Here and there, near and far: How proximity and separation affect scattering in concert halls

Abstract

Classical theory of scattering deals with plane waves, waves which are so far from their source that they form straight fronts of sound, with pressure amplitude and particle velocity exactly in phase. If such waves are much larger than objects they encounter inverse fourth power (Rayleigh) scattering obtains; if the waves are about the same size as the objects resonant (Mie) scattering occurs; if waves are much shorter than the objects specular (Ufimtsev) scattering is the rule. These all affect sound in the far field. But if sources are closer than a wavelength to objects their waves encounter the plane approximation is not valid; pressure amplitude and particle velocity are not in phase, so resonance-like phenomena occur. These occur on stages of concert halls: bass instruments producing waves 2 m or longer always are close to the floor; some, like tympani and viols, can be close to vertical surfaces too. This sort of scat-tering enhances fundamentals of notes with respect to the overtones, strongly affecting the timbre of such instruments.