Question

A cello string has a fundamental frequency of \(65.40 \mathrm{Hz}\) What beat frequency is heard when this cello string is bowed at the same time as a violin string with frequency of \(196.0 \mathrm{Hz} ?\) [Hint: The beats occur between the third harmonic of the cello string and the fundamental of the violin. \(]\)

Short answer

Answer: The beat frequency between the cello string's third harmonic and the violin string's fundamental frequency is \(0.20 \mathrm{Hz}\).

Step-by-step solution

Calculate the Cello String's Third Harmonic Frequency

We are given the fundamental frequency of the cello string as \(65.40 \mathrm{Hz}\). The third harmonic frequency is three times the fundamental frequency: $$ f_{cello\_harmonic\_3} = 3 \times 65.40 \mathrm{Hz} $$ Find the cello's third harmonic by multiplying the given fundamental frequency by three.

Calculate the Beat Frequency

Beat frequency is the absolute difference between the frequencies of two waves. In this case, we're comparing the cello's third harmonic and the violin's fundamental frequencies: $$ f_{beat} = |f_{violin\_fundamental} - f_{cello\_harmonic\_3}| $$ We already have the violin's fundamental frequency given as \(196.0 \mathrm{Hz}\) from the problem. After calculating the cello's third harmonic, we can subtract the cello's value from the violin's to find their absolute difference.

Solve for the Beat Frequency

First, use the result from step 1 to find the third harmonic frequency of the cello string: $$ f_{cello\_harmonic\_3} = 3 \times 65.40 \mathrm{Hz} = 196.20 \mathrm{Hz} $$ Now, substitute the values of the violin's fundamental frequency and the cello's third harmonic frequency into the beat frequency formula: $$ f_{beat} = |196.0 \mathrm{Hz} - 196.20 \mathrm{Hz}| $$ Calculate the absolute difference between the two frequencies to find the beat frequency.

Final Answer

The beat frequency between the cello string's third harmonic and the violin string's fundamental frequency is: $$ f_{beat} = |196.0 \mathrm{Hz} - 196.20 \mathrm{Hz}| = 0.20 \mathrm{Hz} $$ So, the beat frequency heard is \(0.20 \mathrm{Hz}\).