# Bubbly Water as a Natural Metamaterial of Negative Bulk-Modulus

## Abstract

**:**

## 1. Introduction

## 2. Scattering Function and the Oscillator Model for a Single Bubble

#### 2.1. Scattering Function of a Bubble

#### 2.2. The Stiffness Constant and the Mass of the Oscillator Model

## 3. Effective Bulk Modulus and Dispersion Relation

#### 3.1. The Pressure–Volume Relations and the Effective Bulk Modulus

#### 3.2. The Dynamics Equation of the Radial Vibrating Oscillator

#### 3.3. The Effective Bulk Modulus as a Function of Frequency

#### 3.4. Dispersion Relation of Acoustic Modes

#### 3.5. Absorption Effect and the Possibility of Generalization

## 4. Numerical Results

^{1}m. For the purpose of demonstration, in the simulation of lossy media, we take $\delta =0.05$. Figure 4a,b is the results for the lossless media, while Figure 4c,d is the results for the lossy media. Here the dimensionless variable Ka is the product of the wavenumber K in Equation (25) times the lattice constant a of the SC structure. the lattice constant in the SC structure. The real part of the wave number (wave vector) of the mode waves are represented by the blue solid curves, and the red broken lines represent the imaginary part of the wave vector. These dispersion relations are similar to the dispersion relations of the phonon-polariton that mentioned in the previous section. We noticed that the presence of loss or absorption does not destroy the attenuation effect because it is so large.

## 5. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Oscillator model for deriving the effective bulk modulus and effective density. The meaning of the notations are explained in Section 3.

**Figure 2.**The scattering function of a single bubble. The notations in this figure have been defined in Section 2.

**Figure 4.**Dispersion relations for f = 0.001 and f = 0.01, lossless media are shown in (

**a**) and (

**b**). The corresponding results for lossy media are shown in (

**c**) f = 0.001 and (

**d**) f = 0.01.

**Figure 5.**Bulk modulus for lossless cases with f = 0.001 and f = 0.01 are shown in (

**a**) and (

**b**). The corresponding lossy cases are shown in (

**c**) f = 0.001 and (

**d**) f = 0.01.

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**MDPI and ACS Style**

Luan, P.-G.
Bubbly Water as a Natural Metamaterial of Negative Bulk-Modulus. *Crystals* **2019**, *9*, 457.
https://doi.org/10.3390/cryst9090457

**AMA Style**

Luan P-G.
Bubbly Water as a Natural Metamaterial of Negative Bulk-Modulus. *Crystals*. 2019; 9(9):457.
https://doi.org/10.3390/cryst9090457

**Chicago/Turabian Style**

Luan, Pi-Gang.
2019. "Bubbly Water as a Natural Metamaterial of Negative Bulk-Modulus" *Crystals* 9, no. 9: 457.
https://doi.org/10.3390/cryst9090457