Wave speed
Imagine taking photographs of a wave at regular, short intervals. If you picked a crest, you could see it move from frame to frame in the direction of propagation. The propagation of the wave is reflected in this motion of crests (and troughs, and all in-between points), and we can associate a speed with this motion.
Different waves move at different speeds. The most straightforward method to measure the speed of the wave is to pick a crest, follow its motion until it occupies the position at which there was the next crest at the beginning of the observation. From the definition of wavelength it follows that this crest has travelled a distance of one wavelength exactly. At the instant when it reaches the original location of the next crest, there will be a new crest at the location whence it started, which means that exactly one period has passed.
Wave speed
From the arguments above, we can conclude that the speed of the wave is the ratio of the wavelength (the distance the selected crest travelled) to the period (the time this took):
\[v = \frac{\lambda}{T} = \lambda \cdot \frac{1}{T} = \lambda f.\]
Using the angular wave number (\(k = \frac{2\pi}{\lambda},\Rightarrow \lambda = \frac{2\pi}{k}\)) and the angular frequency (\(\omega = \frac{2\pi}{T},\Rightarrow T = \frac{2\pi}{\omega}\)), we can express it in an alternative form:
\[v = \frac{\lambda}{T} = \frac{2\pi / k}{2\pi / \omega} = \frac{\omega}{k}.\]
It is important to note that the speed of the wave only depends on the medium and on the type of the wave, not on the wavelength or the frequency. For example, the speed of sound in air (\(\approx 340\,\mathrm{m}/\mathrm{s}\)) depends on the temperature, humidity and atmospheric pressure, but not on the frequency of the sound. The speed of light in vacuum (\(c \approx 3 \cdot 10^8\, \mathrm{m} / \mathrm{s}\)) is greatest possible speed to our knowledge.
This also means that wavelength and frequency of the wave are not independent for a given type of wave, but one unambiguously determines the other. Knowing the wavelength of light allows one to get the frequency, as they are tied up by the speed of light.
Self-assessment: wave speed
Read the paragraph below and fill in the missing value.