## 2 Method
### Low Frequency Region f1
#### 2.2.3 Optimal Loudspeaker Placement  
The back-propagation method described in Section 2.2.2 yields one source term for each of the \(N=1500\) equivalent sources and \(A=62\) keys. Together, the equivalent sources sample the harpsichord sound field in the region of the sound board. Forward propagation of the source terms approximates the harpsichord sound field in the whole half-space above the sound board. This has been demonstrated, e.g., in [47]. Replacing each equivalent source by one loudspeaker is referred to as _acoustic curtain_, which is the origin of wave field synthesis [39; 48]. In physical terms, this situation is a spatially truncated discrete Rayleigh integral, which is the mathematical core of wave field synthesis [7; 8; 39; 49]. A prerequisite is that all equivalent sources are homogeneous radiators in the half-space above the soundboard. This is the case for the proposed radiation keyboard. For low frequencies, loudspeakers without a cabinet approximate dipoles fairly well. Naturally, single loudspeakers with a diameter in the order of 10 cm are inefficient radiators of low frequencies [50]. However, this situation improves when a dense array of loudspeakers is moving in phase. This is typically happening in the given scenario; when excited with low frequencies, the sound board vibrates as a whole [51], so the loudspeaker signals for the wave field synthesis will be in phase. While truncation creates artifacts in most wave field synthesis setups, referred to as _truncation error_[8; 39; 48], no artifacts are expected in the described setup due to natural tapering: at the boundaries of the loudspeaker array, an acoustic short-circuit will occur. However, the acoustic short-circuit also occurs in real musical instruments, as demonstrated in Figure 9. This is because compressed air in the front flows around the sound board towards the rear, instead of propagating as a wave. The acoustic short-circuit of the outermost loudspeakers acts like a natural tapering window. In wave field synthesis installations artificial tapering is applied to compensate the truncation error. The MEM describes the sound board vibration by \(N=1500\) equivalent source terms. Replacing all equivalent sources by an individual loudspeaker is not ideal, because the spacing is too dense for broadband loudspeakers, and it is challenging to synchronize 1500 channels for real-time audio processing. Audio interfaces including D/A-converters for \(L=128\) synchronized channels in audio-cd quality are commercially available, using, for example, MADI or Dante protocol. In wave field synthesis systems, regular loudspeaker distributions have been reported to deliver the best synthesis results [52]. Covering the complete soundboard of a harpsichord with a regular grid consisting of 128 grid points, yields about one loudspeaker every 12 cm. This is a typical loudspeaker density in wave field synthesis systems and yields a Nyquist frequency of about 1.5 kHz for waves in air [7]. Figure 9: Sound field recorded 50 mm above a grand piano sound board (**left**) and back-propagated sound board vibration according to the MEM (**right**). The black dot marks the input location of the acoustic vibrator. Three exemplary loudspeaker arrays are illustrated in Figure 10.  
Every third equivalent source can be replaced by a loudspeaker with little effect on the sound field synthesis precision below 1.5 kHz. This yields \(W=11\) possible loudspeaker grid positions \(\mathbf{\Theta}^{{}^{\prime}}_{w,l}\). At the ideal location of the loudspeaker grid, all loudspeakers lie near antinodes of all frequencies of all keys. In contrast to regions near the nodes, sound field calculations near the antinodes do not suffer from equipment noise, numerical noise, and small microphone misplacements. Instead, all loudspeakers contribute efficiently to the wave field synthesis.