## 1. Introduction

## 2. Results

#### 2.1. Symmetries in the Dynamics of Constrained Bubble

#### 2.2. Dynamics of a Tethered Bubble

#### 2.3. Bubble Dynamics Close to Interface

#### 2.4. Bubble Oscillations Near an Interface between Two Liquids

## 3. Discussion

## 4. Conclusions

## Funding

## Conflicts of Interest

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**Figure 1.**Schematic illustration of toroidal coordinates. (

**a**) The toroidal coordinates of any point are given by the intersection of a sphere, a torus, and an azimuthal plane. Spheres of different radii that pass through the focal ring are specified by coordinates $\psi $. The surfaces of constant $\eta $ are non-intersecting tori of different radii. The coordinate $\alpha $ is the azimuthal angle about the z axis. (

**b**) The right panel shows circles of constant $\eta $ and $\psi $ observed in the section of the azimuthal plane

**Figure 2.**Schematic illustration of bi-spherical coordinates. (

**a**) A surface, on which the bi-spherical coordinate $\xi $ is constant, represents a sphere of a radius $a/|sinh\xi |$ with center at ($z=acoth{\xi}_{0}$, $x=y=0$). An orthogonal surface, on which the bi-spherical coordinate $\vartheta $ is constant, is formed by the circular arc with center ($x=acot\vartheta $, $z=0$) and radius $a/|sin\vartheta |$ rotating around the axis 0z. The coordinate $\alpha $ is the azimuthal angle about the z axis. (

**b**) Circles of constant $\xi $ and $\vartheta $ in the $(x;z)$ plane are shown in panel (

**b**).

**Figure 3.**Normalized natural frequency as function of distance to boundary $\left(h/{R}_{0}\right)$. Solid circles show the measured values [3]. The dashed line corresponds to case where only monopole component of interaction between bubble and its mirror image is taken into account.

**Figure 4.**Variation of the dimensionless damping coefficient, ${F}_{v}$, entering in Equation (27). The dash-dotted line describes the approximate shape of ${F}_{v}\left(\kappa \right)$, corresponding to large distances.

**Figure 5.**The shape of the bubble at the moments of the largest expansion (dashed line) and the greatest compression (dot-dashed line). For comparison the bubble equilibrium shape is shown by the solid line. All length-dimensional values are normalized to the bubble equilibrium radius ${R}_{0}$. The bubble is located at the distance $h=1.5{R}_{0}$ from the rigid bottom. When calculating the dependencies shown in the figure, the value of the dimensionless amplitude $\zeta \left(0\right)/{R}_{0}=0.2$ was used.

**Figure 6.**Normalized natural frequency as function of distance to boundary $\left(h/{R}_{0}\right)$ and the ratio of densities m. Dash-dotted curve corresponds to bubble above sediment layer ($m=1.95$). Dotted line describes dependence of natural frequency for bubble in blood near artery wall ($m=0.86$). Dashed line corresponds to bubble in sediments ($m=0.51$).

**Figure 7.**Normalized radiation damping factor as function of distance to boundary. Upper solid curve, lower solid curve and thick dashed line correspond to bubble above sediment layer, to bubble in sediments, to bubble in blood near arterial, respectively. Thin dashed lines correspond to case where only monopole component of interaction between bubble and its mirror image is taken into account.

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