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Making Music With Sine Waves

Sine and cosine waves are incredibly important for understanding all sorts of waves in physics.  Musical notes can be thought of in terms of sine curves where we have the basic formula:

y = sin(bt)

where t is measured in seconds.  b is then connected to the period of the function by the formula period = 2π/b.

When modeling sound waves we normally work in Hertz – where Hertz just means full cycles (periods) per second.  This is also called the frequency.  Sine waves with different Hertz values will each have a distinct sound – so we can cycle through scales in music through sine waves of different periods.

For example the sine wave for 20Hz is:

20Hz means 20 periods per second (i.e 1 period per 1/20 second) so we can find the equivalent sine wave by using

period = 2π/b.

1/20 = 2π/b.

b = 40π

So, 20Hz is modeled by y = sin(40πt)

You can plot this graph using Wolfram Alpha, and then play the sound file to hear what 20Hz sounds like.  20Hz is regarded as the lower range of hearing spectrum for adults – and is a very low bass sound.

The middle C on a piano is modeled with a wave of 261.626Hz.  This gives the wave

which has the equation, y = sin(1643.84πt).  Again you can listen to this sound file on Wolfram Alpha.

At the top end of the sound spectrum for adults is around 16,000 – 20,000Hz.  Babies have a ability to hear higher pitched sounds, and we gradually lose this higher range with age.  This is the   sine wave for 20,000Hz:

which has the equation, y = sin(40,000πt). See if you can hear this file – warning it’s a bit painful!

As well as sound waves, the whole of the electromagnetic spectrum (radio waves, microwaves, infrared, visible light, ultraviolet, x rays and gamma rays) can also be thought of in terms of waves of different frequencies.  So, modelling waves using trig graphs is an essential part of understanding the physical world.

If you enjoyed this post you might also like:
Fourier Transforms – the most important tool in mathematics? – how we can use advanced mathematics to understand waves – with applications for everything from WIFI, JPEG compression, DNA analysis and MRI scans.

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