Vibroacoustic Simulation. Alexander Peiffer

target="_blank" rel="nofollow" href="#fb3_img_img_165892ac-ca2f-5a48-bdcf-dde5a6d6b4ce.png" alt="normal upper Pi equals one-half upper R e left-parenthesis bold-italic p bold-italic upper Q Superscript asterisk Baseline right-parenthesis equals one-half upper R e left-parenthesis bold-italic upper Z Subscript a Baseline bold-italic upper Q bold-italic upper Q Superscript asterisk Baseline right-parenthesis equals upper R e left-parenthesis bold-italic upper Z Subscript a Baseline right-parenthesis upper Q Subscript r m s Superscript 2"/> (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.

      2.4.1.4 Point Sources

      A point source is a spherical source with an infinitely small radius. Performing the limit kR→0 for Equation (2.73) leads to the velocity potential for point sources of strength Q

       (2.90)

      The pressure and velocity field of such a source is given by

(2.91)

      and

       (2.92)

      All other relations regarding power and intensity expressions remain. We see that the limit is expressed for kR and not for the wavelength. The reason is that it is the ratio of a characteristic length (in this case the sphere radius) to the wavelength that determines if the geometrical details must be considered or not. In other words, a wave of a certain wavelength doesn’t care about details that are much smaller.

      With the D’Alambert solution for spherical waves (2.66) we can also derive a point source in time domain

       (2.93)

      The point source is of great importance for the solution of the inhomogeneous wave equation in combination with complex boundary conditions. Any source can be reconstructed by a superposition of point sources as shown in Section 2.7.

Performing the limit process with kR→0 and taking the power from equation 2.86 we get the intensity of the point source:

       (2.94)

      and the total radiated power

       (2.95)

      with radiation impedance following from this

       (2.96)

      2.5 Reflection of Plane Waves

A plane wave striking a plane surface is a first example of interaction with obstacles. Imagine a configuration as shown in Figure 2.7. The impedance of the surface is z2, and it is given by using the velocity vz perpendicular to the plane.

      Figure 2.7 Reflection of a plane wave at an infinite surface with impedance Z2. Source: Alexander Peiffer.

      Without loss of generality the wave front is parallel to the y-axis and all properties are functions of x and z. The solution in the half space of z>0 is the superposition of two plane waves.

       (2.97)

      With the following arguments of the exponential function

       (2.98)

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