A general derivation of the classical Doppler effect in 3D space

Abstract

Abstract The classical Doppler effect has already been studied in many textbooks and papers. However, the formula currently used to describe the Doppler effect for a source and a receiver moving with constant velocities is not suitable for practical applications. Indeed, the formula makes use of two angles that depend on the state of the source-receiver system both at the emission instant and at the reception instant of sound, while in most real cases the state of the system is known only at one of the two instants. In this paper, two equivalent formulas are derived for the Doppler shift: one depending solely on the state of the system at the emission instant, and one depending solely on the state of the system at the reception instant. The paper also provides two more general expressions that describe the Doppler shift evolution over time. The expressions derived in this paper are compared with the existing ones in some special cases, obtaining consistent results.