Explain the concepts of fundamental frequency, harmonics and overtones in detail.

Explanation

Fundamental frequency and overtones: Let us now keep the rigid boundaries at x = 0 and x =L and produce a standing waves by wiggling the string (as in plucking strings in a guitar). Standing waves·with a specific wavelength are protluced. Since, the amplitude must vanish at the boundaries, therefore, the displacement at the boundary must satisfy the following conditions
`y (x = 0 , t) = 0` and y ( x= L , t) = 0
Since , the nodes formed are at a distance `(lambda_(n))/(2)` apart , we have `n ((lambda_n)/(2)) = L` , where n is an integer , L is the length between the two boundaries and `lambda_(n)` is the specific wavelength that satisfy the specified boundary conditions . Hence ,
`lambda_(n) = ((2L)/(n)) " " ... (2)`
Therefore, not all wavelengths are allowed. The ( allowed) wavelengths should fit with the specified boundary conditions, i.e., for n =I, the first mode of vibration has specific wavelength `lambda_(1) = 2L`. Similarly for n = 2, the second mode of vibration has specific wavelength
`lambda_(2) = ((2L)/(2)) = L`
For n = 3 , the third mode of vibration has specific wavelength
`lambda_(3)= ((2L)/(3))` and so on .
The frequency of each mode of vibration (called natural frequency) can be calculated.
We have , ` f_(n) = (v)/(lambda_(n)) = n ((v)/(2L)) " " ... (3)`
The lowest natural frequency is called the fundamental frequency.
`f_(1) = (v)/(lambda_(1)) = ((v)/(2L)) " " ... (4)`
The second natural frequency is called the first over tone ,
`f_(2) = 2 ((v)/(2L)) = (1)/(L) sqrt((T)/(mu))`
The third natural frequency is called the second over tone .
`f_(3) = 3((v)/(2L)) = 3 ((1)/(2L) sqrt((T)/(mu)))` and so on.
Therefore , the `n^th` natural frequency can be computed as integral (or integer) multiple of fundamental frequency , i.e.,
`f_(n) = n f_(1)` , where n is an integer `" " .... (5)`
If natural frequencies are written as integral multiple of fundamental frequencies, then the frequencies are called harmonics. Thus, the first harmonic is `f_(1) = f_(1)` (the fundamental frequency is called first harmonic), the second harmonic is `f_(2) = 2f_(1)` , the third harmonic is `f_(3) = 3f_(1)` etc.