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- The phase speed of a wave in the derivation of the non-relativistic Doppler shift does not change between reference frames. Shouldn't the Galilean transformation apply?

Consider the situation where an observer at rest on the ground measures the frequency of a siren which is moving away from the observer at speed ##v_{Ex}##. Let ##v_w## be the speed of the sound wave. Let ##\lambda_0##, ##f_0##, ##\lambda_D##, and ##f_D## be the wavelengths and frequencies measured by the emitter and ground observer. Let T be the wave's period measured by the ground observer. Following the standard non-relativistic doppler shift derivation, ##f_D## = ##\frac{v_w}{\lambda_D}## = ##\frac{v_w}{\lambda_0 + v_{Ex}T}## = ##\frac{v_w}{\frac{v_w}{f_0} + \frac{v_{Ex}}{f_0}}## = ##\frac{f_0}{1 + \frac{v_{Ex}}{v_w}}##.

My question, is why is ##\lambda_0## = ##\frac{v_w}{f_0}##? If the wave speed on the ground is ##v_w##, shouldn't the wave speed as measured by the emitter be calculated using the Galilean transformation? Instead it is the same value as measured by the ground observer.

My question, is why is ##\lambda_0## = ##\frac{v_w}{f_0}##? If the wave speed on the ground is ##v_w##, shouldn't the wave speed as measured by the emitter be calculated using the Galilean transformation? Instead it is the same value as measured by the ground observer.

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