Does wavenumber/wavelength change for a moving observer ?

GOOGLE:

Yes, the wave number k, which is inversely related to wavelength (

$k=2\pi /\lambda$

), does change for a moving observer due to the Doppler effect, resulting in a different observed wavelength and frequency; if the observer moves toward the source, they encounter waves more frequently (higher

$f$

, shorter

$\lambda$

, higher

$k$

), and if moving away, less frequently (lower

$f$

, longer

$\lambda$

, lower

$k$

). 

How it works (Classical Doppler Effect): 

• Observer Moving Towards Source: The observer "catches up" to the wave crests, effectively compressing the wavelength. This leads to a higher observed frequency (
  

  $f^{\prime }$
  ) and a shorter wavelength (
  

  ${\lambda }^{\prime }$
  ).
• Observer Moving Away From Source: The observer moves away from the crests, stretching the wavelength. This results in a lower observed frequency (
  

  $f^{\prime }$
  ) and a longer wavelength (
  

  ${\lambda }^{\prime }$
  ). 

Relationship to Wave Number (

$k$

): 

• The wave number
  

  $k$
  is defined as
  

  $k=2\pi /\lambda$
  (or
  

  $k=\omega /v$
  for angular frequency
  

  $\omega$
  and wave speed
  

  $v$
  ).
• Since the observed wavelength (
  

  ${\lambda }^{\prime }$
  ) changes with the observer's motion, the observed wave number (
  

  $k^{\prime }$
  ) must also change proportionally:
  

  $k^{\prime }=2\pi /{\lambda }^{\prime }$
  . 

In Summary: 

• The observer perceives a different number of waves per unit time (frequency) and distance (wave number) due to their own motion relative to the wave.
• This change in
  

  $k$
  is a direct consequence of the observer's velocity and the Doppler effect, affecting both the spatial (k) and temporal (frequency) aspects of the wave.