Question

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?

Short answer

The beat frequency is approximately 3.31 Hz.

Step-by-step solution

Convert Wavelength to Frequency for First Guitar

To find the frequency of a wave, use the formula: \( f = \frac{v}{\lambda} \), where \( f \) is the frequency, \( v \) is the speed of sound, and \( \lambda \) is the wavelength. Assuming the speed of sound \( v \) is 340 m/s, for the first guitar with a wavelength of 64.8 cm or 0.648 m, we calculate the frequency as: \[ f_1 = \frac{340}{0.648} \approx 524.69 \text{ Hz} \].

Convert Wavelength to Frequency for Second Guitar

Similarly, for the second guitar with a wavelength of 65.2 cm or 0.652 m, calculate the frequency using the same formula: \[ f_2 = \frac{340}{0.652} \approx 521.38 \text{ Hz} \].

Calculate Beat Frequency

The beat frequency is the absolute difference between the two frequencies: \[ f_{\text{beat}} = |f_1 - f_2| = |524.69 - 521.38| = 3.31 \text{ Hz} \].

Key concepts

Wavelength to Frequency Conversion

Converting a wavelength to frequency helps us understand how often the wave oscillates per second. To achieve this, we use the formula: \[ f = \frac{v}{\lambda} \]where:

• \( f \) represents the frequency in hertz (Hz).
• \( v \) is the speed of sound in meters per second (m/s).
• \( \lambda \) is the wavelength in meters (m).

This formula is derived from the basic relationship between wavelength, frequency, and speed, which states that wave speed is the product of frequency and wavelength.
Let's apply this to our guitarists. The first guitar's wavelength is 64.8 cm, which is converted to meters as 0.648 m. Using a standard speed of sound (340 m/s), the frequency is calculated as \( f_1 = \frac{340}{0.648} \approx 524.69 \text{ Hz} \).
Similarly, the second guitar has a wavelength of 65.2 cm or 0.652 m, thus its frequency is calculated as \( f_2 = \frac{340}{0.652} \approx 521.38 \text{ Hz} \). Understanding this conversion is crucial for analyzing sound waves.

Speed of Sound

The speed of sound is the rate at which sound waves travel through a medium, like air. This speed influences how quickly fluctuations in pressure (sound) reach your ears. In air at room temperature, the speed of sound is typically around 340 m/s.
Several factors affect the speed of sound:

• Temperature: Sound travels faster in warmer air because molecules move quicker.
• Medium: Sound moves differently in various substances (faster in water, slower in air).
• Humidity: More moisture means faster sound, as humid air is less dense than dry air.

In musical contexts, assuming the speed of sound as 340 m/s simplifies calculations. This assumption provides accurate results while considering normal ambient conditions. Understanding these factors helps in contexts like determining beat frequencies in musical instruments.

Wave Interference

Wave interference occurs when two or more waveforms overlap in space. This creates effects like constructive interference (waves add up) and destructive interference (waves cancel out). In the context of sound, interference can lead to phenomena like beats.
When two waves with slightly different frequencies play together, they create a fluctuation called a 'beat.' The frequency of this beat is the difference between the two sounds' frequencies:
\[ f_{\text{beat}} = |f_1 - f_2| \]In the example of our guitarists, with frequencies \( f_1 = 524.69 \text{ Hz} \) and \( f_2 = 521.38 \text{ Hz} \), the beat frequency is approximately 3.31 Hz. Listeners perceive this as a rhythmic rise and fall in intensity, making it notable for tuning instruments.
Understanding wave interference is essential in fields like acoustics and helps musicians identify tuning issues through auditory beats.