The solution of Webster equation for adapters of different shape

Pipe adapters are connecting elements made from different materials, which are used for joining pipes in pipeline structures for various purposes. The use of such products occurs in such pipelines as water pipes, heat pipes, gas pipelines, sewage systems, smoke and ventilation pipelines. For each communication, there is a form of such an adapter and usually it depends on the shape and dimensions of the interconnected pipes. In the problems of propagation of intense sound, in concentrators and other waveguide systems, an equation of the Webster type arises: 12∂2p∂t2-c2∂2p∂x2=c2dlnSxdx∂p∂x+εc2ρ∂2p2∂t2+bρ∂3p∂x∂t2"> The Webster equation is also used in calculations of the acoustic field in inhomogeneous media in the approximation of geometric acoustics. This equation also allows us to study the problems of propagation of sound waves of finite amplitude in absorbing media. In the literature, there are several approaches to solving the acoustic wave propagation in a variable cross-section emitter. If the cross section of such a radiator expands, the amplitude decreases, while if the radiator becomes narrower, there is an increase in the energy flux density. Rayleigh and Webster were the first to obtain independently of each other a plane wave approximation for acoustic propagation in a round variable-section emitter. This equation can be solved for some forms such as exponential, conical, parabolic, catenoid, and sine, but in many cases no analytical solution can be found.