A periodic wave advances by one wavelength during one period. Its propagation speed is therefore

$$v=\frac{\lambda}{T}=f\lambda.$$

This relation connects the spatial repetition of the wave with its temporal repetition.

Frequency comes from the source

When a periodic source drives a wave, the source fixes the frequency. The medium and wave type determine the propagation speed, so the wavelength adjusts according to

$$\lambda=\frac{v}{f}.$$

Entering another medium

If a wave crosses into a region where its speed changes, its frequency remains fixed by the oscillation of the source, while its wavelength changes.

This change in speed is the basis of refraction.

Wave speed is not particle speed

The propagation speed describes how quickly the wave pattern or phase moves. It is generally different from the instantaneous speed of the particles or fields that oscillate locally.

What determines speed

For mechanical waves, speed depends on properties such as elasticity, tension and inertia of the medium. Electromagnetic waves have a characteristic speed determined by the electromagnetic properties of the material through which they propagate.