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.

中文翻译：

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3D空间中经典多普勒效应的一般推导

许多教科书和论文已经研究了经典的多普勒效应。然而，目前用于描述以恒定速度移动的源和接收器的多普勒效应的公式并不适合实际应用。事实上，该公式使用了两个角度，这两个角度取决于声源-接收器系统在声音的发射时刻和接收时刻的状态，而在大多数实际情况下，系统的状态仅在其中一个时刻是已知的。两个瞬间。本文针对多普勒频移推导了两个等价公式：一个仅取决于发射时刻的系统状态，一个仅取决于接收时刻的系统状态。该论文还提供了两个更通用的表达式来描述多普勒频移随时间的演变。本文推导的表达式在一些特殊情况下与已有的表达式进行了比较，得到了一致的结果。