Vibroacoustic Simulation. Alexander Peiffer

      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.