# Poroacoustic Traveling Waves under the Rubin–Rosenau–Gottlieb Theory of Generalized Continua

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulations

#### 2.1. Poroacoustic Model Systems

#### 2.2. Finite-Amplitude Equation of Motion: The Case $\mu :=$ const.

#### 2.3. Right-Running Equations of Motion for the Case $\mu :=$ const.

## 3. Comparison of Linearized EoMs: The Cauchy Problem

#### 3.1. The RRG Case: ${a}_{0}>0$

#### 3.2. The BPM Case: ${a}_{0}:=0$

#### 3.3. Remarks: RRG vs. BPM

## 4. Comparison of Right-Running, Weakly-Nonlinear, EoMs: Special Cases with $\mu :=$ const.

#### 4.1. Case (I): Damped KdV (dKdV) Equation

#### 4.2. Case (II): Damped Burgers’ Equation

#### 4.3. Case (III): Damped Riemann Equation

#### 4.4. Numerical Results

## 5. The RRG Case with “Artificial” $\mu $

## 6. Discussion: Possible Follow-On Studies

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Corrections to Ref. [13]

- In Equation (4b), replace ${a}_{1}$ with ${a}_{0}$.
- In Equation (4d), replace the exponent $3/2$ with $-3/2$.
- In Equation (13), delete the factor $\nu $.
- In the caption of FIG. 2, replace (16) with (15).

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**Figure 1.**The dKdV case of IBVP (30). (

**a**–

**c**) correspond to $T={T}_{B}^{*}$, $3.6{T}_{B}^{*}$, and $20{T}_{B}^{*}$, respectively, where ${T}_{B}^{*}\approx 1.382$. Red curves: u vs. X for ${a}_{0}=\left(0.005\right)\sqrt{2}$, $\sigma =0$, $\u03f5\beta =0.12$, $\delta =0.12$, and ${\alpha}^{\U0001f7c9}=0.5$. Blue curves: IC given in Equation (30c).

**Figure 2.**The dBKdV case of IBVP (30). (

**a**–

**c**) correspond to $T={T}_{B}^{*}$, $3.6{T}_{B}^{*}$, and $20{T}_{B}^{*}$, respectively, where ${T}_{B}^{*}\approx 1.382$. Purple curves: u vs. X for ${a}_{0}=\left(0.005\right)\sqrt{2}$, $\sigma =0.005$, $\u03f5\beta =0.12$, $\delta =0.12$, and ${\alpha}^{\U0001f7c9}=0.5$. Blue curves (solid): IC given in Equation (30c). Blue-broken curve: ${u}_{1}(X,20{T}_{B}^{*})$ vs. X (see Equation (31)), where we have set ${\u2647}_{1}\left(20{T}_{B}^{*}\right):=\left(29.9\right)\delta {T}_{B}^{*}$ and ${\psi}_{1}\left(20{T}_{B}^{*}\right):=\left(0.1013\right){T}_{B}^{*}$ based on a series of trial-and-error “visual fits”.

**Figure 3.**The damped Burgers equation case of IBVP (30). (

**a**–

**c**) correspond to $T={T}_{B}^{*}$, $3.6{T}_{B}^{*}$, and $20{T}_{B}^{*}$, respectively, where ${T}_{B}^{*}\approx 1.382$. Green curves: u vs. X for ${a}_{0}:=0$, $\sigma =0.005$, $\u03f5\beta =0.12$, $\delta =0.12$, and ${\alpha}^{\U0001f7c9}=0.5$. Blue curves (solid): IC given in Equation (30c). Blue-broken curve: ${u}_{2}(X,20{T}_{B}^{*})$ vs. X (see Equation (31)), where we have set ${\u2647}_{2}\left(20{T}_{B}^{*}\right):=\left(29.946\right)\delta {T}_{B}^{*}$ and ${\psi}_{2}\left(20{T}_{B}^{*}\right):=0$ based on a series of trial-and-error “visual fits”.

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Jordan, P.M. Poroacoustic Traveling Waves under the Rubin–Rosenau–Gottlieb Theory of Generalized Continua. *Water* **2020**, *12*, 807.
https://doi.org/10.3390/w12030807

**AMA Style**

Jordan PM. Poroacoustic Traveling Waves under the Rubin–Rosenau–Gottlieb Theory of Generalized Continua. *Water*. 2020; 12(3):807.
https://doi.org/10.3390/w12030807

**Chicago/Turabian Style**

Jordan, Pedro M. 2020. "Poroacoustic Traveling Waves under the Rubin–Rosenau–Gottlieb Theory of Generalized Continua" *Water* 12, no. 3: 807.
https://doi.org/10.3390/w12030807