Obviously a lot of things cause air to vibrate, but does air have an actual resonance frequency?
[Physics] resonance frequency to air itself
acousticsairatmospheric sciencefrequencyresonance
Related Solutions
Why does maximum resonance occur at triple the length of air column for the previous maximum resonance?
Because resonance in a pipe that is closed at one end occurs when a standing wave of air is generated within the pipe, and this can only happen if the open end of the air column is a displacement antinode (where the wave is at its max amplitude), while the closed end of the air column is a displacement node (where the wave has zero amplitude). If you try to fit a sinusoid into a pipe of a certain length with these constraints, then you get a sequence of possible standing waves as shown in the following diagram:
As you can see, the first possibility is that one quarter of a wavelength $\lambda$ fits in the pipe, while the second possibility is that three quarters of the wavelength fits in the pipe. In other words, in the second case the pipe must be three times as long as in the first as you indicated.
Also, if the reflected wave is out of phase with the incident wave, will the sound waves really cancel each other out, producing no sound at all?
Yes. The amplitudes of the two waves add together. If the waves are completely out of phase, then at every point in space, the amplitudes of the waves will have opposite sign, so when you add the amplitudes, they cancel.
can I somehow match two light waves out of phase with each other to cancel each other out?
Yes. There is a very famous experiment, the double slit experiment, where this actually happens. In fact, it's not to hard to perform this experiment at home. See the following physics.SE question:
Is it possible to reproduce Double-slit experiment by myself at home?
Re question 1: when you learn this stuff in school you usually simplify the system by modelling it as a simple harmonic oscillator so the amplitude of the system will be given by some equation like:
$$ A(t) = A_0 e^{i\omega_0 t} $$
where $\omega_0$ is the natural frequency of oscillation. Typically you study what happens if you apply a force that also varies sinusoidally with time so:
$$ F(t) = F_0 e^{i\omega t} $$
where the frequency of the applied force, $\omega$, is not necessarily the same as the natural frequency of the oscillator, $\omega_0$. This is what your teacher means by saying that the force has a frequency - they mean the frequency $\omega$. In your teacher's example of a swing the swing has some natural frequency. If you are applying a force periodically, i.e. pushing on the swing in a repetitive way, then the force you apply also varies with time (though it is more like a square wave than a sine wave). The amplitude of the swing is greatest when the frequency with which you push the swing matches the natural frequency of the swing.
Re question 2: when you start learning this stuff you typically start with an undamped simple harmonic oscillator, i.e. the oscillator doesn't lose any energy. If you solve the equations of motion you find that the amplitude goes to infinity when the frequency of the driving force $\omega$ is equal to the natural frequency $\omega_0$. This is because you're putting energy in but the oscillator doesn't lose any energy so the energy just keeps growing.
A real oscillator like a swing loses energy through friction, and we call it a damped harmonic oscillator. The rate at which the oscillator loses energy is related to its amplitude, so as you push your system (the swing in this case) the amplitude increases until the rate of energy loss matches the rate you're putting energy in. So the harder you push your system the more the swing will move. In principle there is no maximum amplitude, though in real life there obviously is since at some point the swing will go over the top and start revolving instead of swinging to and fro. A swing isn't a simple harmonic oscillator! It's only approximately simple harmonic for small swing amplitudes.
Re question 3: Most objects will have a range of resonant frequencies called normal modes. However there are usually many normal modes and the frequencies of these modes are related to the object's shape in a complicated way. The Wikipedia article gives some examples of normal modes, or do a YouTube search for "normal modes" to find loads of videos on the subject - some really impressive!
Best Answer
When you contain air, or any gas for that matter in a bounded volume $V_o$ (container) at static pressure $P_o$ you can define a somewhat linear constant called Elastance . $$E=\gamma\frac{P_o}{V_o}$$
where $\gamma$ is the specific heat ratio of the gas.
The elastance can be thought of as a bulk property of the contained gas that tends to behave as a spring.
Also if you consider gas inside a long slender container we can define another constant, the Inertance
$$I=\frac{\rho l}{A}$$
where $\rho$ i sthe gas density, $l$ is the length and $A$ the cross sectional area of the container respectively. The inertance can be though of as a bulk property of the contained gas that tends to behave as a bulk mass.
So together if we consider these bulk properties acting together within a closed container we can expect to see a natural frequency, resonance of
$$\omega=\sqrt{\frac{E}{I}}$$ given sufficient energy to drive it relative to what energy losses might be also be present.
From these properties, certain geometric assumptions you can derive an expression for the resonant frequency of a Helmholtz Resonator.
So at least by this modeling you need some geometric boundary to establish a resonant condition within the gas - to sustain the properties. Consider the volume $V_o$ larger and larger at constant $P_o$ ; the Elastance and frequency essentially vanish towards zero.
So I don't believe the bulk properties themselves, without a constraining geometry can sustain a resonant system, at least within the domain of acoustic frequencies (sonic and ultrasonic).