Question

What are the first three overtones of a bassoon that has a fundamental frequency of 90.0 Hz? It is open at both ends. (The overtones of a real bassoon are more complex than this example, because its double reed makes it act more like a tube closed at one end.)

Short answer

The first three overtones are 180 Hz, 270 Hz, 360 Hz.

Step-by-step solution

Given Data

The fundamental frequency is 90 Hz.

Calculation of the harmonics

The resonance frequencies of a tube open at both the ends are,

${\mathrm{f}}_{\mathrm{n}}=\mathrm{n}\frac{\mathrm{v}}{2\mathrm{l}},\mathrm{n}=1,2,3,....$

The fundamental frequency is

${\mathrm{f}}_1=90\mathrm{H}\mathrm{z}$

The second harmonic is,

$\begin{array}{r}{\mathrm{f}}_2=2\times \frac{\mathrm{v}}{2\mathrm{l}}\\ =2\times 90\\ =180\mathrm{H}\mathrm{z}\end{array}$

The second harmonic is,

$\begin{array}{r}{\mathrm{f}}_3=3\times \frac{\mathrm{v}}{4\mathrm{l}}\\ =3\times 90\\ =270\mathrm{H}\mathrm{z}\end{array}$

The third harmonic is,

$\begin{array}{r}{\mathrm{f}}_4=4\times \frac{\mathrm{v}}{4\mathrm{l}}\\ =4\times 90\\ =360\mathrm{H}\mathrm{z}\end{array}$

Hence, the first three overtones are 180 Hz, 270 Hz, 360 Hz.