# Nonlinear Behavior of High-Intensity Ultrasound Propagation in an Ideal Fluid

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Nonlinear Wave Propagation and Shock Formation

- Change of the wave propagation speed due to drift with velocity $u$.
- Change in local sound speed from ${c}_{0}\mathrm{to}{c}_{u}$.

## 3. Nonlinear Interactions Within the Acoustic Mode

#### 3.1. Head Shock

^{3}, and the specific heat ratio $\gamma $ for the fluid is 1.4. The particle speed behind the head shock is ${u}_{r1}$ is $212.39\mathrm{m}/\mathrm{s}$. The unknown ${P}_{r1}$ can be calculated using Equation (10):

^{**}. Shocks always move at supersonic speed as observed from ahead of the shock, and subsonic speed as observed from behind the shock [10]. (** Calculations for ${\rho}_{r1}$ and ${U}_{1}$ are shown in Appendix B).

#### 3.2. Rarefaction

#### 3.3. Tail Shock

^{***}. (*** Calculations for ${P}_{2}$, ${\rho}_{2}$, and ${U}_{2}$ are shown in Appendix C).

## 4. Decaying of N-Wave

#### 4.1. Shock Wave Overtaken by a Rarefaction Wave

- A transmitted shock and a reflected rarefaction wave
- A transmitted shock and a reflected compression wave that steepens into a shock wave
- A transmitted rarefaction wave and a reflected rarefaction wave
- A transmitted rarefaction wave and reflected compression wave that steepens into a shock wave

#### 4.2. Shock Wave Overtaking a Rarefaction Wave

_{r12}–P

_{r23}) and transmitted shocks (P

_{1}and P

_{2}). After the interaction, the state behind the head shock and ahead of forwarding rarefaction have the same pressure. Thus, the forward rarefaction has attenuated the head shock to a Mach wave, just as its tail reaches the head shock. Similarly, in the case of the tail shock overtaking the backward rarefaction, states (Pr

_{22}–P

_{r23}) and (P

_{2}) are similar pressure. Thus, the incident tail shock has been attenuated to a Mach wave, just as it reaches the head of the backward rarefaction [16]. Thus, after the interaction pressure jump across head shock becomes nearly $1\left(101,269/101,325=0.99~1\right)$ which before the interaction was $2.26\left(229,029/101.325\right)$, similarly pressure jump across tail shock becomes unity $\left(99,954/100,527=0.9943~1\right)$ which before the interaction was $2.46\left(99,954/40,055\right)$.

## 5. Discussions of Results

## 6. Summary and Conclusions

## 7. Suggestions for Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## Appendix B

## Appendix C

## References

- Gallego-Juárez, J.A.; Graff, K.F. (Eds.) Introduction to Power Ultrasonics. In Power Ultrasonics: Applications of High-Intensity Ultrasound; Woodhead Publishing: Cambridge, UK, 2015; pp. 1–6. [Google Scholar] [CrossRef]
- Sapozhnikov, O.A. High-intensity Ultrasonics Waves in Fluids: Nonlinear Propagation and Effects. In Power Utrasonics Appl. High-Intensity Ultrasound; Gallego-Juárez, J.A., Graff, K.F., Eds.; Woodhead Publishing: Cambridge, UK, 2015; pp. 9–35. [Google Scholar] [CrossRef]
- Pierce, A.D. Acoustics: An Introduction to Its Physical Principle and Applications; Acoustical Society of America: Woodbury, NY, USA, 1994. [Google Scholar]
- Blackstock, D.T.; Hamilton, M.F. Progressive Waves in Lossless and Lossy Fluids. In Nonlinear Acoustics; Hamilton, M.F., Ed.; Academic Press: San Diego, CA, USA, 1998. [Google Scholar]
- Murnaghan, F. Finite Deformation of an Elastic Solid. Am. J. Math.
**1937**, 59, 235–260. [Google Scholar] [CrossRef] - Bjørnø, L. Introduction to nonlinear acoustics. Phys. Procedia
**2010**, 3, 5–16. [Google Scholar] [CrossRef][Green Version] - Leighton, T. What is Ultrasound? Prog. Biophys. Mol. Biol.
**2007**, 93, 3–83. [Google Scholar] [CrossRef] [PubMed] - Pierce, A.D. Entropy Acoustic, and Vorticity Mode Decomposition as an Initial step in Nonlinear Acoustic Formulations. In Nonlinear Acoustics at the Beginning of the 21st Century; Moscow State University: Moscow, Russia, 2002; pp. 11–19. [Google Scholar]
- Brunet, T.; Jia, X.; Johnson, P.A. Transitional nonlinear elastic behaviour in dense granular media. Geophys. Res. Lett.
**2008**. [Google Scholar] [CrossRef][Green Version] - Courant, R.; Friedrichs, K.O. Supersonic Flow and Shock Waves; Springer: New York, NY, USA, 1948. [Google Scholar]
- Landau, L.D.; Lifshits, E.M. Fluid Mechanics; Pergamon Press: New York, NY, USA, 1987. [Google Scholar]
- Bukiet, B. Application of Front Tracking to Two-Dimensional Curved Detonation Fronts. Soc. Ind. Appl. Math.
**1988**, 9, 80–99. [Google Scholar] [CrossRef] - Zhenting, Z. Long-Term Management and Condition Assessment of Concrete Culvert; New Jersey Institute of Technology: Newark, NJ, USA, 2017. [Google Scholar]
- Igra, O. One-Dimensional Interactions. In Handbook of Shock Waves; Ben-Dor, G., Igra, O., Tov, E., Eds.; Academic Press: San Diego, CA, USA, 2001; Volume 2. [Google Scholar]
- Glass, I.I.; Heuckroth, L.E.; Molder, S. On the One-Dimensional Overtaking of a Shock Wave by a Rarefaction Wave; Institute of Aerophysics, University of Toronto: Toronto, ON, Canada, 1959. [Google Scholar]
- Bremner, G.F.; Dukowicz, J.K.; Glass, I.I. On the One-Dimensional Overtaking of a Rarefaction Wave by a Shock Wave; Institute of Aerophysics, University of Toronto: Toronto, ON, Canada, 1960. [Google Scholar]

**Figure 2.**Schematic representation of acoustic distortion and shock formation during nonlinear propagation [Adopted based on figures in Blackstock and Hamilton (1998) and Leighton (2007)].

**Figure 5.**Overtaking of a shock wave by a weak rarefaction wave [Adopted based on a figure in Glass et al (1959)].

**Figure 6.**Overtaking of a rarefaction wave by a weak shock wave [Adopted based on a figure in Glass et al (1959)].

Power Level, W (watt) | Frequency, f (kHz) | Intensity, I (W/m ^{2}) | Amplitude, A (10 ^{−6} m) | Particle Speed, V (m/s) |
---|---|---|---|---|

1500 | 20 | 9,477,702.96 | 1691 | 212.39 |

1500 | 500 | 9,477,702.96 | 67.6 | 212.39 |

1500 | 2000 | 9,477,702.96 | 16.9 | 212.39 |

500 | 20 | 3,152,941.98 | 976.3 | 122.61 |

500 | 500 | 3,152,941.98 | 39.1 | 122.61 |

500 | 2000 | 3,152,941.98 | 9.76 | 122.61 |

150 | 20 | 94,772.6 | 534.7 | 67.22 |

150 | 500 | 94,772.6 | 21.4 | 67.2 |

150 | 2000 | 94,772.6 | 5.34 | 67.22 |

50 | 20 | 315,924.2 | 308.7 | 38.78 |

50 | 500 | 315,924.2 | 12.3 | 38.78 |

50 | 2000 | 315,924.2 | 3.08 | 38.78 |

Particle Speed, V(m/s) | P_{1}(Pa) | P_{r1}(Pa) | P_{r}(Pa) | P_{r2}(Pa) | P_{2}(Pa) | P_{r12} = P_{r13}(Pa) | P_{r22} = P_{r23}(Pa) | Head Shock, U_{1}(m/s) |
---|---|---|---|---|---|---|---|---|

212.39 | 101,325 | 229,029 | 101,161 | 40,055 | 99,954 | 101,269 | 100,527 | 490.8 |

122.61 | 101,325 | 164,678 | 102,005 | 60,027 | 101,058 | 102,004 | 101,152 | 421.7 |

67.22 | 101,325 | 132,853 | 102,220 | 76,414 | 101,281 | 102,228 | 101,294 | 383 |

38.78 | 101,325 | 118,629 | 102,258 | 86,225 | 101,317 | 102,260 | 101,319 | 364 |

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**MDPI and ACS Style**

Kewalramani, J.A.; Zou, Z.; Marsh, R.W.; Bukiet, B.G.; Meegoda, J.N. Nonlinear Behavior of High-Intensity Ultrasound Propagation in an Ideal Fluid. *Acoustics* **2020**, *2*, 147-163.
https://doi.org/10.3390/acoustics2010011

**AMA Style**

Kewalramani JA, Zou Z, Marsh RW, Bukiet BG, Meegoda JN. Nonlinear Behavior of High-Intensity Ultrasound Propagation in an Ideal Fluid. *Acoustics*. 2020; 2(1):147-163.
https://doi.org/10.3390/acoustics2010011

**Chicago/Turabian Style**

Kewalramani, Jitendra A., Zhenting Zou, Richard W. Marsh, Bruce G. Bukiet, and Jay N. Meegoda. 2020. "Nonlinear Behavior of High-Intensity Ultrasound Propagation in an Ideal Fluid" *Acoustics* 2, no. 1: 147-163.
https://doi.org/10.3390/acoustics2010011