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

Explanation

Keep the rigid boundaries at x=0 and x=L and produce a standing waves by wiggling the strig (as in plucking strings in a guitar). Standing waves with a specific wavelengths are produced. Since, the amplitude must vanish at the boundaries, therefore, the displacement at the boudary.
`y(x=0,t) =0` and `y(x=L, t)=0`.........(1)
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 wavelengths that satisfy the specified boundary condition. Hence,
`lambda_(n) =((2L)/n)`........(2)
For n=1, 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) =3(v/(2L)) =3(1/(2L)sqrt(T/mu))`
and so on.
Therefore, the nth natural frequency is equal to integral (or integer) multiple of fundamental frequency, i.e.,
`f_(n) nf_(1)`, where n is an integer ............(5)
If natural frequencies are written as integral multiple of fundamental frequencies. Thus, the first harmonics 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.