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We know that the speed of light depends on the density of the medium it is travelling through. It travels faster through less dense media and slower through more dense media.

When we produce sound, a series of rarefactions and compressions are created in the medium by the vibration of the source of sound. Compressions have high pressure and high density, while rarefactions have low pressure and low density.

If light is made to propagate through such a disturbance in the medium, does it experience refraction due to changes in the density of the medium? Why don't we observe this?

Actually this effect has been discovered in 1932 with light diffracted by ultra-sound waves.
In order to get observable effects you need ultra-sound
with wavelengths in the μm range (i.e. not much longer than light waves),
and thus sound frequencies in the MHz range.

See for example here:

• 
  

  On the Scattering of Light by Supersonic Waves
  
  by Debye and Sears in 1932

  
  
  

  
  

  

  
  

  
  


• 
  Propriétés optiques des milieux solides et liquides soumis aux
  vibrations élastiques ultra sonores
  
  (Optical properties of solid and liquid media subjected to ultrasonic elastic vibrations)
  
  by Lucas and Biquard in 1932
  
  

  
  

  Résumé : Dans cet article sont décrites les principales propriétés optiques présentées par les milieux solides et liquides, soumis à des vibrations élastiques ultra sonores dont les fréquences s'étagent de 600 000 à 30 millions de période par seconde. Ces ultra sons ont été obtenus par la méthode de Langevin à l'aide de quartz piézo-électriques excités en haute fréquence. Dans ces conditions, et suivant les valeurs relatives des dimensions des longueurs d'onde élastiques, des longueurs d'onde lumineuses, et de l'ouverture du faisceau lumineux traversant le milieu étudié, différents phénomènes optiques son t observables. Dans le cas des longueurs d'onde élastiques les plus petites allant jusqu'à quelques dixièmes de mm, on observe des figures de diffraction lumineuse analogues à celles d'un réseau lorsque les rayons lumineux incidents cheminent parallèlement aux plans d'ondes élastiques.
  

  
  

  
  


• 
  The diffraction of light by high frequency sound waves: Part I
  
  by Raman and Nagendra Nathe in 1935
  
  

  
  

  A theory of the phenomenon of the diffraction of light by sound-waves of high frequency in a medium, discovered by Debye and Sears and Lucas and Biquard, is developed.

  
  

  
  

I have seen it with standing waves in water, a PhyWe demonstration experiment. The frequency 800 kHz, which gives a distance between nodes of about a millimeter. The standing wave is in a cuvette, between the head of a piezo hydrophone transducer and the bottom. When looking through the water, one sees the varying index of refraction as a "wavyness" of the background.

I could not find a description of this online, but I found this about demonstration experiments in air: https://docplayer.org/52348266-Unsichtbares-sichtbar-machen-schallwellenfronten-im-bild.html

A few factors contribute to this:

• Air has low index of refraction therefore optical effects arising from its mechanical pressure will be weak;


• Even loud sounds have low mechanical pressure. Wolfram Alpha database lists 200 pascals as pressure of jet airplane at 100 meters, which works out as ~0.5% pressure difference between peak and trough;


• Waves do not cause harsh boundary between high and low pressures;


• Sources of loud sounds typically cause other phenomena that obscure this. Combustion creates light and heat, and rapid pressure release can force water in the air to become opaque.

Even with all that, it is possible to magnify the effect using distant point light and either by merely observing refracted patterns or creating a setup where half of the refocused image is blocked. Using the second technique it is possible to observe clap of hands.

You can see the effect of density change on refractive index due to heating of air. For a simple example, light a candle and look through the air column directly above the flame. The flame heats air which rises, but the flow is turbulent, so you'll see objects on the other side of the air column shimmer as the stream of hot air wavers from side to side.

You can see this effect when you look across a paved surface on a hot sunny day.

You won't see this effect with sound, at least not at typical listening levels because the density changes are too small (as noted in one of the other answers).