## 2 Theoretical Analysis of the Directivity Response
### Far Field Radiation of Plane Waves  
Theoretically, the propagation of structure-borne sound by bending waves and the subsequent sound radiation can be represented in the wavenumber space (also referred to as k-space), enabling the consideration of the ratio of the bending wavelength \(\lambda_{B}\) to the acoustic wavelength \(\lambda_{A}\) according to [PERSON] (1975). The wavenumbers in the three-dimensional space are related to the wavenumber of the plane airborne sound wave:  
\[k^{2}=k_{x}^{2}+k_{y}^{2}+k_{z}^{2}. \tag{1}\]  
where \(k_{x}\) and \(k_{y}\) represent the wavenumbers of the bending wavelength in the x direction and y direction, respectively, and \(k_{z}\) is the wavenumber for the radiated sound field perpendicular to the vibrating surface and is a dependent value. The parameter \(k\) defines the propagating acoustic wavenumber with the propagating speed of sound \(c_{A}\) defined as follows:  
\[k_{x}=\frac{2\pi}{\lambda_{B,x}}=\frac{\omega}{c_{B,x}}\,\ k_{y}=\frac{2\pi}{ \lambda_{B,y}}=\frac{\omega}{c_{B,y}}\,\ k=\frac{2\pi}{\lambda_{A}}=\frac{ \omega}{c_{A}}. \tag{2}\]  
The ratio of \(\lambda_{B}\) and \(\lambda_{A}\) is frequency dependent, caused by the dispersion of bending waves. Therefore, three interesting cases exist: subsonic radiation (\(\lambda_{A}>\lambda_{B}\)); coincidence (\(\lambda_{A}=\lambda_{B}\)); and supersonic radiation (\(\lambda_{A}<\lambda_{B}\)). Depending on the ratio of the bending wavenumber and the acoustic wavenumber, different panel areas are sources for the acoustic far field (1975). To visualize this dependency, a simple comparison of a mode without a nodal line (mode 1 \(\times\) 1) and a mode with nodal lines (mode 1 \(\times\) 7) is made, as schematically shown in Figure 3 and by numerical simulation in Figure 4.  
For the special case of \"subsonic radiation\" or \"below coincidence\" \(k_{x},k_{y}>k\), the following relation results for the wavenumber \(k_{z}\):  
\[k_{z}=\pm j\sqrt{k_{x}^{2}+k_{y}^{2}-k^{2}} \tag{3}\]  
For this special case, the wave field decays with an increase in distance (evanescent wave), and the bending waves on the surface cancel each other, which results in inefficient radiation into the far field. By assuming an infinite panel with subsonic wave components no radiation would occur. In the case of a finite panel, uncancelled sources on the panel exist, as shown in Figure 3. If the bending wavelength or the distance between the \"individual  
Figure 3: Theoretical comparison of the radiation behavior below and above the coincidence of two different mode shapes. If a mode shape with nodal lines occurs, the coupling to the air depends on the ratio of the bending wavelength \(\lambda_{B}\) to the acoustic wavelength \(\lambda_{A}\). sources\" on the surface is smaller than the acoustic wavelength, sound sources at the edges will remain, which have no corresponding source to cancel--as shown in Figure 4.  
When \(\lambda_{B}<<\lambda_{A}\), the mode \(1\times 7\) radiates less sound because an acoustic short circuit dominates the near-field region. Nevertheless, the radiation corresponds to that of a monopole. At this frequency, the \(1\times 1\) mode is almost an ideal monopole with a uniform spherical radiation pattern. As the frequency increases and the acoustic wavelength decreases (\(\lambda_{B}<\lambda_{A}\)), the directivity response becomes more complex. In this region, the acoustic wavelength is no longer large enough to eliminate all local sources on the panel due to the acoustic short circuit. As a result, the edges remain as sources for the acoustic far field. The sound pressure distribution of mode \(1\times 7\) indicates that the sources are located at the left and right edges of the panel. Their interference generates radiation to the front, and to a large extent, to the side. If the bending wavelength corresponds to the acoustic wavelength (\(\lambda_{B}=\lambda_{A}\)), then coincidence occurs.