## 1 Introduction  
Flat panel loudspeakers are a promising alternative to conventional loudspeakers. Invisible integration, wide and diffuse radiation and improved room interaction are the advantages of a flat panel loudspeaker design [1]. Furthermore, larger and more powerful devices can be integrated without disturbing customers' views or the aesthetics of a room [2]. In addition to invisible integration, the customer of an audio device expects that an audio signal can be reproduced with sufficient amplitude and quality [3]. The acoustic quality can be expressed in terms of the frequency limits, maximum sound pressure level (SPL), flatness of the pressure response, harmonic and nonlinear distortions and directivity response [4, 5]. This study focuses on improving the directivity response of flat panel loudspeakers. [PERSON] [6] examined numerous studies regarding the influence of loudspeaker directivity on perceived sound quality. A central result is that the spatial uniformity of reflections as a function of frequency is the most important factor. The perceived sound quality of a loudspeaker is a combination of direct sound, slightly delayed and attenuated early reflections, and a progressively delayed reverberant sound field [7]. [PERSON] [8] developed a model based on objective measurements to assess the subjective preference of loudspeakers. He concluded that the uneven radiation of sound energy into a room produces discoloration and unusual spatial artifacts. In addition, loudspeakers with inhomogeneous directivity response are rated worse than loudspeakers with a more uniform response. Resonance in loudspeakers affects the temporal and frequency-dependent radiation [7]. Therefore, a resonance-based speaker system will result in a strong frequency-dependent directivity response, which has been shown in an earlier study [9]. Especially in the higher frequency range (>1 kHz), flat panel loudspeakers tend to radiate more sound off-axis(\(\pm 90^{\circ}\)) than into the listening window (LW, \(\pm 30^{\circ}\)), which can cause coloration and a worse subjective preference rating. An example of such a frequency-dependent directivity response is shown in Figure 1, which presents a comparison of the Genelec 8250A studio monitor (left) and a simple flat panel loudspeaker prototype with a single exciter (right). The studio monitor represents a cone loudspeaker with uniform directivity response at low frequencies. If the acoustic wavelength \(\lambda_{A}\) is much larger than the size of the loudspeaker (width = 286 mm), then the radiation can be described as a monopole radiation. With increasing frequency, \(\lambda_{A}\) becomes shorter, and the loudspeaker can start to beam if \(\lambda_{A}\) is half the size of the loudspeaker dimensions. More sound pressure is radiated to the front than to the side angles. The directivity response of flat panel loudspeakers is much more complicated. At low frequencies, the directivity is also close to monopole directivity. With increasing frequency, the directivity is strongly mode dependent. Even with dimensions that are three times larger (width = 855 mm), the loudspeaker stops beaming to the front for higher frequencies. In contrast, with rising frequency and a much shorter \(\lambda_{A}\) than the speaker dimensions, more sound pressure radiates off-axis. This behavior can also be visualized with the directivity index (DI). In this study, the DI is defined as the difference in the frequency response of the LW (\(\pm 30^{\circ}\)) to the total radiation in the front hemisphere (FH, \(\pm 90^{\circ}\)), as plotted in Figure 2. At low frequencies (\(<\)100 Hz), the first eigenmode dominates, and the directional characteristics correspond to that of a monopole with a DI value of zero. In the mid-frequency range (100 Hz-1 kHz), the DI is not constant but fluctuates around the value of zero. At higher frequencies (\(>\)1 kHz), the DI tends towards a negative value, and more sound radiates off-axis than on-axis.