Phase Speed of Wave in non-relativistic Doppler Shift Derivation

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Summary:
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.

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