acoustics - Apparent frequency and wavelength in the Doppler effect

For a transverse wave(or for pressure waves required to produce longitudinal waves), the motion perpendicular to the direction of propagation of the wave is governed by an equation like $y = Asin(\omega t)$ in case of harmonic waves(here $\omega$ is angular frequency of simple harmonic motion). The time period ($T$) of the (harmonic) wave is then $2*\pi/\omega$. And frequency$(\nu)$ is $1/T$. Also, velocity of propagation of a wave $v = \nu*\lambda$.

Now the Doppler effect shows us that if the source of the wave or the observer is in motion the frequency$(\nu)$ of the wave changes(Here I am talking about longitudinal waves propagating in a stationary medium). My initial doubt was how the frequency and the time period of the wave could change if $\omega$ did not change. However, this animation (scenes 4 and 5 of 8) helped me realise(?) that, in the case of the observer moving and the source remaining stationary, the wave itself does not change, but the apparent frequency of the wave as seen by the observer changes.

Also, as there is relative velocity between the source and the wave I thought that the apparent change in frequency would be due to the apparent change in velocity and wavelength $(v=\nu*\lambda)$ which means that $\lambda$ would be constant.

However, in the case where the source is moving with respect to the observer, which is stationary, (scene 7 of 8) the state of motion of the observer is the same as in the case where both the observer and the source are stationary(scene 3 of 8). This must mean that there is an inherent change in the wavelength and frequency of the wave, visible to any stationary observer. However, according to my initial arguments, how can frequency of the wave change if there is no change in $\omega$?

So my final questions are:-

1.In the case where the observer is moving, is it true that the apparent wavelength does not change from the case where both source and observer are stationary? And in the case where the source is moving, is it true that both wavelength and frequency are changing from the normal case? And if so, why is it that the relative motion is not what matters?

2.In the case where the source is moving, how can frequency change when $\omega$ remains constant?

Edit: Clarified that I am talking about longitudinal waves propagating in a stationary medium.

Answer

The animations that you linked refer to sound waves - that is, there is a stationary medium involved in the transmission of the sound. This means that it matters whether the source or the observer is moving relative to the medium.

Answering your two specific questions: yes, if the observer is moving, the wavelength of the waves does not change (they are generated by the source and will propagate in the same way regardless of whether somebody is there to observe them). When the source is moving, the wavelength is increased (as the animation makes clear) so while the source frequency is unchanged, the frequency observed at a particular point in space does change. However, an observer traveling in the same direction as the source and at the same speed, would observe the original frequency (the distance between them doesn't change). In that scenario, the wavelength is still changed - because the sound waves are traveling towards the observer faster than the speed of sound in the medium.