In Optics lecture we took a formula for the speed of a wave which is:
$$ v=\frac{\omega}{k} $$
where $\omega$ is number of complete vibrations per second:
$$ \omega=\frac{2\pi}{\tau} $$
and:
$$ k = \frac{2\pi}{\lambda} $$
Now since the sound is a wave, does that mean that its velocity is not constant in air?
Take your expression:
$$ v=\frac{\omega}{k} \tag{1} $$
We'll write $\omega=2\pi f$, rather than $\omega = 2\pi/\tau$, where $f$ is the frequency of the wave and substitute for $\omega$ and $k$ in equation (1) to get:
$$ v = f\lambda \tag{2} $$
The wavelength $\lambda$ is the distance the wave moves in one cycle, and the frequency is the number of cycles per second, so if we multiply these together we get the distance moved per second. And the distance moved per second is of course just the speed. That's why the equation (2) applies.
So equation (1) and its equivalent form equation (2) aren't telling us anything very exciting. They just tell us there is a relationship between the frequency, wavelength and velocity without telling us anything about which if any of these are constant. As it happens the velocity of sound in an ideal gas is given by:
$$v = \sqrt{\gamma\frac{P}{\rho}} $$
where $P$ is the pressure and $\rho$ is the density of the gas. $\gamma$ is a constant called the adiabatic index. Or with some messing around we can rewrite this in terms of the temperature:
$$ v = \sqrt{\gamma RT} $$
So (for an ideal gas) the speed of sound is determined by the temperature, and it's constant when the temperature is constant.
There is an addition needed here. The speed of a wave is constant through a homogeneous isotropic medium. Sound wave is a mechanical wave which propagates as compression and rarefaction through the medium. It actually pressurizes and depressurizes certain region of medium. The medium has several properties upon which the velocity of sound relies up on. If these properties are same throughout the medium (i.e., the medium is isotropic), then the velocity of sound wave is constant through a given medium.
Yes, the velocity of sound through air is a constant at a given temperature. In dry air at 20 °C (68 °F), the speed of sound is 343.2 metres per second
In general, the speed of sound in gas depends on composition, temperature and pressure. If those three are constant the speed of sound is constant.
The speed of sound in a perfect gas does not depend on frequency. In real gases, however, the speed of sound depends (slightly) upon frequency, this is called dispersion (have a look at Is speed of sound really constant?). As well, the speed of sound being constant depends on small displacements.
