What is the formula of ratio of pressure between nodes and antinodes of a standing air wave? On the internet I haven't found any formula related to pressure ratio of nodes and antinodes in a standing air wave. And what is is the dependency of that ratio on the frequency of standing wave.
Also, can you please give the link to the the research paper mentioning this formula since I'm using this in my research.
 A: A one-dimensional standing acoustic wave (without higher harmonics) can be characterized by a pressure profile of the form
$$P(x,t) = P_0 + \delta P_{\rm sound} \sin(kx)\sin(\omega t + \varphi_0)\,,$$
where $k = \omega/c_{\rm s}$ and the origin of the time and space coordinate is chosen so that $x=0$ is at a node, and $t=0$ corresponds to a momentarily flat pressure profile. $P_0$ is then the ambient pressure, and in an ideal monatomic gas we have the speed of sound $c_{\rm s} =\sqrt{P/\rho}$ with $\rho$ the mass density. The wavenumber $k$ must be chosen so that the wave "fits" into your resonator, and that also determines $\omega$. $\delta P_{\rm sound}$ is simply the amplitude of the sound wave and is determined by how much energy you pump into the resonator.
Now you see that the pressure is varying in time, but a node is defined as the points where the pressure stays constant, or $P=P_0$ (in other words, pressure caused by the sound wave is zero at nodes). This happens when $kx=0,\pi,2\pi,...$. On the other hand, an antinode is characterized by maximum variation of pressure and this happens at $kx = \pi/2,3\pi/2,5\pi/2,...$. In that case we have $P = P_0 + \delta P_{\rm sound} \sin(\omega t) $. You can now easily see that the maximum ratio of antinode-node pressure is reached at $\omega t = \pi/2,3\pi/2,...$ and it reads
$$\frac{P_{\rm antinode}}{P_{\rm node}}|_{\rm max} = 1 + \frac{\delta P_{\rm sound}}{P_{0}}\,.$$ 
