Chapter 8 Approximation methods

Chapter 8Approximation methods Relevant methods in acoustical problems:

Websters horn -regular perturbation method

Multiple scales- singular perturbation method that describes problems in which in the problem several length scales act in the same direction

3. Helmholtz resonator with non-linear dissipation-an application of multiple scale technique which allows us to investigate the small non-linear terms that will be seen to represent a small damping.Slowly varying ductsReflection at an isolated turning point- a turning point is described as a point where the wave number vanishes.7. Ray acoustics in temperature gradient- When a sound wave propagates in free space through a medium that varies on a much larger scale than the typical wave length, the same ideas ofmultiple scales may be applied. In contrast to the duct, where the wave is confined by the duct walls, the waves may now freely refract and follow curved paths7. Refraction in shear flow

8. Matched asymptotic expansions Very often it happens that a simplifying limit applied to a more comprehensive model gives a correctapproximation for the main part of the problem, but not everywhere: the limit is non-uniform.9. Duct Junction - very simple problem that can be solved with matched asymptotic expansions is the reflection andtransmission of low-frequency sound waves through a junction of two ducts with different diameter.10. Co-rotating line-vortices - In an inviscid infinite 2D medium a stationary line vortex producesa time-independent velocity and pressure field. Two of such vortices,however, move in each others velocity field. Two equallystrong and equally orientated vortices rotate around a common centre,and produce a fluctuating velocity and pressure field (for a fixedobserver).Websters horn( also known as Websters horn)

A regular perturbation method,known as method of slow variation,since the typical axial length scale is much greater than the transverse length scale.

Websters horn equation

8.8 Matched asymptotic expansions