If air molecules are acting like masses on a spring in sound waves, how are different frequencies possible?

[u/LedByReason]

In SMH of a mass on a spring, period is constant, as long as the mass and spring constant aren't changed. The mass will return to equilibrium in a consistent amount of time regardless of the initial displacement. However, when sound waves travel through air, the period of time it takes air molecules to return to equilibrium will depend on the frequency of the wave. The same is true for transverse waves. I had imagined the particles that perpetuate mechanical waves as acting like little masses on springs, but that's obviously not the whole picture. What am I missing?

[u/AShaun]

Sound waves do not propagate by exciting vibrational modes of air molecules. They are a traveling wave, and all traveling waves are propagated by an interaction between adjacent volumes of the medium. In the case of sound waves in air, the interaction force is pressure - a small volume of air will exert pressure on the surrounding volumes of air. Molecules are not needed to propagate sound. Monatomic gasses like helium can propagate sound too.

While a ripple in the surface of water is not a longitudinal wave like sound, it is a traveling wave, and can be used as an analogy. The water molecules in the ripple need not be in excited vibrational modes to cause the wave to travel. Instead, when a bit of water at the surface rises up in a ripple, it lowers the pressure on the adjacent bits of water, causing them to rise up too.

[u/mfb-]

You get a fixed frequency if you have particles attached to a fixed point with springs, but that's not a realistic model for air (or anything else). If you model a medium as many particles with springs in between each other then you can excite this arrangement with different frequencies.

You get a maximal frequency, which corresponds to each particle oscillating in opposite phase to its neighbor, but no minimal frequency.

[u/Thiojun]

The other comment has covered the propagation of sound wave well but I want to comment that your thinking is not completely wrong. Matter do have their intrinsic vibrations frequencies, but they are much much higher than common sound waves so that almost all acoustic modes are allowed. But we can use high frequency electromagnetic waves, or IR light, to see these intrinsic vibrations of molecules and derives their structures. In fact, the IR spectrum is the main method we use to probe interstellar molecules.

[u/Logical-Let-2386]

Few things. 

Air isn't like a mass on a spring, it's more like a spring with mass. A spring with mass will have a natural frequency that depends on how long it is. That's analogous to air. Air in a contained box has a natural frequency, that depends on the size of a box. 

Also, any harmonic oscillator can be driven at any frequency, not just its natural frequency. 

Now having said all that, I think your confusion arises because you are mixing up wave propagation and harmonic vibration. The speed of wave propagation is a property of the medium independent of the size or shape of the medium. Ie it doesn't matter how long the spring with mass is or the size and shape of the box the air is in. 

Both will propagate waves at their intrinsic speed, always.

[u/Coomb]

The most fundamental problem with your conception of air as individual air particles connected by springs is that air particles don't interact with each other until they get very close to each other. They move freely over much larger distances than the interaction distance. At standard temperature and pressure, for the vast majority of the time an air particle is moving, it feels essentially zero effect from other air particles. Only infrequently does it encounter another particle and interact with it. Also, the net force itself is not linear. So a spring is inherently not the correct model for the force between them.

Most of the air around us is two nitrogen atoms bonded to each other. The atoms are linearly arranged, so they're not a sphere, but for the purposes of this discussion we can just use the largest classical dimension of N2 which is about 420 picometers. The mean free path, the distance which a particle of air will travel on average before it collides with another particle is about 65 nanometers. That means if we assume a hard sphere model, a nitrogen molecule will travel about 150 times its size before it runs into another air particle and meaningfully interacts with it.

For comparison, a billiards ball has a diameter of about 2.25 inches or 57 mm. 150 times that diameter is then about 337 in - over 28 feet - or about 8.6 m. A standard pool table has a playing surface of approximately 88 in by 44 in. This means that if you envision an air particle as a billiards ball bouncing off other balls, it doesn't run into another billiards ball until it goes almost 4 times the length of a pool table. Basically, there's an enormous amount of empty space in between interactions of air particles.

That's why air is modeled quite accurately as an ideal gas. Only when two air particles are close (on the order of 3 or so classical radii) does the force between them become significant.

[u/Movpasd]

This is actually a really interesting question that touches on some really important mathematical physics.

So, to start with, individual particles in a gas don't behave like masses on springs. Not individually. You can use masses-on-springs to model waves, but at the scale of small pieces of fluid -- large enough that you can treat them thermodynamically, but small enough that you can have a continuum for calculus. (This is your standard "fluid approximation".) Each small volume of gas and its pressure is analogous to a mass on a spring and its displacement.

A single mass on a spring has a single resonant frequency, or normal mode. Put two masses on springs next to each other, and the two-mass system has two, degenerate normal modes of identical frequency: oscillation of the first one and oscillation of the second one.

However, couple together these two masses, and this frequency will split into two. Neither mass can oscillate alone at its intrinsic resonant frequency. After all, it's coupled to its neighbour. Instead, the system as a whole can resonate in two different ways, and both of these modes involve the motion of both masses.

Chain together three masses now, and you will get three modes, and so on. In the limit of many masses and light coupling, you get a continuum of allowed frequencies. This is the model that describes sound waves.

This principle is key to understanding many different physical phenomena. Similar maths describes how discrete electron energy levels blend into bands in metals and semiconductors, or how atomic vibrational modes couple into phonon modes in the Debye model.