were extended to cover general three-dimensional viscous flow in the current study.

When the velocity field is described by solutions of the Helmholtz equation, the gradient of the viscous stress tensor for incompressible flow is proportional to the velocity field. The product of a solution to the Helmholtz equation and any time function is also a solution to the Helmholtz equation. By substituting the product into the NS equations, these equations can be decomposed into the Euler equations and a linear ordinary differential equation for the time function. Based on Shin’s (2022) research, this study solved the Euler equations for general flow. Some exact solutions to NS equations exist. The Taylor–Green vortex (Taylor and Green, 1937) and Antonio’s solution (Antuono, 2020) are two interesting examples. The Taylor–Green vortex was an unsteady flow of a decaying vortex in Cartesian coordinates. Antonio’s solution is considered a three-dimensional generalization of the classic two-dimensional Taylor–Green vortex. For verification, the Taylor–Green vortex and Antonio’s solutions are special cases of the solutions presented in this study. Substituting the results into the vorticity equations confirmed that the vorticity equations were satisfied.

This study adjusted Shin’s results (2022, 2023) to comply with NS equations. The process through which waves evolve into swells was elucidated. The modified wave theory was verified by comparing the decay modulus with the experimental data of Shiau and Rumer (1974) and the theoretical solution presented by Dean and Dalrymple (1984). The comparison results showed a high level of agreement.

### 2. Navier-Stokes Equations for Incompressible Flow

Tensor notation was used for simplicity, and the following rule was adopted when using tensor notation. Whenever the subscript appeared twice as a letter in a grouping, the expressions of this grouping were summed, with the repeated index taking on successively the values 1, 2, and 3. The non-repeated letter subscripts or numerical subscripts remain fixed during the process (Dym and Shames, 1973). The partial derivatives of a tensor are denoted using commas and indices as
∂ ( ) ∂ x i = ( ) , i where

*x*is the position vector; the conservation of mass is presented as follows: where_{i}*v*is the velocity field, which is a vector function of position,_{i}*x*, and time,_{i}*t*. The conservation of momentum is presented as follows:A time derivative will be denoted with the ‘over-dot’ notation. Therefore,

*v̇*is the local acceleration._{i}*ρ*is the density of the fluid, which is a constant;*p*=*p*(*x*,_{i}*t*) is pressure field;*μ*is kinematic viscosity, which is a constant;*f*is body force per unit mass, which is a conservative force. The NS equations for an incompressible fluid are composed of Eqs. (1) and (2). The vorticity is defined as follows: where_{i}*∊*is the Levi–Civita symbol. Taking the inner product of the vorticity and the Levi-Civita symbol results in_{ipq}Taking the inner product of the velocity

*v*and the deformation tensor_{p}*v*_{q}_{,}*, the convective acceleration in Eq. (2) is expressed as*_{p}Replacing

*p = j, q = i,*and*i = k*, Eq. (5) can be presented as follows because*v*Ω_{p}*=*_{i}∊_{ipq}*v*Ω_{p}*=−*_{i}∊_{qip}*v*Ω_{p}*:*_{i}∊_{qpi}Substituting Eq. (6) into Eq. (2), the alternative form of the momentum equation is represented as follows:

Because

*f*is a conservative force, it has potential energy. The first term on the right-hand side represents the outer product of the velocity and vorticity. Therefore, the term is zero when the vorticity and the velocity are parallel to each other._{i}### 3. Decomposition of the Navier-Stokes Equations

A definitive clue to solving the NS equations can be found in the solution of a single-degree-of-freedom damped harmonic oscillator or long waves with bottom friction (Dean and Dalrymple, 1984). In the oscillator, the damping force is proportional to the velocity

*ż*of the object because it is due to fluid viscosity: −*cż*, where*c*is the damping coefficient. Newton’s second law for a damped harmonic oscillator is presented as follows: where*ω*is the undamped angular frequency of the oscillator, and*ζ*is the damping ratio. The displacement is expressed as where A is the amplitude,*ϕ*is the phase, and*e*^{−}*is the exponential decay of the amplitude. Eq. (9) indicates that the damping force reduces the natural frequency, decreases the amplitude, and eventually eliminates the oscillation. Considering that the gradient of viscous stress in Eq. (7) arises from a Newtonian fluid, similar to the oscillator, the solutions to the NS equations are presented in a similar form to Eq. (9). If the velocity field is a solution to the Helmholtz equation, i.e.,*^{ξωt}*v*_{i}_{,}*=*_{jj}*bv*, the gradient of viscous stress is presented as_{i}*μv*_{i}_{,}*=*_{jj}*μbv*, where_{i}*b*is a constant with the unit of inverse area. Therefore, like the damped harmonic oscillator, the gradient of viscous stress is proportional to the velocity field. Eq. (9) is based on the assumption that the velocity field for viscous flow is represented as follows: where*F*(*t*) is a function of time, which is a dimensionless quantity, and*u*is a velocity field for inviscid flow to satisfy the Helmholtz equation. Examples of_{i}*u*include the water particle velocities presented by Shin (2022, 2023)._{i}**Theorem 1.**When the velocity fields are represented as a product of a time function and solutions of the Helmholtz equation, the NS equations can be decomposed into the Euler equations and an ordinary equation for the time function.

The gradient of viscous stress is presented as

*μv*=_{i,jj}*μbv*=_{i}*μbF*(*t*)*u*, and the local acceleration is presented as follows:_{i}Substituting Eq. (12) and

*μv*_{i}_{,}*=*_{jj}*μbF*(*t*)*u*into Eq. (7), Eq. (7) is converted to the Euler momentum equation, provided that the following condition is satisfied:_{i}The constant, b, must be negative because the flow must eventually dissipate. Eq. (14) shows similarities to the exponential decay of the amplitude, as expressed in Eq. (9). Eq. (7) is presented as follows:
where the vorticity,
Ω k def = F ( t ) Ω ¯ k . Therefore,
Ω ¯ k = ∊ k p q u q , p .

**Theorem 2.**When the velocity field is a solution of the Helmholtz equation, then the vorticity is also a solution of the Helmholtz equation.

**Proof.**Ω

_{i}_{,}

*=(*

_{jj}*∊*

_{ipq}v_{q}_{,}

*)*

_{p}_{,}

*=(*

_{jj}*∊*

_{ipq}v_{q}_{,}

*)*

_{jj}_{,}

*=(*

_{p}*∊*)

_{ipq}bv_{q}_{,}

*=*

_{p}*b*(

*∊*

_{ipq}v_{q}_{,}

*)=*

_{p}*b*Ω

_{i}As a result, the velocity field and the vorticity field were determined using the Helmholtz equation. The next step was to determine the pressure field using Eq. (15). “ 0 <

*F*(*t*) < 1”, and it has no dimensions. Hence, it is considered a non-dimensional perturbation parameter. Using the perturbation method, Eq. (15) can be solved in the following chapters for two cases:*u*is steady flow and_{i}*u*is unsteady flow._{i}### 4. Pressure Field for Steady Flow

When the velocity field,

*u*is independent of time, Eq. (10) is presented as follows:_{i}Therefore, the local acceleration in Eq. (15) is zero. The pressure field is presented as the sum of the dynamic pressure and static pressure. The pressure field must be presented by a quadratic function with respect to

*F*(*t*) because the dynamic pressure is proportional to the square of the velocity, and static pressure is proportional to the body force, as follows.The first and second terms mean the static and dynamic pressures, respectively. Substituting Eqs. (16)–(17) into Eq. (15), the result is presented as

Eq. (18) is decomposed with two equations because Eq. (18) is valid for all

*t*, and*F*(*t*) ≠ 0, using the perturbation method, as follows. The zero-order perturbation equation is presented as follows:Integrating Eq. (19), the potential energy of the body force is determined as follows:

The static pressure field is determined using Eq. (21).

*u*is a solution of the Helmholtz equation. Hence,_{i}*u*_{j}_{,}*=*_{pp}*bu*Therefore, the last term of Eq. (20) is presented as follows:_{j}.Using Eq. (4),

Where Ω ¯ k , k = 0 . Therefore,
Ω ¯ k Ω ¯ i , k = ( Ω ¯ k Ω ¯ i ) , k . Let’s introduce a quantity,
E , i def = ( Ω ¯ k Ω ¯ i ) , k . Therefore,
E = ∫ ( Ω ¯ k Ω ¯ i ) , k dxΩ ¯ k is also a solution of the Helmholtz equation and is determined by the velocity field, the energy, E can be easily determined. Substituting Eq. (24) into Eq. (20), the result is presented as follows.

*δ*is the Kronecker delta. Since divergence of curl is zero, we have_{ij}_{i}is a type of kinetic energy. BecauseThe dynamic pressure field is determined using Eq. (26). The second term represents the kinetic energy associated with irrotational motion. In contrast, the third term represents the kinetic energy related to the vortex. The total kinetic energy is the linear sum of the two terms because the constant, b is negative, which is consistent with the law of conservation of energy. The ratio, r, is defined as

*b*, is negative. Therefore 0≤

*r*≤1.

The energy E is referred to as an anti-vortex energy because it reduces the kinetic energy due to the vortex like frictional energy. Two cases of anti-vortex energy are considered as follows:

### 4.1 Anti-Vortex Energy Zero Case

Therefore,

*r*= 1 for two-dimensional flow. This condition is not generally valid in three-dimensional flow. Examples include the Beltrami flows considered in the next section.### 4.2 Beltrami Flow

Beltrami flows are flows in which the vorticity vector and the velocity vector are parallel to each other. The outer product of the velocity and the vorticity is zero, i.e.,
∊ i j k u j Ω ¯ k = 0 . Applying it to Eq. (24),
Ω ¯ k Ω ¯ k , i = Ω ¯ k Ω ¯ i , k . Therefore,

*r*= 0 for a Beltrami flow. In this case, Eq. (26) represents Bernoulli’s principle for irrotational flow, and Beltrami flow is like irrotational flow.### 5. Pressure Field for Unsteady Flow

When the velocity field,

*u*is unsteady flow, the velocity field can be expressed as Eq. (10). A typical example is progressive water waves (Shin, 2022, 2023). Using the moving coordinate system proposed by Dean (1965), an unsteady flow can be converted into a steady flow, as presented in the previous Chapter. The moving coordinate system is defined as_{i}*G*(

_{i}*t*) has a length unit. In a damped harmonic oscillator, angular frequency is modified with a damping ratio, “

*ζ*,” as shown in Eq. (9). Therefore, the function

*G*(

_{i}*t*) shows that viscosity also affects the angular frequency of water waves. The first term in Eq. (15) must be a quadratic term of

*F*(

*t*), it is assumed that the function

*G*(

_{i}*t*) is defined as where

*C*is the phase speed for water waves or a reference speed for other flow, which is a constant. The relative velocity with respect to the moving coordinate system is presented as

_{j}Using the coordinate system, the velocity field in Eq. (10) can be expressed as

The pressure field can be presented by a quadratic function with respect to

*F*(*t*) as follows: where the velocity field*u*is a solution of the Helmholtz equation. Using Eqs. (29), (31), and (33), the local acceleration in Eq. (15) is presented as_{i}The convective acceleration in Eq. (15) is presented as

Differentiating Eq. (35), the gradient of the pressure is presented as

The zero-order perturbation equation is presented as the same in Eq. (19), and the second-order perturbation equation is presented as

Because

*dX*=_{i}*dx*, integrating the above equation with respect to_{i}*dx*gives Bernoulli’s principle as follows:_{i}The dynamic pressure field was determined using Eq. (41). Substituting Eq. (14) into Eq. (30) and integrating the result results in
where

*B*is an integral constant. For inviscid flow,*G*(_{j}*t*)=*C*as proposed by Dean (1965); hence, there is a condition to determine the constant._{j}t### 6. Results and Verification

Some exact solutions to the NS equations exist for specific conditions (Antuono, 2020; Ethier and Steinman, 1994; Taylor and Green, 1937; Wang, 1991). For verification, the Taylor–Green vortex and Antonio’s solution were proven to be specific cases of the general solutions presented in this study.

### 6.1 General Form of the Velocity Field

A function, u i = ∑ q = 1 N A q e a q x 1 − G 1 ( t ) e c q x 2 − G 2 ( t ) e d q x 3 − G 3 ( t ) , where b = a q 2 + c q 2 + d q 2 to satisfy

*f*(*x*)=*e*, where “^{ax}*a*” is a complex number in this section, satisfies that*f*”(*x*)=*a*^{2}*f*(*x*). Therefore, the general solution to the Helmholtz equation is presented with a linear combination, i.e.,*a*,_{q}*c*, and_{q}*d*are complex numbers in this section, and_{q}*u*_{i}_{,}*=*_{jj}*bu*. The coefficients_{i}*A*are determined to satisfy Eq. (11), boundary conditions, and initial conditions. The vorticity is also presented with the same form. Three samples are presented in the following sections._{q}### 6.2 Vorticity Transport Equation

Differencing Eq. (2) with respect to

*x*and taking the inner product of the result and the Levi-Civita symbol,_{q}*∊*with respect to the indices_{rqi}*i*and*q*, the vorticity transport equation can be derived as follows.The left-hand side is the material derivative of the vorticity vector. The first term on the right-hand side describes the stretching or turning of vorticity caused by the flow velocity gradients. Eq. (45) can be used as a criterion to judge the validity of the solution because the velocity field has already been determined in the previous section. Substituting Eq. (10) into Eq. (45) and using Theorem 2, the result is as follows.

The unsteady case is only considered because unsteady flow contains steady flow. When

*u*is unsteady, using the moving coordinate system, Eq. (46) can be expressed as_{j}The first term on the right-hand side in Eq. (47) represents vortex stretching and turning, which is zero for two-dimensional flows. Therefore, Eq. (47) is presented for two-dimensional flows as follows since
Ω ¯ 1 = Ω ¯ 2 = 0 .

A stream function can be defined as

*U*_{1}=*ψ*_{,2}and*U*_{2}=−*ψ*_{,1}. Therefore, Ω_{3}=*ψ*_{,}*. Ω*_{ii}_{3}=*ψ*_{,}*=*_{ii}*bψ*because the stream function is also a solution of Helmholtz equation. Substituting it into Eq. (48), gives### 6.3 Taylor–Green Vortex

The Taylor–Green vortex solution is used to test and validate the temporal accuracy of NS algorithms (Chorin, 1968; Kim and Moin, 1985). Taylor–Green vortex solution (Taylor and Green, 1973) is presented as follows. In the domain 0≤

*x*_{1},*x*_{2}≤2*π*, the velocity field is expressed as where*F*(*t*)=*e*^{− 2}*. The pressure field is expressed as and the vorticity is expressed as*^{μt}
Eqs. (51)–(54) were cited from Taylor and Green (1937). Therefore, Ω ¯ 1 = 0 ,
Ω ¯ 2 = 0 , and
Ω ¯ 3 = Ω ¯ 3 ( x 1 , x 2 ) . Therefore,
Ω ¯ k Ω ¯ i , k = 0 , and

*v*_{1,}*=− 2*_{jj}*v*_{1}and*v*_{2,}*=− 2*_{jj}*v*_{2}. The velocity is a solution of the homogeneous Helmholtz equation. Therefore, this study for*b*=−2 gives the Tayor–Green vortex. Ω_{3,}*=−2Ω*_{jj}_{3}, which is a solution of the Helmholtz equation. Because the flow is 2-D,*r*= 1. Substituting the velocity and vorticity into Eq. (26), the dynamic pressure for*b*=− 2 is calculated as follows:For

*Q*= 1/4, the above gives Eq. (53). Therefore, the Taylor–Green vortex is a special case for steady flow in Sec. 4.1.### 6.4 Viscous Three-Dimensional Periodic Solution

Antuono’s (2020) solutions for three-dimensional problems are defined on a three-dimensional torus 0≤

*x*_{1},*x*_{2},*x*_{3 }≤*L*and are characterized by positive and negative helicity, respectively. The solution with positive helicity is expressed asIf φ = - 3 k , and the dynamic pressure is given by Eq. (26) for

*i*= 1, then*j*= 2, and*l*= 3; if*i*= 2, then*j*= 3, and*l*= 1; if*i*= 3, then*j*= 1, and*l*= 2. The solution is a sample for steady flow in Sec. 4.2. The velocity field is presented with a linear combination of three trigonometric functions with respect to three variables,*x*_{1},*x*_{2}and*x*_{3 }*.*The velocity field is a solution of the Helmholtz equation for*b*=− 3*k*^{2}because the linear combination is a solution of the Helmholtz equation. The vorticity field is parallel to the velocity because both solutions belong to the class of the Beltrami flow presented in Sec. 4.2. For the case with positive helicity, the solution is Ω*=*_{i}*ϕu*where_{i}*r*= 0. Antuono’s (2020) solutions are also special cases of the solutions presented in Sec. 4.2.### 6.5 Two-Dimensional Progressive Water Waves

Using the conventional coordinate system considered in Shin (2022, 2023), the phase is defined as

*β*=*kx*−*ωt*where k is the wave number. Therefore,*kG*corresponds to*ωt*=*k**Ct*for 2-D inviscid flow. Using Eqs. (42) and (44),Using power series expansion, ξ def = − μ b / ω

*kG*=*ωt*+*O*((*μbt*)^{2}), where*O*stands for Big-O notation. Therefore, for inviscid flow, Eq. (55) gives*ω**t*as defined in Shin (2022, 2023). “*μb*” has a unit of angular frequency. Thus, a dimensionless quantity called the “damping ratio” is defined as*.*Using the damping ratio, the phase for viscous flow is represented asMultiplying

*e*^{−}*by the velocity field in Shin (2022, 2023) gave a solution to the NS equation. Multiplying*^{ξωt}*e*^{− 2}*to the dynamic pressure in Shin (2022, 2023) gives the pressure field for viscous flow. In the conventional coordinate system, the kinematic free surface boundary condition (KFSBC) is presented as where*^{ξωt}*v*_{1}and*v*_{2}are horizontal and vertical velocities, respectively, for viscous flow, and*η*is the wave profile. The KFSBC stipulates that there must be no flow crossing any fluid interface. The free surface, which is a fluid interface, remains unaffected by the viscosity. Consequently, the kinematic boundary condition should remain unaltered by viscosity. Using Eq. (10), the KFSBC is presented as where*u*_{1}and*u*_{2}are the horizontal and vertical velocities, respectively, for inviscid flow. The dimensionless elevation of the wave profile is defined as*γ*=*k*(*η*+*h*) in Shin (2022, 2023), where*h*is the water depth. Using the phase defined in Eq. (56), andSubstituting Eqs. (59) and (60) into Eq. (58), results in Shin (2022, 2023) because

*e*^{−}*≠0. The KFSBC in Shin (2022, 2023) is still valid for viscous flow. Applying the dynamic boundary condition on the free surface to Eq. (35), the wave height for viscous flow is expressed as where*^{ξωt}*H*is the wave height at_{o}*t*= 0. Waves emanating from a remote source are referred to as swells, whereas waves produced in the nearby vicinity are termed seas. Specifically, waves formed due to the wind-induced forced oscillation of water are generally identified as seas. In contrast, those originating from the natural oscillation of water post the cessation of wind are labeled as swells. Eqs. (56) and (61) indicate that the height of swell waves decreases when wind energy no longer influences wave formation, leading to a longer wave period and an increase in wavelength over time. The quantity “*b*” represents the strength of vorticity,*σ*defined in Shin (2022).*b*=− (*kσ*)^{2}because the length is normalized by the wavelength in Shin (2022, 2023). Using*ξω*=−*μb*, the damping ratio is determined as follows:A logarithmic decrease is used to determine the damping ratio. Using Eq. (61), the damping ratio can be expressed as
where

*H*_{1}, and*H*_{2}represent the wave height of any two successive peaks. Although the strength of vorticity,*σ*, is determined using the average water particle velocities on the free surface and the seabed in Shin (2022, 2023), it can also be determined using Eqs. (62) and (63), which indicate that the currents considered in Shin (2022, 2023) are generated by the same source as the waves, but they differ from the Stokes drift. This study has proven that viscosity is the source of circulation and current in fluids.
Dean and Dalrymple (1984) provide the decay modulus for long waves with frictional damping as follows:
where P def = μ 2 / g k 2 h 5 . Shiau and Rumer (1974) carried out a series of experiments to examine the decay of seiches in a basin. The experiments were conducted in very shallow water (0.15 <

*P*is the Proudman number,*h*< 8.5 cm). The theoretical values of*α*in Eq. (64) were comparable to the experimental data in Fig. 1.The velocity fields for viscid flows,

*v*, were checked. The velocity fields for inviscid flows,_{i}*u*, were compared with experimental data by Le Méhauté et al. (1968) in Shin (2022, 2023). Therefore, according to Eq. (10), the function,_{i}*F*(*t*) was checked instead of the velocity field,*v*, as follows. Eq. (61) is valid for long waves because Shin (2022, 2023) is valid even for ultra-shallow water waves._{i}*α*= 2*ξωT*= 4*πξ*from Eq. (61) because the decay modulus is defined by the relative reduction in amplitude over one wave period. Using Eq. (62), the decay modulus is expressed asNote that μ = P g k 2 h 5 . Using Eq. (11) in Shin (2023), Eq. (65) is expressed as
where τ 12 = ρ ω μ u 1 , where μ . The present study used a Newtonian fluid model in which shear stress is linearly related to the strain rate, expressed as

*ω*^{2}=*gk*^{2}*h*for shallow water waves and*u*is the average speed in the horizontal direction over a period on still water and_{s}*u*is the average speed in the horizontal direction over a period on the seabed._{b}*ū*=_{s}*u*/_{s}*C*and*ū*=_{b}*u*/_{b}*C*, where*C*is the celerity. Upon comparing Eq. (64) and Eq. (66), two distinct differences emerge between these equations. Eq. (64) is based on long-wave theory. Owing to the relatively small water depth in relation to the wavelength, variations in velocity in the vertical direction were disregarded, and only the average velocity was considered in Eq. (64). Consequently, the decay modulus in Eq. (64) is not dependent on the velocity distribution. On the other hand, the decay modulus in Eq. (66) is closely associated with the velocity distribution. Dean and Dalrymple (1965) used a quadratic friction law where the shear stress is directly proportional to the square of the velocity. The decay modulus in Eq. (64) was calculated assuming that bottom shear stress is presented by*u*_{1}is the horizontal velocity. Therefore, the shear stress is directly proportional to*τ*_{12}=*ρμ*(*u*_{1,2}+*u*_{2,1}), where*x*_{1}is the horizontal direction and*x*_{2}is the vertical direction. Therefore, the shear stress is directly proportional to*μ*. Consequently, this leads to a change in the exponent of*P*from 1/4 to 1/2. When the same constitutive equation and the average concept considered in the long wave theory are used, the average shear stress gradient is as follows.For the velocity ratio,
1 - U ¯ s 1 - U ¯ b = cos ( 1 4 ) = 0.97 , Eqs. (64) and (68) are equivalent to one another, as shown in Fig.1. Eqs. (64) and (68) are compared in Fig. 1. The comparison results indicated a significant level of concordance between Eq. (64) and Eq. (68). In contrast, this indicates a distinction between Eq. (64) and Eq. (66). The distinction elucidates the rationale behind Dean and Dalrymple’s (1965) choice to use a quadratic friction law instead of a Newtonian fluid model. Eqs. (67) and (68) express the applicability of this study even for non-Newtonian fluids, provided that the linearization of shear stresses is achieved using the technique described by Dean and Dalrymple (1984).

### 7. Conclusions

NS equations play a crucial role in elucidating various scientific and engineering phenomena. They are used in modeling ocean currents, water waves, and other water flows. They help with the design of the structure. Despite the wide range of practical applications, the equations are mainly solved using numerical or experimental methods, while there are some exact solutions to specific problems.

General solutions to NS equations for incompressible flow are presented in this study. The solutions were calculated by combining the solutions of the Helmholtz equation and the solution of a single-degree-of-freedom damped harmonic oscillator. They are analytical exact solutions to NS equations.

The velocity fields and the vorticity fields were presented with solutions of the Helmholtz equations. When the velocity fields are expressed as the product of the solution of the Helmholtz equation and a univariate function of time, the NS momentum equation is decomposed into the Euler momentum equations and an ordinary differential equation for the univariate function of time. The Euler momentum equations were converted to Bernoulli’s principle to determine the pressure field. The time function was determined by solving the ordinary differential equation. Consequently, the general solutions were derived through the linear combinations of the solutions to the Helmholtz equation. The solutions presented in this study are functions of class

*C*^{∞}because solutions of the Helmholtz equations are smooth (Lin, 1988).The Taylor–Green vortex solution and the solutions proposed by Antuono (2020) are particular cases of these solutions, and these solutions satisfy the vorticity equations. The solutions presented by Shin (2022, 2023) were enhanced to integrate viscid water waves with shear currents.

As time elapses or as a swell moves away from its generating area, its height decreases while the period and wavelength increase. Eventually, the swell dissipates entirely because of the removal of the energy source. This phenomenon was substantiated mathematically in the present study. The decay modulus was compared with the experimental data from Shiau and Rumer (1974) and the solution proposed by Dean and Dalrymple (1984). The agreement was excellent. Unlike Dean and Dalrymple (1984), the study showed that the decay modulus is influenced by the velocity gradient in the depth direction.

The technique presented in this study can also be applied to non-Newtonian fluid, provided that the linearization of shear stresses is achieved through the technique described by Dean and Dalrymple (1984).