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Resonance occurs when an object vibrates at a frequency that matches the natural frequency of another object, causing it to vibrate as well. In the context of a tube with a tuning fork, resonance happens when the frequency of the sound wave matches one of the natural frequencies of the air column inside the tube. This creates standing waves where certain points called nodes (with no movement) and antinodes (with maximum movement) are formed.
For a tube with one open end and one closed end, like in the exercise, resonance is set up at odd harmonics (1st, 3rd, 5th, etc.). These standing wave patterns allow sound waves to reinforce each other, amplifying the sound at these specific lengths.
Resonance is an essential concept because it explains how sound can be amplified in musical instruments and in engineering applications. When an instrument or a structure resonates, it can produce loud and rich sounds or beneficial vibrational properties.

Harmonics are integral multiples of the fundamental frequency of a system. They determine the pattern and positions where standing waves occur in the air column. In a tube with one end closed, which resembles the setup in the exercise, only odd harmonics are allowed.

• 1st Harmonic (=1): The simplest standing wave, where the air column length is \(\frac{\lambda}{4}\).
• 3rd Harmonic (=3): Consists of 3/4th the wavelength in the column.
• 5th Harmonic (=5): Involves 5/4th of the wavelength.
• And so on.

These harmonics directly dictate where resonance, and consequently constructive interference, can occur, amplifying sound at each harmonic frequency. The natural resonant frequencies are crucial for designing musical instruments, as they determine the notes that can be played.

The speed of sound is a fundamental property, determining how fast sound waves travel through a medium, such as air. At room temperature, this speed is approximately 343 m/s in air. In the problem, this value facilitates the calculation of the sound wavelength when the frequency of the tuning fork, 900 Hz, is known.
The formula to relate speed, wavelength, and frequency is given by \( v = f \cdot \lambda \) where \( v \) is the speed of sound, \( f \) is the frequency, and \( \lambda \) is the wavelength. Here, knowing the speed and frequency, we derive the wavelength as \( \lambda = \frac{v}{f} = \frac{343 \text{ m/s}}{900 \text{ Hz}} \approx 0.381 \text{ m} \).
The speed of sound can vary based on factors such as temperature, humidity, and air pressure, but the given speed provides a standard reference point for the calculations in most exercises.

The wavelength of a sound wave is the physical distance between two successive peaks (or troughs) of the wave. It is crucial as it determines the length of the air column required for each resonant frequency in the tube. In the given problem, the wavelength was calculated using the speed of sound and the frequency of the source.
When standing waves are formed in a tube, the relationship between the wavelength and the air column length is governed by the harmonic number. For example, the first harmonic occurs at a length equal to \(\frac{\lambda}{4}\), the third at \(\frac{3\lambda}{4}\), and so on. These small fractions of the wavelength are used to find the specific positions inside the tube where resonance occurs.
Understanding wavelengths allows for the designing of various acoustic and audio devices to enhance or reduce sound wave propagation and to align with the desired resonant frequencies.