Question

What is the wavelength of middle "C" on a piano as it travels through air at standard temperature and pressure?

Answer 1

Wavelength `=1.3m` rounded to one place of decimal

Explanation:

Let's find out frequency of note Middle C to begin with.

A modern piano has 88 keys and is tuned to twelve-tone equal temperament.

It has its 49th key, the fifth A, also called A4, tuned to a frequency of 440 Hz (referred to as A440).

Due to twelve-tone equal interval, frequency of each successive key is derived by multiplying frequency (also called pitch) of lower key (or dividing pitch of higher key) by a factor of the twelfth root of two.

General expression which gives the frequency `f` of the `n^{th}` key is
`f(n) = (root12(2))^{n-49} times 440 ,text{Hz}`

Given note is Middle C, also called C4, is the 40th key. Inserting this value in general expression we obtain its pitch as

`f(40) = (root12(2))^{40-49} times 440` ,
Remembering that `(root12(2))` can also be written as `2^(1/12)`
`f(40) = (2^(1/12))^{40-49} times 440`
`= (2)^({40-49}/12) times 440`
`=261.626` `text{Hz}`, rounded to three places of decimal.

Taking speed of sound at STP (Temp `0"^@text{C}` and pressure 1 bar) as 331.5 metres per second, we find the required wavelength from

`v=nxxlambda`
`331.5=261.626xxlambda`
`=1.267`
`=1.3m` rounded to one place of decimal