It is well known that physics is in practically everything we do in everyday life. At times we hear sounds produced by some instruments – wind instruments, for example. They look like tubes, open at both ends or open at one and closed at the other.
The pressure waves produced at one end occur due to a device called an embouchure . The jet of air that enters the tube is directed against the mouth, thus it is funneling, determining the vibration that gives rise to the waves.
Closed tubes
We can see in the figure below that at the end of the mouth the longitudinal standing wave forms only a belly and a node at the closed end. In this type of tube, or rather in all vibration modes, there is only an increase in the number of intermediate nodes. Let’s see in the figure:
According to the figure we see that the distance between a belly and a consecutive node is equivalent to a quarter of the wavelength, so we have (λ/4). As the vibration frequency is given by f = v/λ , we can establish that:
In a closed tube, the natural frequencies are odd multiples of the ratio ( v/4L ), as observed in the following equation:
For i = 1 we have the fundamental frequency, for i = 3 we have the third harmonic, for i = 5 we have the fifth harmonic, etc. Remember that a closed tube does not emit even-order harmonics.
Open tubes
We can see in the figure below that the longitudinal standing wave formed has a belly at both ends. The simplest way to vibrate corresponds to a node at the center point. We can see that with each new mode of vibration, another intermediate node appears.
The distance between two consecutive antinodes is equal to half a wavelength, that is, ( λ/2 ), we have that the frequency is given by f = v/λ . In the equation, v is the speed of the wave inside the tube. In this way, we can establish that:
In an open tube, the natural frequencies of vibration are given by the following equation:
For N = 1 we have the fundamental frequency, for N = 2 we have the second harmonic, for N = 3 we have the third harmonic, and so on.
