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

Explanation

Let us know keep the rigid boundries at x=0 and x=L and produce a standing wavves bt wiggling the string (as in plucking atrings in a guitar). Standing waves with a specofic wavelength are produced . Since the amplitude must vanish at the boundries, therefore, the displacement at tge boundry must satisfy the following conditions
y(x=0,t)=0 and y(x=L,t)=0
Since, the nodes formed are at a distance `((lamda_(n))/(2))` apart , we have `n((lamda_(n))/(2))=L`, where n is an ineger, L is the length between the two boundaries and `lamda_(n)` is the specific wavelength that satisfy the specified boundry condition Hence
Therefore, not all wavelengths are allowed, The (allowed) wavelengths should fit with specified boundary condition for n=1, the first mode vibration has specified wavelength `lamda_(1)=2L`. Similarly for n=2 the second mode of vibration has specific wavelength.
`lamda_(2)=((2L)/(2))=L`
The lowest natural frequency is called the fundamental frequency.
`f_(1)=(v)/(lamda_(1))=((v)/(2L))`
The second frequency is called the first over tone.
`f_(2)=2((v)/(2L))=(1)/(L)sqrt(T/mu)`
The third frequncy is called the second over tone .
`f_(3)=3((v)/(2L))=3((1)/2Lsqrt(T/mu))`
If natural frequency 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 harmoinc is `f_(2)=2f_(1)`, the third harmonic is `f_(3)=3f_(1)` etc.