The Bessel functions can be approximated by a series in 2kR taking the first series term of both functions (Jacobsen, 2011)
This expression is valid for ka<0.5. From the imaginary part we get for the mass
Assuming a cylindrical volume V=πR2lc of the fluid above the piston we can calculate the length of the moving mass cylinder to be
meaning that at low frequencies the piston is moving a fluid layer of 0.85 times the radius acting as an inertia without radiation.
2.7.4 Power Radiation
For the radiated power calculation of the piston we took the pressure at the piston surface and integrated the pressure–velocity product over the surface. Due to the fact that the velocity is constant the surface integral involves mainly the pressure as a space-dependent property. In case of vibrating structures with complex shapes of vibration the velocity distribution over the surface is not homogeneous, and we need a more detailed approach.
In the above equation a function with argument (r−r0) is multiplied by the velocity function for r0 and integrated over the two-dimensional space. Mathematically, this can be interpreted as a two-dimensional convolution in space
Thus, when we apply the two-dimensional Fourier transform to the Rayleigh integral the result is the product of the Fourier transform of the vibration shape vz(r0) and the Green’s function in wavenumber space leading to
So, we have replaced the expensive convolution operation by a multiplication. This simplification is at the cost of two-dimensional Fourier transforms that are required to get the expressions in wavenumber domain.
The time averaged intensity of a sound field is given by the product of pressure and velocity (2.45). As the velocity is not uniform over the surface we perform a surface integration over the vibrating area to get the total radiated power
Thus, for the determination of radiated power a double area integral is required that may become computationally expensive.
In the above expression we can also switch to the wavenumber domain. In this case the area integration is replaced by an integration over the two-dimensional wavenumber space.
The double integral is replaced by a single two-dimensional wavenumber integration. Thus, once the shape function is available the power calculation in wavenumber space is much faster than in real space (Graham, 1996).
2.7.4.1 Radiation Efficiency
The radiation efficiency is a quantity that relates the power of a plane wave to the radiated power of a vibrating surface with same surface averaged velocity. The definition of the radiation efficiency was motivated by experimental procedures because it allows the estimation of the radiated power from the measurements of the vibration velocity. The squared average velocity of a vibrating surface is
