Superposition Of Waves

When a string vibrates in 'n' loops, its length 'L' is equal to 'n' times half the wavelength ( ). Given:
Length of the string, m
Number of loops,
Substituting the values: The wavelength of the wave is 2 m.
The distance between a node and an adjacent antinode is always one-quarter of a wavelength ( ).

In the context of standing waves produced in a medium (like a string or an air column), harmonics and overtones describe the different frequencies at which the medium can naturally vibrate.
Fundamental Frequency ( ): This is the lowest possible frequency at which a system can vibrate. It is also called the first harmonic.
Harmonics: Harmonics are all the integer multiples of the fundamental frequency. The series of frequencies is given by . The first harmonic is the fundamental frequency itself, the second harmonic is , the third is , and so on. All possible vibrational modes are harmonics.
Overtones: Overtones are any frequencies that can be produced by the system that are higher than the fundamental frequency. The first overtone is the next possible frequency above the fundamental, the second overtone is the next frequency after that, and so on.
Relationship:
For a system where all harmonics are present (e.g., a string fixed at both ends, an open organ pipe), the first overtone corresponds to the second harmonic, the second overtone corresponds to the third harmonic, and the overtone corresponds to the harmonic.
For a system where only odd harmonics are present (e.g., a closed organ pipe), the first overtone corresponds to the third harmonic, the second overtone corresponds to the fifth harmonic, etc.

End correction ( ) accounts for the fact that the antinode at the open end of a pipe is not formed exactly at the edge, but slightly outside it. The corrected length is for a closed pipe and for an open pipe.
Let and be the frequencies of two successive resonating modes.
(i) For a pipe open at both ends:
The fundamental frequency is .
Let the pipe resonate with a tuning fork of frequency at length ,
and with a tuning fork of frequency at length .
Then, and . (ii) For a pipe closed at one end:
The fundamental frequency is . Let the pipe resonate with a tuning fork of frequency at two successive lengths, (for the fundamental mode) and (for the third harmonic). For the first resonance: .
For the second resonance (3rd harmonic): The corrected length is effectively , so is not the way. Instead, we use the fact that the distance between two consecutive nodes is .
Let the first resonating length be and the second be .
Then and .
Subtracting the first equation from the second: Substitute back into the first equation: A simpler experimental method uses two different frequencies and one length L. Let's re-derive for two lengths.
From and : The second derivation is correct for closed pipes as well.