Vibroacoustic Simulation - Alexander Peiffer

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1 plus j k r Over r EndFraction right-parenthesis normal upper Phi equals StartFraction bold-italic upper Q left-parenthesis omega right-parenthesis Over 4 pi r squared EndFraction left-parenthesis StartFraction 1 plus j k r Over 1 plus j k upper R EndFraction right-parenthesis e Superscript minus j k left-parenthesis r minus upper R right-parenthesis"/> (2.75)

      2.4.1.1 Field Properties of Spherical Waves

      The acoustic impedance z is according to Equation (2.38)

      In contrast to the plane wave, the specific acoustic impedance is not real. It contains a resistive and a reactive part. When the resistive part is dominant the pressure is in phase with the velocity. When the reactive part dominates, the velocity is out of phase to the pressure. The out of phase component does not generate any power in the sound field as it was the case for moving a mass or driving a spring. The motion is partly introduced into the local kinetic energy, and this part can be recovered as it is the case for an oscillating mass. For the acoustic field of a spherical source the reactive field represents the near-field fluid volume that is carried by the sphere motion but not emitting a wave.

      Figure 2.6 Reactance and resistance of specific acoustic impedance of a pulsating sphere. Source: Alexander Peiffer.

      There are two limit cases in Equation (2.76):

      1 kr≪1; the wave length λ is much larger than distance r.

      2 kr≫1; the wave length λ is much smaller than distance r.

       k r much-less-than 1 long right double arrow bold-italic z equals j rho 0 c 0 k r equals j rho 0 r omega (2.77)

      and a resistive part equal to plane waves for (ii)

       k r much-greater-than 1 long right double arrow bold-italic z equals rho 0 c 0 (2.78)

      2.4.1.2 Field Intensity, Power and Source Strength

      The time averaged radiated intensity is

       mathematical left-angle upper I left-parenthesis r right-parenthesis mathematical right-angle Subscript upper T Baseline equals StartFraction ModifyingAbove upper Q With caret squared k squared rho 0 c Over 32 pi squared r squared left-parenthesis 1 plus k squared upper R squared right-parenthesis EndFraction equals StartFraction upper Q Subscript r m s Superscript 2 Baseline k squared rho 0 c Over 16 pi squared r squared left-parenthesis 1 plus k squared upper R squared right-parenthesis EndFraction (2.79)

      The total radiated power can now be evaluated from Equation (2.54) and the integration surface 4πr2

       mathematical left-angle normal upper Pi mathematical right-angle Subscript upper T Baseline equals 4 pi r squared mathematical left-angle upper I left-parenthesis r right-parenthesis mathematical right-angle Subscript upper T Baseline equals StartFraction upper Q Subscript r m s Superscript 2 Baseline k squared rho 0 c 0 Over 4 pi left-parenthesis 1 plus k squared upper R squared right-parenthesis EndFraction (2.80)

      The mean square pressure can be derived from (2.74) and expressed by the intensity using (2.45)

      Replacing the intensity in (2.81) gives the rms pressure in the spherical sound field due to power

      2.4.1.3 Power and Radiation Impedance at the Surface Sphere

      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

       bold-italic upper Z Subscript a Baseline equals StartFraction bold-italic p Subscript normal s normal u normal r normal f Baseline Over bold-italic upper Q Subscript normal s normal u normal r normal f Baseline EndFraction equals StartFraction bold-italic p Subscript normal s normal u normal r normal f Baseline Over upper A Subscript s Baseline bold-italic v Subscript normal s normal u normal r normal f Baseline EndFraction upper A Subscript s Baseline equals 4 pi upper R squared (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)

       bold-italic upper Z Subscript a Baseline equals StartFraction j rho 0 c 0 k upper R Over 4 pi upper R squared left-parenthesis 1 plus j k upper R right-parenthesis EndFraction (2.84)

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

       bold-italic z Subscript a Baseline equals 4 pi a squared bold-italic upper Z Subscript a Baseline equals StartFraction j rho 0 c 0 k upper R Over 1 plus j k upper R EndFraction (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

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