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The fundamental frequency, often known as the first harmonic, refers to the lowest frequency produced by an oscillating system, such as a pipe resonator. It serves as the base frequency upon which overtones or harmonics build.
An important formula to remember when dealing with pipe resonators is the equation for frequency, given by:

• \[ f_1 = \frac{v}{\lambda} \]

In this equation, \( f_1 \) is the fundamental frequency, \( v \) is the speed of sound, and \( \lambda \) is the wavelength.
In simpler terms, when you pluck a string of a given length, its vibration creates a sound wave that travels at the speed of sound, producing the fundamental frequency.
Understanding the fundamental frequency is crucial as it sets the stage for determining overtones, and hence, the harmonics of the system.

An open pipe resonator is a type of tube with both ends open, allowing air to move freely in and out. This design supports both even and odd harmonics, multiplying the sound experience.
For an open pipe, the length \( L \) of the pipe plays a key role in determining its fundamental frequency. Using the equation:

• \[ \lambda_1 = 2L \]

One can calculate the fundamental wavelength, and thus, the fundamental frequency using our earlier mentioned formula.
The open pipe's harmonics can be seen in the formula:

• \( f_n = n \times f_1 \)

where \( n \) represents the order of the harmonic.
In such resonators, the first overtone is twice the fundamental frequency, the second overtone is thrice, and so on. These harmonics create the rich sounds utilized in musical instruments like flutes and organ pipes.

A closed pipe resonator is a tube that is open at one end and closed at the other. This unique setup produces sound waves that only reflect odd harmonics, unlike their open-ended counterparts.
For closed pipes, the fundamental wavelength is given by:

• \[ \lambda_1 = 4L \]

where \( L \) is the length of the pipe.
The fundamental frequency is also determined using our standard formula, but the harmonics here follow a distinct pattern:

• \( f_n = (2k-1) \times f_1 \)

where \( k \) is an integer starting from 1, representing only odd-numbered harmonics.
This means the pipe supports only the first, third, fifth harmonics, and so on, making instruments like clarinets and bassoons sound unique compared to open pipe instruments.

The speed of sound is a critical component in the propagation of sound waves through a medium. On a standard day at sea level, the speed of sound in air is approximately 343 meters per second.
Factors affecting the speed of sound include:

• Temperature: Sound travels faster in warmer air because molecules move more quickly.
• Medium: Sound waves travel at different speeds through air, water, and solid materials due to their density and elasticity.
• Humidity: More humid air increases the speed of sound because water vapor is less dense than dry air.

In calculating frequencies of resonators, the speed of sound allows us to translate the physical properties of the pipe into the musical qualities of pitch and tone.
Understanding the speed of sound's dependency on the environment aids in accurately predicting and designing musical instruments and audio arrays.