Question

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

Answer 1

Hint: The wavelength of any periodic wave is the interval between a given point in the wave and the corresponding point in the next step of the wave, commonly expressed by the Greek letter lambda$(\lambda )$. It's also known as the distance travelled by sound in a single duration or time.

Complete step by step answer:
In any medium, the sound velocity $v$ (see speed of sound) equals the frequency $n$ times the wavelength ($v = n\lambda $). If you know the velocity and frequency, you can calculate the wavelength by dividing the velocity by the frequency $\lambda = \dfrac{v}{n}$. The $88$ keys on a modern piano are tuned to twelve-tone equal temperament. The fifth $A$, also known as $A_4$, is the $49th$ key, and it is tuned to $440Hz$ (referred to as $A440$).

The frequency of ${n^{th}}$ key is given by –
$f(n) = {(\sqrt[{12}]{2})^{n - 49}} \times 440,Hz$
The given note is Middle C (also known as $C_4$), which is the $40th$ key. We get its pitch by putting this value into a general expression
$f(40) = {(\sqrt[{12}]{2})^{40 - 49}} \times 440,Hz$
$\Rightarrow f(40) = {(2)^{\dfrac{{40 - 49}}{{12}}}} \times 440,Hz$
$\Rightarrow f(40)= 261.626Hz$
Assuming speed of sound at STP ($0^\circ C$ and pressure $1$ bar) as $331.5$ metres per second, wavelength is calculated as –
$\lambda =\dfrac{331.5}{261.6} \\
\therefore \lambda = 1.3\,m$

Hence,the wavelength is $1.3\,m$.

Note: Due to the twelve-tone equivalent interval, the frequency of each subsequent key is calculated by multiplying the frequency (also known as pitch) of the lower key by a factor of the twelfth root of two (or dividing the pitch of the higher key by a factor of the twelfth root of two).