Two guitarists attempt to play the same note of wavelength 64.8 cm at the same time, but one of the instruments is slightly out of tune and plays a note of wavelength 65.2 cm instead. What is the frequency of the beats these musicians hear when they play together?

Verified step by step guidance

1. First, understand the concept of beats. Beats occur when two sound waves of slightly different frequencies interfere with each other, resulting in a fluctuating sound intensity. The beat frequency is the absolute difference between the frequencies of the two waves.
2. Convert the wavelengths of the notes played by the guitarists into frequencies. Use the formula: $f=\frac{v}{\lambda }$, where $f$ is the frequency, $v$ is the speed of sound in air (approximately 343 m/s), and $\lambda$ is the wavelength.
3. Calculate the frequency of the first note using its wavelength of 64.8 cm. Convert the wavelength from centimeters to meters by dividing by 100, then apply the formula: $f=\frac{343}{0.648}$.
4. Calculate the frequency of the second note using its wavelength of 65.2 cm. Similarly, convert the wavelength to meters and apply the formula: $f=\frac{343}{0.652}$.
5. Determine the beat frequency by finding the absolute difference between the two frequencies calculated in the previous steps. Use the formula: $f$_beat=|f₁-f₂|, where $f$₁ and $f$₂ are the frequencies of the two notes.

Key Concepts

Wave Interference

Wave interference occurs when two waves meet while traveling along the same medium. The principle of superposition states that the resultant wave is the sum of the individual waves. In this context, the interference between the two sound waves of slightly different wavelengths results in a phenomenon known as beats.

Beat Frequency

Beat frequency is the frequency at which the amplitude of the resultant wave fluctuates due to the interference of two waves with slightly different frequencies. It is calculated as the absolute difference between the frequencies of the two waves. This concept explains the periodic variation in sound intensity heard when the two guitarists play together.

Relationship Between Wavelength and Frequency

The frequency of a wave is inversely proportional to its wavelength, given by the equation f = v/λ, where f is frequency, v is the speed of sound, and λ is the wavelength. Understanding this relationship is crucial for calculating the frequencies of the notes played by the guitarists, which are needed to determine the beat frequency.