A specific cavitation phenomenon occurs inside the stems of trees. The internal pressure in tree conduits can drop down to significant negative values, which causes the nucleation of bubbles. The bubbles exhibit high-frequency oscillations just after their nucleation. In the present study, this phenomenon is modeled by taking into account acoustic waves produced by bubble oscillations. A dispersion equation is derived, which is then used to calculate the resonance frequency and the attenuation coefficient of the bubble oscillations. Radiation damping is found to be predominant in comparison with viscous damping, except for very small bubbles. A typical number of oscillation cycles before the complete damping of the oscillation is found to be of the order of 10, as observed for cavitation bubbles in biomimetic synthetic trees.

Orthodontists are often concerned about the prognosis of root-filled teeth, particularly when extractions are required for orthodontic treatment. This review provides guidance on assessing the quality of root fillings, as well as the factors which affect the prognosis of root-filled teeth. The implications of previous traumatic injuries and the likelihood of root resorption during orthodontic tooth movement are discussed.

In this paper, we consider the classical Riemann problem for a generalized Burgers equation,with a spatially dependent, nonlinear sound speed, ℎ 𝛼 (𝑥) ≡ (1 + 𝑥 2 ) −𝛼 with 𝛼 > 0, which decays algebraically with increasing distance from a fixed spatial origin. When 𝛼 = 0, this reduces to the classical Burgers equation. In this first part of a pair of papers, we focus attention on the large-time structure of the associated Riemann problem, and obtain its detailed structure, as 𝑡 → ∞, via the method of matched asymptotic coordinate expansions (this uses the classical method of matched asymptotic expansions, with the asymptotic parameters being the independent coordinates in the evolution problem; this approach is developed in detail in the monograph of Leach and Needham, as referenced in the text), over all parameter ranges. We identify a significant bifurcation in structure at 𝛼 = 1 2 .