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 `(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 λ_n is the specific wavelength that satisfy the specified boundary conditions. Hence,

`lambda_"n" = ((2"L")/"n")`    ...(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

`lambda_2 = ((2"L")/2)` = L

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

`lambda_3 = ((2"L")/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 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.