## 2 Compound Sound Sources  
### Compound Sources in Noise Control  
The sound field in small enclosed spaces is governed by a strong spatial and frequency non-uniformity of the acoustic energy, due to the distinct and intense modes. There is a significant impact on the sound quality of the audio content, causing the feeling of distortion and sound coloration. The acoustic response of a source is determined substantially by its position, which affects its coupling with the modal field. Therefore, in the usual case that a steady-state low-frequency industrial source operates, the noise reduction level is critically dependent on the location of secondary sources, and it is more effective when their number increases. However, the physical size of the usually big-volume low-frequency sources may be a constraint on their positioning.  
The directivity in low frequencies, as its principles were presented by [PERSON] (1973) concerning the gradient loudspeakers [PERSON] (1973), can be utilized to partially overcome the spatial constraints. In multipole topologies or compound sources, the directivity results from the radiated wave superposition of the elementary sources and depends on their setup and driving parameters. A system consisting of multiple components can be effective for equal coverage in the low-frequency range by adapting to a defined listening area [PERSON] (1973). Each component of the subwoofer array is driven by a dedicated signal, adjusted in amplitude and phase.  
The control of the directivity pattern can have an advantageous effect on noise attenuation [PERSON] (1973). The feasibility of active directivity control for the reduction of sound radiation to a certain direction from the near to the far field has been presented with numerical simulations and experiments with a planar error microphone array ([PERSON] _et al._, 2016). In the free field, the cancellation of a primary monopole sound field using secondary multipole sources has been studied ([PERSON] _et al._, 2016). A numerical study on the types of secondary sources for sound radiation control concluded that arrays of monopoles are suitable for controlling a primary source ([PERSON] _et al._, 2016). The compound sources can be used to substitute secondary monopoles, reduce the number of control channels, and improve the noise reduction performance of a system for different primary sound fields, as presented in ([PERSON] _et al._, 2016) for frequencies higher than 200 Hz and in free-field conditions.  
The efficiency of the source decreases for low-frequency radiation as the order increases in the free field ([PERSON] _et al._, 2016). On the contrary, in closed or semi-closed spaces, room boundaries positively affect the directivity of the sources and improve the noise attenuation performance ([PERSON] _et al._, 2016; [PERSON] _et al._, 2016; [PERSON] _et al._, 2016). The spatial radiation of a compound source can be adapted in such a way that it is coupled with the modal noise field and produces a destructive effect of the latter. The noise reduction also depends on the orientation angle of the source configuration, based on the mechanism that a surface can convert a vertical dipole into a longitudinal quadrupole ([PERSON] _et al._, 2016). At low frequencies, if a dipole is aligned parallel to a reflective surface, the maximum pressure amplification can be achieved, acting as a gradient source. The defined directivity causes the excitation of modes in the direction of the dipole axis ([PERSON] _et al._, 2016).  
In the present study, two compound source configurations are applied, aiming at the coupling of the modal field of the investigated room: the longitudinal and the lateral quadrupoles ([PERSON] _et al._, 2016). The topology of each one is depicted in Figure 1. They consist of two dipoles with their monopoles radiating out of phase at a small distance \(d\), so that \(d\ll\lambda\), and \(d\ll r\), where \(\lambda\) is the wavelength and \(r\) the observation distance ([PERSON] _et al._, 2016).