The one-dimensional bubble: Theory, experiment, and relevance to exposure of divers to low frequency sound, and of lung to lithotripsy
Abstract
In this paper the oscillation dynamics of a linear gas pocket in a liquid are analysed. These include several features of potential use at undergraduate level for illustrating characteristics beyond those normally encountered in expositions of simple harmonic oscillators (for example, the simple linear spring-bob system). The oscillator in question is unusual firstly in that, as with all gas bubbles in liquid, the inertia is invested primarily in the surrounding liquid, rather than in the pocket itself. In other words the radiation mass is far larger than the mass of the trapped gas. This is demonstrable even in the simplest analysis, that of linear undamped small-amplitude oscillations. Secondly, the oscillator is nonlinear. The equation of motion of a damped, forced, one-dimensional bubble is obtained, the nonlinearity arising through the dependence of the oscillator stiffness (or alternatively the gas compressibility) on the amplitude of oscillation. In reduced forms, this equation can be solved analytically for the natural frequency of undamped oscillations in the linear limit, and also in the nonlinear regime of finite-amplitude free oscillation, where the fundamental frequency is found to be amplitude-dependent. Whilst analytical solutions of the undamped, unforced form of the equation of motion can be obtained in phase space, the full nonlinear damped forced equation must be solved numerically. These solutions are compared with those of the linear undamped analysis. Experimental measurements of the phase and amplitude response of a laboratory one-dimensional bubble are made, and compared with theory. Viscous effects, which for three dimensional bubbles in water are second-order and often justifiably neglected in simple analyses based on the gas stiffness and liquid inertia, are in the case of one-dimensional water bubbles comparable with these other effects (i.e. stiffness and inertia), and cannot be neglected. The analysis is used to predict: (1) the response of air bubbles within the ear canals of divers to low frequency underwater sound (from several tens of hertz to several hundred, such as is being used increasingly in the ocean for climate monitoring); and (2) the theoretical potential of bubbles in blood to cause hemorrhage of blood vessels, particularly within the lung, when subjected to high-amplitude ultrasonic pulses.
- Publication:
-
NASA STI/Recon Technical Report N
- Pub Date:
- May 1995
- Bibcode:
- 1995STIN...9529817L
- Keywords:
-
- Aerospace Medicine;
- Bubbles;
- Diving (Underwater);
- Exposure;
- Gas Dynamics;
- Gas Pockets;
- Harmonic Oscillators;
- Mathematical Models;
- One Dimensional Flow;
- Physiological Responses;
- Resonant Frequencies;
- Sound Waves;
- Undamped Oscillations;
- Underwater Acoustics;
- Blood Vessels;
- Clinical Medicine;
- Compressibility;
- Equations Of Motion;
- Free Vibration;
- Frequency Response;
- Linear Systems;
- Lungs;
- Middle Ear;
- Nonlinear Equations;
- Phase Shift;
- Stiffness;
- Viscous Flow;
- Fluid Mechanics and Heat Transfer