Wave spacing at the observer
Observed crests are uniformly spaced at $\lambda_{\rm obs}$.
Observed crests are uniformly spaced at $\lambda_{\rm obs}$.
Connectors show how each rest line shifts under $v_r$.
Ripples compress ahead and stretch behind because air carries the disturbance.
Each crest travels at $c$ in vacuum. The spacing is set at emission and remains uniform when observed.
Compare the non-relativistic approximation with the exact relativistic relation and watch where they diverge.
$$\lambda_{\rm obs} = \lambda_0\left(1 + \frac{v_r}{c}\right),\qquad \nu_{\rm obs} = \frac{\nu_0}{1 + v_r/c}$$
$$\lambda_{\rm obs} = \lambda_0\sqrt{\frac{1+\beta}{1-\beta}},\qquad \nu_{\rm obs} = \nu_0\sqrt{\frac{1-\beta}{1+\beta}},\qquad \beta = \frac{v_r}{c}$$
$$z = \frac{\lambda_{\rm obs} - \lambda_0}{\lambda_0},\qquad \beta = \frac{(1+z)^2 - 1}{(1+z)^2 + 1}$$