How does the frequency of the fundamental mode of the...

Name: Problem 11, Chapter 12: Advanced Engineering Mathematics
Uploaded: 2022-02-07T04:03:57-08:00
Duration: 1 min 20 s
Channel: Manik Pulyani
Description: How does the frequency of the fundamental mode of the vibrating string depend on the length of the string? On the mass per unit length? What happens to the string if we double the tension? Why is a contrabass larger than a violin?

How does the frequency of the fundamental mode of the vibrating string depend on the length of the string? On the mass per unit length? What happens to the string if we double the tension? Why is a contrabass larger than a violin?

Key Concepts

Vibrating String Equation

This equation, typically expressed as f = (1/(2L))?(T/?), provides the fundamental relationship between the string’s tension (T), length (L), and mass per unit length (?) with its fundamental frequency (f). It serves as the foundation for understanding how changes in these physical properties affect the pitch of a string's vibration.

Dependence on String Length

The fundamental frequency of a vibrating string is inversely proportional to its length. This means that as the length of the string increases, the frequency decreases. A longer string requires more time to complete one oscillation, resulting in a lower pitch, while a shorter string vibrates faster and produces a higher pitch.

Dependence on Mass per Unit Length

The frequency is inversely proportional to the square root of the mass per unit length. A string with a greater mass per unit length vibrates more slowly, leading to a lower frequency, whereas a lighter string vibrates faster, resulting in a higher pitch. This relationship highlights the impact of the string’s density on its vibrational characteristics.

Effect of Tension on Frequency

Tension in a string directly influences its vibration frequency through the wave speed, which is proportional to the square root of the tension. Increasing the tension increases the wave speed and consequently the frequency. For instance, doubling the tension boosts the frequency by a factor of ?2, demonstrating how tension adjustments can change the pitch.

Musical Instrument Scaling

The size of a musical instrument, such as a contrabass compared to a violin, is designed according to the desired pitch range. Lower pitches require longer vibrating strings, hence larger instruments like the contrabass are built with longer strings to naturally produce lower frequencies. This principle guides the design and proportional scaling of stringed musical instruments.

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