Question

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

Answer 1

Fundamental frequency and overtones: Let us now keep the rigid boundaries at x = 0 and x = L and produce standing waves by wiggling the string (as in plucking strings in a guitar). Standing waves with a specific wavelength are produced. Since, the amplitude must vanish at the boundaries, therefore, the displacement at the boundary must satisfy the following conditions

x(x = 0, t) = 0 and y(x = L, t) = 0

Since, the nodes formed at a distance ${\displaystyle \frac{{\lambda }_{\text{n}}}{2}}$ apart, we have ${\displaystyle \text{n}\left(\frac{{\lambda }_{\text{n}}}{2}\right)}$ = L, 

where n is an integer, L is the length between the two boundaries and λ_n is the specific wavelength that satisfy the specified boundary conditions. Hence,

${\displaystyle {\lambda }_{\text{n}}=\left(\frac{2\text{L}}{\text{n}}\right)}$    ...(2)

Therefore, not all wavelengths are allowed. The (allowed) wavelengths should fit with the specified boundary conditions, i.e., for n = 1, the first mode of vibration has a specific wavelength λ₁ = 2L. Similarly for n = 2, the second mode of vibration has a specific wavelength

${\displaystyle {\lambda }_2=\left(\frac{2\text{L}}{2}\right)}$ = L

For n = 3, the third mode of vibration has specific wavelength

${\displaystyle {\lambda }_3=\left(\frac{2\text{L}}{3}\right)}$ and so on.

The frequency of each mode of vibration (called natural frequency) can be calculated.

We have, ${\displaystyle {\text{f}}_{\text{n}}=\frac{\text{v}}{{\lambda }_{\text{n}}}=\text{n}\left(\frac{\text{v}}{\text{2L}}\right)}$    ...(3)

The lowest natural frequency is called the fundamental frequency.

${\displaystyle {\text{f}}_1=\frac{\text{v}}{{\lambda }_1}=\left(\frac{\text{v}}{\text{2L}}\right)}$    ....(4)

The second natural frequency is called the first over tone.

${\displaystyle {\text{f}}_2=2\left(\frac{\text{v}}{\text{2L}}\right)=\frac{1}{\text{L}}\sqrt{\frac{\text{T}}{\mu }}}$

The third natural frequency is called the second over tone.

${\displaystyle {\text{f}}_3=3\left(\frac{\text{v}}{\text{2L}}\right)=3\left(\frac{1}{\text{2L}}\sqrt{\frac{\text{T}}{\mu }}\right)}$ and so on.

Therefore, the n^th natural frequency can be computed as integral (or integer ) multiple of the fundamental frequency, i.e.,

f_n = nf₁ where n is an integer …(5)

If natural frequencies are written as an integral multiple of fundamental frequencies, then the frequencies are called harmonics. Thus, the first harmonic is f₁ = f₁ (the fundamental frequency is called first harmonic), the second harmonic is f₂ = 2f₁, the third harmonic is f₃ = 3f₁ etc.