## 2 Compound Sound Sources
### Investigations with Quadrupole Control Sources  
In room acoustics, a wave-dominated field can be predicted accurately using a wave-based computational method, such as the finite element method (FEM) [44]. Simulations in a typical room model with finite elements are presented in order to investigate the advantage of employing a quadrupole over a simple dipole as a control source. The model dimensions are (x, y, z) \(=\) (4 m, 3.5 m, 2.8 m), as depicted with the control positions (CP) in Figure 2. Both primary and secondary sources were modeled as point sources, introducing pressure waves with a mass flow rate \(\dot{m}\) = \(\rho_{0}Q\) (kg/s) and monopole strength \(Q\) (m\({}^{3}\)/s) as volume velocity [45]. A combination of point sources is included as a forcing term in the acoustic wave equation for the steady-state response as  
\[\
abla^{2}\hat{p}+\left(\frac{\omega}{c_{0}}\right)^{2}\hat{p}=-\frac{i\omega \rho_{0}}{V}\hat{Q}=-\frac{i\omega}{V}\dot{\hat{m}}, \tag{1}\]  
where \(\hat{Q}\) is the strength of a point mass source operating at the forcing frequency \(f=\omega/2\pi\) (Hz). The noise field minimization with a stepped sweep of the parameters is examined in a rectangular area of 0.9 m\({}^{2}\)(z = 0.5 m). The adjustable driving parameters are the moments of the dipoles through the strength or amplitude of monopoles and their relative polarity. The simulations are carried out at the room's natural frequencies of 49 Hz (0, 1, 0) and 65 Hz (1, 1, 0). For the quadrupole point sources, \(d=20\) cm (\(D=40\) cm), and for the dipole, \(d=40\) cm, so that the acoustic centers have the same geometry length. The metric for the control evaluation is the noise level reduction using the average sound pressure level (SPL) \(L_{\mathrm{av}}\) determined by (2), where \(L_{i}\) are the sample levels. The control performance with the highest attenuation in the area is given in Figure 2.  
\[L_{\mathrm{av}}=10\,\mathrm{lg}\left(\frac{1}{n}\sum_{i=1}^{n}10^{L_{i}/10}\right) \tag{2}\]  
The noise attenuation using a dipole source, either at CP 1 or CP 2, is limited. For the area under control, the average noise level reduction is satisfyingly higher with a quadrupole. The attenuation of 49 Hz with the longitudinal source at CP 2 reaches up to 55 dB. Instead of using the longitudinal source for the case of 65 Hz at CP 2, the highest noise attenuation is attained with a lateral quadrupole. This example indicates the flexibility of using a quadrupole source. It offers a more adjustable control field, while many different areas can be controlled along with the investigated area, as seen in the case of 49 Hz at CP 2. The second dipole offers more degrees of freedom to adjust the radiation pattern of a quadrupole while varying its driving parameters and coupling with a modal field. A more uniform sound field is obtained after the modal field control. In the case of 49 Hz at CP 2 or 65 Hz at CP 1, it seems that the control is almost global.  
It should be noted that the limited noise reduction which is attained with one dipole at a specific area presupposes higher source amplitude than using a quadrupole. These investigations assume point sources and the results may seem very optimistic for such low frequencies. However, they can be a guideline for the following noise control measurements with quadrupoles as secondary sources.  
Figure 1: Compound sources, with \(d\) as the distance between the two poles of a dipole component, \(D\) as the distance between the dipoles, and the encircled minus and plus symbols determining the 180\({}^{\circ}\) phase difference of the dipole components. (**a**) Longitudinal and (**b**) lateral quadrupole.