Q62PE - College Physics (Urone)

Question

Three adjacent keys on a piano (F, F-sharp, and G) are struck simultaneously, producing frequencies of 349, 370 and 392 Hz. What beat frequencies are produced by this discordant combination?

Step-by-Step Solution

Answer

The beat frequencies produced by the discordant combination are 21 Hz, 22 Hz and 43 Hz.

Step 1: Identification of the given data

The give data is listed as below,

 

• The frequency of first key is, f₁ = 349 Hz
• The frequency of second key is, f₂ = 370 Hz
• The frequency of third key is, f₃ = 392 Hz

Step 2: Definition of beat frequency

The beat frequency is defined as the difference between the frequency of two similar waves. If f₁ and f₂ are the frequencies of two similar waves then the beat frequency is expressed as follows,

 

\[{{\rm{f}}_{\rm{B}}}{\rm{ = }}\left| {{{\rm{f}}_{\rm{2}}} - {{\rm{f}}_{\rm{1}}}} \right|\]

Step 3: Determination the beat frequencies produced by the discordant combination

Write the expression for the first pair to get the beat frequency.

 

\[{f_{B1}} = \left| {{f_1} - {f_2}} \right|\]

 

Substitute all the values in the above expression.

 

\[\begin{aligned}{f_{B1}} = \left| {{\rm{349}}\;{\rm{Hz}} - {\rm{370}}\;{\rm{Hz}}} \right|\\ = {\rm{21}}\,\;{\rm{Hz}}\end{aligned}\]

 

Write the expression for the second pair to get the beat frequency.

 

\[{f_{B2}} = \left| {{f_2} - {f_3}} \right|\]

 

Substitute all the values in the above expression.

 

\[\begin{aligned}{f_{B2}} = \left| {{\rm{370}}\;{\rm{Hz}} - {\rm{392}}\;{\rm{Hz}}} \right|\\ = {\rm{22}}\,\;{\rm{Hz}}\end{aligned}\] 

 

Write the expression for the third pair to get the beat frequency.

 

\[{f_{B3}} = \left| {{f_3} - {f_1}} \right|\]

 

Substitute all the values in the above expression.

 

\[\begin{aligned}{f_{B3}} = \left| {{\rm{392}}\;{\rm{Hz}} - {\rm{349}}\;{\rm{Hz}}} \right|\\ = 43\,\;{\rm{Hz}}\end{aligned}\]

 

Thus, the beat frequencies produced by the discordant combination are 21 Hz, 22 Hz and 43 Hz.