Читать книгу Vibroacoustic Simulation - Alexander Peiffer - Страница 69

The characteristic impedance of the sphere exactly at the surface at radius R can be translated into the radiation impedance of the sphere as a volume source. The radiation impedance is defined as the ratio of pressure to source strength at the vibrating surface

(2.83)

If we assume a constant harmonic surface velocity vR we get for the radiation impedance of the breathing sphere and according to the acoustic impedance (2.76)

(2.84)

The acoustic radiation impedance is the ratio pressure and normal velocity at the sphere’s surface

(2.85)

We can now use this impedance to eliminate either p or vr. The power transmitted by a vibrating sphere using Equation (2.54) over the surface of the sphere

(2.86)

(2.87)

or for constant source pressure

(2.88)

It is instructive to see the mechanical properties considering the limit cases from above and extract the mass that is moved by the surface. From Newtons’s law a force given by F=4πR2p leads to an in-phase acceleration of jωvr of a mass m

hence

(2.89)

For kR≪1 we get: m=4πR3ρ0=3Vsphρ0. Thus, at low frequencies the source surface motion carries three times the fluid volume of the sphere. This motion near the source is called an evanescent wave, because it is oscillatory motion of fluid that does not radiate.