## 2 Theoretical Analysis of the Directivity Response
### Far Field Radiation of Individual Modes  
In the following section, the directivity of individual modes is investigated. The simulation model described in the previous section is now employed to determine the DI of individual modes. This is a purely numerical analysis, in which only one corresponding eigenmode was extracted and solved in the frequency range between 20 Hz and 5 kHz. For a real panel, the total surface vibration is the superposition of all driven modes and not only one mode is radiating. In this example, only symmetric (odd) modes are investigated since the antisymmetric (even) modes are not excited due to the central exciter position. For this purpose, microphones were placed at a distance of 2 m in 2 steps on the central horizontal plane, and from this, the DI is calculated according to Equation (8). The DI is determined from the difference in the averaged SPL in the LW (\(\pm\)30\({}^{\circ}\)) relative to the entire angular range (\(\pm\)90\({}^{\circ}\)). Figure 5 shows the DIs for different panel eigenmodes.  
Mode 1 \(\times\) 1 behaves as a monopole radiator since the panel moves in phase at all times. As a result, the DI is never negative. With increasing frequency, beaming occurs, which yields a positive, nonzero DI. All other eigenmodes exhibit a characteristic progression of the DI and always have both positive and negative DIs. This result is due to the interference of individual sources on the panel. All eigenmodes have a DI of zero at low frequencies; i.e., the loudspeaker radiates evenly in all directions as a monopole. This effect can be explained by the significantly larger wavelength of the airborne sound wave than the path difference between the individual sources. All eigenmodes have the first peak in the DI at approximately 350 Hz. This peak is produced by the interference of the outermost two sources, which are almost equally spaced for all eigenmodes. The strong dip, which occurs in all modes (except mode 1 \(\times\) 1), corresponds to a quasi coincidence frequency. At this frequency, the bending wavelength \(\lambda_{B}\) in the x direction coincides with the wavelength of the air. As the mode order increases, \(\lambda_{B}\) decreases, and \(f_{c}\) is shifted to higher frequencies. The dip becomes deeper and more sound is radiated off-axis. Furthermore, the maxima and minima of different eigenmodes are observed on vertical lines since similar source spacings occur among the different eigenmodes.  
Every eigenmode, except mode 1 \(\times\) 1, generates a negative DI due to the constructive interference of individual wave paths outside the LW. For a real panel, the first eigenmode, in particular, dominates at lower frequencies, emitting sound relatively homogeneously, similarly to a monopole radiator. With increasing frequency, higher eigenmodes also drive sound radiation. If the bending wavenumber \(k_{B}\) of these eigenmodes is close to the acoustic wavenumber \(k\), then an increased amount of off-axis radiation occurs.  
Figure 5: Overview of the mode shapes and DIs of modes with increasing wavenumber \(k_{x}\) in the x direction. The results are calculated by solving only one individual mode in the simulation software.