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Harmonics are the integral multiples of the fundamental frequency of a system, describing the different standing wave patterns that can form on a medium such as a string or an air column. When you pluck a string, it can vibrate in different patterns, producing different harmonics.

* **Fundamental Frequency (1st Harmonic)**: It's the lowest frequency at which a string vibrates, creating a standing wave pattern with a single antinode and nodes at the ends.
* **2nd Harmonic**: This involves a standing wave pattern that includes one more node and antinode, with double the fundamental frequency.
* **3rd Harmonic**: At this stage, the string forms three distinct antinodes and has a frequency three times that of the fundamental frequency. As noted in the original exercise, a frequency of 36 Hz, which is three times the fundamental frequency of 12 Hz, corresponds to the third harmonic. The pattern for this will display three loops, indicating three antinodes.

These harmonics and their respective frequencies help in understanding the succession of patterns formed as the frequency increases. They are essential in musical instruments where the quality of sound is richer due to multiple harmonics being heard at once.

Wave speed measures the rate at which the wave propagates through a medium. In a string fixed at both ends, the wave speed is determined by both the tension in the string and the mass density of the string. Interestingly, wave speed remains constant for a given medium and conditions, regardless of frequency.

Wave speed is calculated using the formula:
\[ v = f \times \lambda \]
where \( v \) is the wave speed, \( f \) is the frequency, and \( \lambda \) is the wavelength of the wave.

The frequency specifically refers to how many cycles occur per second, measured in hertz (Hz). Wavelength, on the other hand, is the physical length of one cycle of the wave. The fundamental wavelength in a string fixed at both ends is twice the length of the string, as the wave has to return to its starting point to complete one full cycle.

In the problem given, we've determined the wave speed to be 24 m/s using a frequency of 12 Hz and a wavelength of 2 meters.

The fundamental frequency, often called the first harmonic, is the lowest frequency at which a system resonates. Imagine a string of a musical instrument; when plucked, it vibrates at this fundamental frequency to produce the basic note.

To find the fundamental frequency, one can analyze the differences between known consecutive harmonics. For instance, in the original exercise, the frequencies were 24 Hz (second harmonic) and 36 Hz (third harmonic). The difference between these frequencies, 12 Hz, gives us the fundamental frequency.

It's essential for understanding how different standing waves and harmonics derive from a simple frequency. This knowledge helps in tuning musical instruments and understanding wave behaviour in various mediums.

By knowing the fundamental frequency, other harmonics and their respective standing wave patterns can be predicted accurately, making it a crucial concept in wave mechanics and acoustics.