# Nonlinear Wave Evolution in Interaction with Currents and Viscoleastic Muds

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Model

#### 2.1. Nonlinear Wave–Current Interaction Model

#### 2.2. Model for Surface Wave Evolution over Viscoelastic Mud

#### 2.2.1. Macpherson Model

#### 2.2.2. Liu and Chan Model

#### 2.2.3. Comparison between Viscoelastic Mud Models

## 3. Model Results

#### 3.1. Model Validation

#### 3.2. Effect of Currents on Propagation of Monochromatic Waves over Mud

#### 3.3. Effects of Currents on Propagation of Random Wave Spectra over Mud

#### 3.4. Propagation of Cnoidal and Random Wave Spectra over Mud of Arbitrary Depth

## 4. Discussion

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Surface wave damping rate as a function of frequency for different shear moduli of mud in the presence of co-propagating current with $U=+0.15$ m/s (solid line), without current (dot line), and in the presence of counter-propagating current with $U=-0.15$ m/s (dashed line), $\zeta =100$, $h=1.00$ m, ${d}_{m}=0.12$ m, and ${\rho}_{m}=1111$ kg/m${}^{3}$.

**Figure 4.**Comparison between the attenuation rates from the present model and laboratory experiments of Zhao et al. [19], U > 0 indicates experiments where current are in the same direction, and U < 0 indicates the opposite.

**Figure 6.**Comparison between the attenuation rates from the present model and laboratory experiments of An and Shibayama [53].

**Figure 7.**Propagation of cnoidal wave spectrum over mud with shear moduli of G = 0–200 Pa. Blue-solid-x line: the initial spectrum at $x=0$, black-solid line: the spectrum with $U=+0.15$ m/s, black-dashed line: the spectrum for $U=-0.15$ m/s, and black-dot line: the spectrum with $U=0$, at the end of mud patch (x = 800 m), $h=1.00$ m, ${d}_{m}=0.12$ m, ${\rho}_{m}=1111$ kg/m${}^{3}$, and $\zeta $ = 100.

**Figure 8.**Evolution of a cnoidal wave spectrum with subharmonic interactions deactivated. Wave and mud parameters and water depth are the same as those in Figure 7.

**Figure 9.**Spatial variation of cnoidal wave H over viscous ($G=0$) and viscoelastic mud with shear moduli of G = 50–200 Pa. dot-line: $U=0$, solid-line: $U=$ +0.15 m/s, and dashed-line: $U=$−0.15 m/s. The mud patch is located at x = 300–800 m, $\zeta $ = 100, h = 1.00 m, ${d}_{m}=0.12$ m, and ${\rho}_{m}=1111$ kg/m${}^{3}$.

**Figure 10.**Variation of surface wave damping rate with frequency for different values of mud shear modulus. Solid line: $Fr=+0.15$ m/s, dot line: $Fr=0$, and dashed line: $Fr=-0.15$ m/s, $\zeta =100$, $h=2.00$ m, ${d}_{m}=0.20$ m, and ${\rho}_{m}=1111$ kg/m${}^{3}$.

**Figure 11.**Evolution of random wave spectra with peak frequency of ${f}_{p}=0.0625$ Hz for two values of mud shear modulus of $G=0,100$ and 200 Pa (${U}_{r}=2.08$, $h=2.00$ m, ${d}_{m}=0.20$ m, and ${\rho}_{m}=1111$ kg/m${}^{3}$, and $\zeta $ = 100). In (

**a**–

**c**): dot-line is initial spectra at $x=0$, solid-line is spectra at $x=21{L}_{p}$ for $Fr=+0.15$, and dashed-line is spectra at $x=21{L}_{p}$ for $Fr=-0.15$ (${L}_{p}$ is the wavelength of spectral peak). (

**d**–

**i**): energy density at spectral peak (dot-line), second (dashed line), and third (dashed-dot line) harmonic of the peak, and subharmonic of the peak (${f}_{p}/2$) (solid line).

**Figure 12.**Spatial variation of random wave ${H}_{rms}$ over viscoelastic mud with shear moduli of G = 0–200 Pa with $Fr=$ +0.15 (solid line), $Fr=$ 0 (dot line), and $Fr=$−0.15 (dashed line). Simulation parameters are the same as in Figure 11.

**Figure 13.**Spatial variation of random wave height, ${H}_{rms}$, over viscoelastic mud with shear moduli of G = 0–200 Pa in presence of currents with $Fr=$ +0.15 (solid line), $Fr=$ 0 (dot line), and $Fr=$−0.15 (dashed line). Simulation parameters are the same as those in Figure 12 but the spectral peak frequency is $0.28$ Hz.

**Figure 14.**Variation of surface wave damping rate with frequency for mud with shear moduli $G=0$, 100 and 200 Pa. Solid line: LC model, dashed line: Macpherson [17], $\zeta =100$, $h=0.8$ m, ${d}_{m}=0.4$ m, and ${\rho}_{m}=1111$ kg/m${}^{3}$.

**Figure 15.**Propagation of cnoidal wave spectrum over mud with shear moduli $G=0,100,200$ Pa, $h=0.80$ m, ${d}_{m}=0.40$ m, and ${\rho}_{m}=1111$ kg/m${}^{3}$.

**Figure 16.**Spatial variation of cnoidal wave height, H, over viscous ($G=0$) and viscoelastic mud with shear modulus of $G=200$ Pa, as normalized by incident wave height ${H}_{0}$, solid-line: LC, dashed-line: Macpherson [17]. The mud patch is located at x = 300–800 m, $\zeta $ = 100, h = 0.80 m, ${d}_{m}=0.40$ m, and ${\rho}_{m}=1111$ kg/m${}^{3}$.

**Figure 17.**Spatial variation of random wave ${H}_{rms}$ over viscous ($G=0$) and viscoelastic mud with shear modulus of $G=200$ Pa, as normalized by incident wave height ${H}_{0}$, $\zeta $ = 100, h = 0.80 m, ${d}_{m}=0.40$ m, and ${\rho}_{m}=1111$ kg/m${}^{3}$.

**Figure 18.**Evolution of random wave spectrum over mud with LC and Macpherson [17] mechanisms, shear moduli $G=0,200$ Pa, $h=0.80$ m, ${d}_{m}=0.40$ m, ${\rho}_{m}=1111$ kg/m${}^{3}$, and $\zeta $ = 100.

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**MDPI and ACS Style**

Sharifineyestani, E.; Tahvildari, N. Nonlinear Wave Evolution in Interaction with Currents and Viscoleastic Muds. *J. Mar. Sci. Eng.* **2021**, *9*, 529.
https://doi.org/10.3390/jmse9050529

**AMA Style**

Sharifineyestani E, Tahvildari N. Nonlinear Wave Evolution in Interaction with Currents and Viscoleastic Muds. *Journal of Marine Science and Engineering*. 2021; 9(5):529.
https://doi.org/10.3390/jmse9050529

**Chicago/Turabian Style**

Sharifineyestani, Elham, and Navid Tahvildari. 2021. "Nonlinear Wave Evolution in Interaction with Currents and Viscoleastic Muds" *Journal of Marine Science and Engineering* 9, no. 5: 529.
https://doi.org/10.3390/jmse9050529