“Sinusoidal waves (or sine waves for short) have turned out to be essential to understanding how our world works.”
Sinusoidal Waves as Sound
We’ve studied the graph of
in some detail in class. You might wonder, “Why do we care? Does this ever show up in real life?” The answer is most definitely yes. Sinusoidal waves (or sine waves for short) have turned out to be essential to understanding how our world works…
Of course, when you listen to your stereo you hear more than one note at a time. How does that work? Simple: just take the sine functions for all of the notes and add them together. I’ve done that here with the notes A, C#, and E. (If you have any experience as a musician, you’ll recognize that we’re forming a major chord here.)

Play the chord
There are some interesting things about this picture. First of all, the graph is no longer a sine curve, but there’s definitely a pattern to it. Moreoever, the pattern repeats, so this is still a periodic function.
Whenever you see an “oscilloscope,” for example when you play music using certain programs on a computer, you’re really seeing a whole bunch of sine waves added together. Who knew?
Cancelling Sounds
You may have seen advertisements for “noise reducing” headphones. (If not, you can see a description of one here.) Here’s a rough idea of how they work. First they figure out what kind of noise is reaching your ears. Essentially this means it does “curve fitting” like we did in class; it figures out the sine waves of the ambient sounds. Then it creates additional noise whose sine waves cancel out the ambient sounds. It’s strange, but true: you add together the original noises and new noises, and it all just seems to disappear!
“No, all waves in nature aren’t sine waves. Quite the contrary… …a violin string vibrates in a pattern close to a saw wave. Nature is messy.” ~ HermanVonElsewhere
Sawtooth wave
“…a sawtooth wave’s sound is harsh and clear and its spectrum contains both even and odd harmonics of the fundamental frequency. Because it contains all the integer harmonics, it is one of the best waveforms to use for subtractive synthesis of musical sounds, particularly bowed string instruments like violins and cellos, since the slip-stick behavior of the bow drives the strings with a sawtooth-like motion.” ~ Wikipedia
Harmonic
“In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the fundamental frequency of a periodic signal.” ~ Wikipedia