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The fundamental frequency of a string is the lowest frequency at which it naturally vibrates. This occurs when the string oscillates in a single segment, often resulting in the string's purest sound. The formula to calculate the fundamental frequency of a vibrating string is:

• \( f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \), where:
• \( f \) is the frequency,
• \( L \) is the length of the string,
• \( T \) is the tension applied to the string,
• \( \mu \) is the linear mass density.

The equation indicates that the fundamental frequency is inversely proportional to the length and depends on the square root of the tension and mass density.
Hence, even a small change in tension or mass density can lead to significant frequency changes, affecting the pitch of the string.

Tension is a critical factor in the physics of string instruments. It refers to the tightening force applied along a string, which affects its vibration and resulting sound.
In the frequency formula, tension is represented by \( T \).
When a string's tension is increased, it impacts its vibrational properties in the following ways:

• The frequency increases, making the pitch higher.
• The vibrations become quicker, leading to a sharper sound.

When tension is multiplied by a factor, such as four, the effect on frequency can be observed mathematically. Using the example from the exercise, when tension increases fourfold, the new frequency \( f' = \frac{1}{2L} \sqrt{\frac{4T}{\mu}} \) simplifies to \( 2f \).
This illustrates that quadrupling the tension results in doubling the frequency.

The mass density, also known as linear mass density, is a measure of the mass of the string per unit length, symbolized by \( \mu \). It plays a crucial role in determining the vibrational characteristics of the string. Mass density is influenced by:

• The material composition of the string.
• The thickness of the string—thicker strings typically have a higher mass density.

In the formula for fundamental frequency, the mass density \( \mu \) appears in the denominator within the square root. Thus, a higher mass density results in a lower frequency, assuming tension and length remain constant. This means that for strings of the same length and tension, a thicker string will produce a lower pitch than a thinner one.
Understanding mass density is crucial for instrument design and playing, as it helps musicians choose the right strings to achieve desired sound qualities.