Assessment of modal density and free path distribution in central-planned halls

Central-planned halls are highly widespread in the historical architectures of the Western world, such as rotundae, Christian baptisteries, and Roman tombs. In such halls, whispering galleries, flutter echoes, and sound focusing are the acoustic phenomena mainly investigated by scholars. Instead, modal behaviour and free path distribution are generally less treated in literature. The present study explores the modal density at low frequencies and the relationship with the most recurrent free path lengths in three historical nearly circular spaces, here assessed as case studies. Acoustic measurements allowed the collection of objective experimental data, i.e., room impulse responses and the resulting room acoustics criteria. Wave-based numerical models allowed for the investigation of the eigenfrequencies distribution, while the free paths trend has been experienced through ray-based models. The main outcomes of both analyses show the prominence of the circular modes, rather than the diametral and the elevation ones. Moreover, the mean free path calculated using ray-tracing proves to be higher than the theoretical value commonly assumed for any kind of shape. The consequent longer reverberations compared to halls with other shapes and the same volume justify the significant support historically provided to sound signals by circular halls.