Sound in a Vacuum Chamber Pt. 1: Prediction

As a fun demo, I have placed a bluetooth speaker into a vacuum chamber, and then pumped the air out of the chamber. Before pumping the chamber down, you can hear the bluetooth device, albeit faintly; after removing all of the air, it becomes inaudible. This is as should be expected, since the absence of air means that there is no medium for the sound waves to travel through (the device is suspended by a thin wire so that it is not touching any of the chamber walls).

This short demonstration is little different from the cellphone under a vacuum bell-jar demonstration which is described by Mr. Christian Villa in a short write-up for The Physics Teacher. In this demonstration, a cellphone is placed under a bell-jar. A call is placed to the phone, which lights up and then it can be heard ring, although "the jar's thickness alone considerably attenuated the sound." The bell jar is then evacuated of air, and the cell phone is again called, and it again lights up, but this time no sound can be heard. Air is then re-introduced to the jar, the phone is called a third time, and now when the screen lights up the ringtone can be heard.

I wanted to take this idea a step farther by measuring the sound intensity as a function of pressure. The predicted relationship between intensity and ambient pressure is

Where I is the intensity,  is frequency of sound,  is the amplitude of the sound wave particle displacement,  is speed of sound, and  is density of medium in which sound is traveling. The speed of the sound wave is dependent on density and bulk modulus, the frequency is determined by the source frequency, and the amplitude is determined by the source amplitude. Ultimately, this means that the intensity is directly proportional to the ambient density of the air, so via the ideal gas law, , so the sound wave intensity should be linearly proportional to the ambient pressure: we expect I = a P, where P is the pressure and a is a constant which accounts for frequency, amplitude, speed, etc. Thus, doubling the pressure of the chamber should double the intensity of the sound emitted from a speaker, assuming no other changes.