17.4 Normal modes of a standing sound wave  (Page 5/9)

Resonance in a tube open at both ends

Another source of standing waves is a tube that is open at both ends. In this case, the boundary conditions are symmetrical: an antinode at each end. The resonances of tubes open at both ends can be analyzed in a very similar fashion to those for tubes closed at one end. The air columns in tubes open at both ends have maximum air displacements at both ends ( [link] ). Standing waves form as shown.

The relationship for the resonant wavelengths of a tube open at both ends is

Based on the fact that a tube open at both ends has maximum air displacements at both ends, and using [link] as a guide, we can see that the resonant frequencies of a tube open at both ends are

where $f_1$ is the fundamental, $f_2$ is the first overtone, $f_3$ is the second overtone, and so on. Note that a tube open at both ends has a fundamental frequency twice what it would have if closed at one end. It also has a different spectrum of overtones than a tube closed at one end.

Note that a tube open at both ends has symmetrical boundary conditions, similar to the string fixed at both ends discussed in Waves . The relationships for the wavelengths and frequencies of a stringed instrument are the same as given in [link] and [link] . The speed of the wave on the string (from Waves ) is $v=\sqrt{\frac{F_T}{\mu }}.$ The air around the string vibrates at the same frequency as the string, producing sound of the same frequency. The sound wave moves at the speed of sound and the wavelength can be found using $v=\lambda f.$

Summary

• Unwanted sound can be reduced using destructive interference.
• Sound has the same properties of interference and resonance as defined for all waves.
• In air columns, the lowest-frequency resonance is called the fundamental, whereas all higher resonant frequencies are called overtones. Collectively, they are called harmonics.