Why is speed of sound in a gas independent of change in pressure?

I was reading about the speed of sound in gases. It is clear that the change in pressure and volume of a gas, when sound waves are propagated through it are adiabatic hence $$v=\sqrt{\frac{B}{\rho}} = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\frac{\gamma P V}{M}},$$ so why isn't the speed of sound in gas not affected by a change in pressure of gas?

In my textbook, the explanation is that $$PV$$=constant, and so $$\sqrt{\frac{\gamma P V}{M}}$$ is not affected by change in pressure. However, as the pressure change in sound wave is adiabatic, it is $$PV^\gamma =$$ constant, not $$PV$$.

The speed of sound, $$c=\sqrt{\dfrac{\gamma P}{\rho}}$$, depends on the pressure $$P$$ and the density $$\rho$$.
However the ideal gas equation, $$PV = MRT \Rightarrow P = \left( \dfrac MV\right)RT =\rho RT$$, still holds so $$P \propto \rho$$ and thus $$\dfrac P \rho$$ is constant.
• $P\propto \rho$ is only true when $T$ is constant, however we assume the sound propagation to be an adiabatic process, which implies that $T$ does not necessarily need to be a constant. How do you justify the answer taking the above argument into account?
• Maybe. It might just work well because the temperature difference is small. However, even in this case, using $PV^{\gamma}=\rm constant$ would still be more accurate. I don't see any reason to not use it, and instead use $PV=\rm constant$.