The first thing to say about breathing in helium is: DON'T DO IT! There are reported cases of people trying to have fun with the effect of this gas, but actually dying instead. It's not that helium is poisonous; it's actually the least reactive of all the chemical elements. The problem is simply that helium displaces the air from your lungs, and you need the oxygen in air to live. That said lets talk science!
In order to understand the effect of helium on speech, it is useful to talk a bit about how we make sounds to each other. When you choose to speak, you begin by causing the vocal chords in your throat to vibrate (wiggle up and down). That action, in turn, causes the air in you throat and mouth to experience moving waves of alternating compression (increased density) and rarefaction (decreased density). These are actually sound waves. When someone hears a sound, it is its frequency, or pitch - rather than its wavelength that he or she perceives. If you think about it a bit, you will recognize that, for a fixed wavelength of sound, the frequency must depend on how fast the waves are moving. The faster they move, the higher the frequency. We'll come back to that point in a moment.
The next thing to know about speech is that your wiggling vocal chords don't make sound of just one frequency. There are a whole bunch of sound waves of different frequency, all produced at once. Some of the frequencies come out louder than others, because of a phenomenon called resonance. Resonance really means the matching of one thing with another. You encounter it in a playground if you try to push a small child on a swing. If you time the pushing motion of your arms just right, you can use the energy produced by your muscles with maximum efficiency to propel the swing higher and higher, whereas if you push at the wrong time the effect is much less pleasant for all involved! The matching of your pushes with the motion of the swing is an example of resonance. In the case of sound waves, resonance can occur if the wavelength of the sound for air in a pipe (or some multiple of the wavelength) exactly matches the length of the pipe. All of the pipes that you see in a church organ have different sizes so that they will be in resonance with sound of different wavelengths, and hence frequencies. The equivalent of the organ pipe when you speak is the combination of your throat and mouth - sometimes called the vocal tract. It's more sophisticated than a simple pipe, because you can change its shape, but nevertheless it does emphasize some of the sound waves produced by your vocal chords by being in resonance with them. It is the emphasized frequencies that you hear most obviously when someone talks to you.
Finally we come to the helium. The sound waves in a gas require the particles (atoms or molecules) of the gas to get pushed about. Not surprisingly, lighter particles can be pushed about more easily, and so sound waves travel faster through gases consisting of lighter particles. Helium atoms are, on average, only about one quarter as massive as the nitrogen and oxygen molecules that are the primary constituents of air, and so sound travels much faster through helium. But remember how wavelength and frequency are related to how fast the waves are travelling? By increasing the speed of the sound, we have increased the frequencies of the sounds that are in resonance with the vocal tract, and hence the overall perception of a listener is that the pitch of the speech has gone up.
And in the process has made someone sound like a Munchkin!
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