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1、Thus far we have developed a set of differential equations that describe the local behavior of sound waves.We become acquainted with some other important types of wave behavior - spherical waves.Our emphasis here is on the waves themselves and in the following chapter we will pursue the interaction
2、of these waves with their sources., 牛牛文库文档分享,Thus far we have developed a s,Spherical spreading describes the decrease in level when a sound wave propagates away from a source uniformly in all directions., 牛牛文库文档分享,Spherical spreading describes,Sound levels are therefore constant on spherical surfac
3、es surrounding the sound source. Sound levels decrease rapidly as sound spreads out from a sphere with a radius of r0 to a larger sphere with a radius r., 牛牛文库文档分享,Sound levels are therefore con,Spherical waves are represented in complex form by,The wave diminish in amplitude as the distance from th
4、e source increase.,the particle velocity is not in phase with the pressure., 牛牛文库文档分享,Spherical waves are represente,The Acoustic impedance,Separating za into real and imaginary parts, we have,The first term is the acoustic resistance, and the second term is the acoustic reactance., 牛牛文库文档分享,The Aco
5、ustic impedance Separat,The acoustic impedance of spherical wave Real , imaginary parts and phase, 牛牛文库文档分享,The acoustic impedance of sphe,We emphasize again that the acoustic impedance depends on the frequency , as well as on the medium, the wave type, and the geometric position.Neither the frequen
6、cy nor the distance alone is so important as is the dimension combination kr. For kr 1, Za is predominantly resistive, meaning that energy fed into such waves propagates away from the source, never to return., 牛牛文库文档分享,We emphasize again that the ac,The larger the value of kr, the less curved are th
7、e wave fronts and the more nearly do these waves act just like plane waves.But for kr 1 the reactive component is large, and it is positive ( masslike )This means that pushing on such a wave is mainly a matter of storing kinetic energy in motion of the liquid, where it can be recovered later by the
8、source. (like energy in an electrical inductor ), 牛牛文库文档分享,The larger the value of kr, th,The intensity and power of spherical waves,There is an alternative view that suggests a simpler insight into the spherical wave.If we had considered the ratio u/p instead of p/u , the resulting admittance would
9、 be a much simpler function., 牛牛文库文档分享,The intensity and power of sph,For a harmonic spherical wave, the acoustic intensity is :,Notice that the formula has been found to be exactly true for both plane and spherical waves., 牛牛文库文档分享,For a harmonic spherical wave,The acoustic intensity can be express
10、ed in another form :,The average rate at which energy flows through a closed spherical surface of radius r surrounding a source of symmetrical spherical waves is :, 牛牛文库文档分享,The acoustic intensity can be,The average rate of energy flow through any spherical surface surrounding the origin is independ
11、ent of the radius of the surface, a conclusion that is consistent with conservation of energy in a lossless medium.,Since, 牛牛文库文档分享,The average rate of energy flo,A diverging spherical wave has a peak acoustic pressure of 2Pa at a distance of 1m from the source at standard atmospheric pressure and t
12、emperature. What is its intensity at a distance of 10m from the source?,Example, 牛牛文库文档分享,A diverging spherical wave has,Assume the source is emitting a constant amount of energy to the sound wave. For diverging spherical waves, the area of the wavefront increases as the waves are traveling farther
13、and farther from the source. Hence intensity of such waves diminishes with distance of propagation.At a distance of 1m from the source,Where p0 =1.21kg/m3 is the density of air, and c=343m/s is the speed of sound in air at standard atmospheric pressure and temperature, 牛牛文库文档分享,Assume the source is
14、emitting,At a distance of 10m from the source, the effective sound pressure will change but the power radiated will remain the same., 牛牛文库文档分享,At a distance of 10m from the,3-10 Cylindrical Waves,Sound waves radiate from something acting more or less as a line source rather than a point source. Then
15、 it is natural to describe such waves in cylindrical coordinates,The simplest solution using these coordinates will be when p does not depend on, 牛牛文库文档分享,3-10 Cylindrical WavesSound,A simple approximation for spreading loss in a medium with upper and lower boundaries can be obtained by assuming tha
16、t the sound is distributed uniformly over the surface of a cylinder having a radius equal to the range r and a height H equal to the depth of the ocean., 牛牛文库文档分享,A simple approximation for spr,In this case , the pressure is a function of radial distance and time, Then the wave equation simplifies t
17、o,If the waves are harmonic, such waves are represented in complex form by,The equation (3-10-3) is called the Bessel equation with the general solution, 牛牛文库文档分享,In this case , the pressure is,Here J0(kr) is the zero order Bessel functionN0(kr) is the zero order Neumann function,Introduce Hankel fu
18、nction,For the diverging waves, 牛牛文库文档分享,Here J0(kr) is the zero order,When kr 1,The diverging waves are, 牛牛文库文档分享,When kr 1The diverging waves,The acoustic intensity is, 牛牛文库文档分享,The acoustic intensity iswww.n,The relative intensity level decrease less rapidly for cylindrical than for spherical spr
19、eading.,The total power crossing a cylinder surrounding the source equals the intensity times the area of the cylinder:, 牛牛文库文档分享,The relative intensity level d,The acoustic impedance,When kr 1,When kr 1, 牛牛文库文档分享,The acoustic impedance When kr,When kr 1, the acoustic impedance,In this limit the aco
20、ustic impedance again approaches p0c0, since very large cylindrical wave fronts will be practically indistinguishable from plane waves.We may remark that , like the spherical waves, cylindrical waves take on a much simpler form when kr becomes large.The pressure amplitude of cylindrical waves approa
21、ch, 牛牛文库文档分享,When kr 1, the acoustic imp,Homework,Textbook P276 3-8, 3-10Spherical acoustic waves of frequency 125Hz are emitted from a small source. At a radial distance of 1.5m from the source, what is the phase angle between acoustic pressure and particle velocity? Find the magnitude of the acoustic impedance at this point., 牛牛文库文档分享,HomeworkTextbook P276 3-8,
