A flute player hears four beats per second when she compares her note to a 523 Hz tuning fork (the note C). She can match the frequency of the tuning fork by pulling out the 'tuning joint' to lengthen her flute slightly. What was her initial frequency?

Verified step by step guidance

1. Step 1: Understand the concept of beats. Beats occur when two sound waves of slightly different frequencies interfere with each other. The beat frequency is equal to the absolute difference between the two frequencies: \( f_{beat} = |f_1 - f_2| \). In this case, \( f_{beat} = 4 \, \text{Hz} \).
2. Step 2: Identify the known values. The frequency of the tuning fork is \( f_{tuning} = 523 \, \text{Hz} \), and the beat frequency is \( f_{beat} = 4 \, \text{Hz} \). The initial frequency of the flute, \( f_{flute} \), is unknown.
3. Step 3: Set up the relationship between the frequencies. Since the beat frequency is the absolute difference between the flute's frequency and the tuning fork's frequency, we can write: \( f_{beat} = |f_{flute} - f_{tuning}| \). Substituting the known values: \( 4 = |f_{flute} - 523| \).
4. Step 4: Solve for \( f_{flute} \). The absolute value equation \( |f_{flute} - 523| = 4 \) has two possible solutions: \( f_{flute} = 523 + 4 \) or \( f_{flute} = 523 - 4 \). This gives \( f_{flute} = 527 \, \text{Hz} \) or \( f_{flute} = 519 \, \text{Hz} \).
5. Step 5: Determine which frequency matches the scenario. The flute player adjusts her instrument by lengthening it, which lowers the frequency. Therefore, her initial frequency must have been \( f_{flute} = 527 \, \text{Hz} \), as pulling out the tuning joint reduces the frequency to match the tuning fork's \( 523 \, \text{Hz} \).

Key Concepts

Beats Frequency

Beats frequency occurs when two sound waves of slightly different frequencies interfere with each other, resulting in a periodic variation in amplitude. The frequency of the beats is equal to the absolute difference between the two frequencies. In this case, the flute player's note and the tuning fork's frequency create a beat frequency of four beats per second, indicating that her initial frequency is either slightly higher or lower than 523 Hz.

Frequency and Wavelength Relationship

Frequency refers to the number of cycles of a wave that occur in one second, measured in Hertz (Hz). The relationship between frequency and wavelength is described by the equation v = fλ, where v is the speed of sound, f is the frequency, and λ is the wavelength. When the flute player adjusts the length of her flute, she changes the wavelength and thus the frequency of the sound produced, allowing her to match the tuning fork.

Tuning Fork and Musical Pitch

A tuning fork produces a specific pitch when struck, which is determined by its frequency. The standard tuning fork for the note C has a frequency of 523 Hz. Musicians use tuning forks to ensure their instruments are in tune, as the frequency of the sound produced by an instrument must match the frequency of the tuning fork for harmonious sound. The flute player's ability to adjust her instrument's length to match this frequency is crucial for achieving the correct pitch.