Question

A string of a violin produces 2 beats per second when sounded along with a standard fork of frequency \(400 . \mathrm{Hz}\). The beat frequency increases when the string is tightened. a) What was the frequency of the violin at first? b) What should be done to tune the violin?

Short answer

Answer: The initial frequency of the violin string was 398 Hz. To tune the violin, the string should be tightened until its frequency reaches 400 Hz to match the frequency of the tuning fork.

Step-by-step solution

Understand the beat frequency

Beat frequency is the difference between the frequencies of two waves. In this case, the beat frequency is created when the violin string is played along with a standard tuning fork. The beat frequency is given as 2 beats per second (Hz).

Use the beat frequency formula to find the initial frequency

The formula for beat frequency is: Beat frequency = |f1 - f2| where f1 and f2 are the frequencies of the two waves. In this case, we are given the beat frequency (2 Hz) and the frequency of the tuning fork (400 Hz). Let the initial frequency of the violin string be f_violin. Then we have: 2 = |f_violin - 400|

Solve for the initial frequency of the violin string

We have two possibilities for the initial frequency: 1. f_violin - 400 = 2, which implies f_violin = 402 Hz 2. 400 - f_violin = 2, which implies f_violin = 398 Hz Since we are given that the beat frequency increases when the string is tightened, we can infer that the initial frequency of the violin string must be 398 Hz (lower than the tuning fork). Therefore, the initial frequency of the violin string is 398 Hz. a) The frequency of the violin at first was 398 Hz.

Determine the action needed to tune the violin

We are given that the beat frequency increases when the string is tightened, which means the frequency of the violin string also increases as it approaches the frequency of the tuning fork (400 Hz). To tune the violin, we need to adjust the string to match the frequency of the tuning fork. b) To tune the violin, we should tighten the string until the frequency of the violin string reaches 400 Hz and matches the frequency of the tuning fork. At this point, there will be no beats as both the sources have the same frequency.

Key concepts

Violin Tuning

When tuning a violin, it's important to match the string's pitch to a standard reference. A common reference is a tuning fork, often with a frequency of 440 Hz, also known as the concert pitch A. For specific exercises like our example, the tuning fork is 400 Hz. When a violinist tunes a string, they adjust the tension of the string by either tightening or loosening it. This changes the frequency of the string's vibration.

This process involves listening for beats—pulses of sound difference that help find a match.

• If the beats disappear as tension is adjusted, it indicates that the string's frequency matches the desired reference pitch.
• If beats increase when a string is tightened, it means the string frequency is going farther from the target.

To tune effectively, the player must gently adjust and listen for a reduction in beats until they vanish completely.

Frequency Matching

Frequency matching is crucial in music to ensure that all instruments sound in harmony. In the context of our violin exercise, frequency matching involves adjusting the violin's string until its frequency equals that of the reference tuning fork. This eliminates the beat frequency.

The beat frequency is the difference between two frequencies. The equation for beat frequency is \(|f_1 - f_2|\), where \(f_1\) and \(f_2\) are the frequencies of the two sources. So, a beat frequency results when there's a slight difference between the two frequencies.

• A smaller beat frequency indicates closer matching.
• A zero beat frequency means perfect alignment.

Thus, tuning involves making minute adjustments to match the frequencies and produce a clear, beat-free note.

Sound Waves

Sound waves are the carriers of musical notes and tones. In a violin, sound is produced when the bow moves across a string, causing it to vibrate. These vibrations travel through the air as waves, reaching our ears. Each wave has a specific frequency, which we perceive as pitch.

In our exercise, the tuning fork creates a sound with a fixed frequency, and as the violin string vibrates, it creates its own set of waves. Through playing the two together, the musician can adjust the string's tension to change its frequency, aiming to get both sets of waves in sync to eliminate beat frequencies.

• Higher tension results in higher frequency.
• Lower tension results in lower frequency.

By understanding this physical process, musicians can precisely tune their instruments for optimal performance.

Physics of Music

The physics of music explains how sound is produced and perceived in musical contexts. Vibrations in strings, air, or even electronic signals generate sound waves, which have specific characteristics like frequency, amplitude, and speed. Frequency dictates the pitch we hear.

For instance, in a violin tuning scenario, understanding physics helps in knowing how to adjust string tension for the correct pitch. It’s an application of physical principles like tension and wave behavior.

• Beats arise when two close frequencies produce interference, creating a pulsing effect.
• Matching frequencies eliminates the interference, resulting in one clear tone.

Hence, the study of sound waves and the physics of music helps musicians achieve precision in tuning through hands-on experience and scientific knowledge.