Q40E - University Physics with Modern Physics

Question

Two guitarists attempt to play the same note of wavelength $\mathbf{64}\mathbf{.}\mathbf{8}\mathbf{cm}$   at the same time, but one of the instruments is slightly out of tune and plays a note of wavelength  $\mathbf{65}\mathbf{.}\mathbf{2}\mathbf{ }\mathbf{cm}$ instead. What is the frequency of the beats these musicians hear when they play together?

Step-by-Step Solution

Answer

The beats have the frequency  $3.29\text{\hspace{0.17em}Hz}$ .

Step 1: Beat frequency formation

The beat frequency is the frequency that is equal to the difference in the frequency of the waves, forming the beat.

  ${\mathbf{f}}_b\mathbf{=}|{\mathbf{f}}_1\mathbf{-}{\mathbf{f}}_2|$

Step 2: Calculate the two frequencies

The values of wavelengths are given and the velocity of sound in air is $344\text{\hspace{0.17em}m}/\text{s}$.

 

  $\begin{array}{c}{f}_1=\frac{v}{{\lambda }_1}\\ =\frac{344\text{\hspace{0.17em}m}/\text{s}}{0.65\text{\hspace{0.17em}m}}\\ =530.86\text{\hspace{0.17em}Hz}\end{array}$

$\begin{array}{c}{f}_2=\frac{v}{{\lambda }_2}\\ =\frac{344\text{\hspace{0.17em}m}/\text{s}}{0.65\text{\hspace{0.17em}m}}\\ =527.60\text{\hspace{0.17em}Hz}\end{array}$

Step 3: Calculate the beats frequency

$\begin{array}{c}{f}_b=|f_1-f_2|\\ =|530.86\text{\hspace{0.17em}Hz}-527.61\text{\hspace{0.17em}Hz}|\\ =3.29\text{\hspace{0.17em}Hz}\end{array}$

So, the beat frequency is  $3.29\text{\hspace{0.17em}Hz}$.