Guide d'onda en

From "Physics, Waves, Music": a website about the physics of waves and sound, acoustics of musical instruments, musical scales, harmony and music.

AI-translated from Italian. Original page: Guide_d'onda

What they are

  • A waveguide is a constrained path into which waves are sent. It essentially consists of a tube open at the ends, whose walls reflect the waves and trap them inside the tube itself.
  • There are waveguides of different types, depending on the nature of the waves that are to be "guided".

Some examples:

  • electromagnetic waves:
    • microwave guides are metal tubes into which microwaves are radiated. They usually have a rectangular cross-section.
    • Optical fibres are waveguides for light. They consist of a tube of transparent material (such as glass, or certain polymers), surrounded by a cladding of a material with a lower refractive index. Internal reflection occurs in this case because of the phenomenon of the critical angle.
An optical fibre bent into a loop, glowing red in the dark
Optical fibre carrying a red laser beam
  • sound waves:
    • the speaking tube once used for internal communication on ships, aircraft, submarines and other vehicles[1]
    • the ear canal (the cavity leading from the outer ear to the eardrum). See anatomy of the auditory system.
Three black metal pipes fixed to a wall, ending in funnel-shaped mouthpieces
Speaking tube in the military museum of Fort McArthur, San Pedro, California

What they are for

The very name of these devices indicates that they force a wave to follow a certain path, instead of propagating freely in the surrounding space. The example of optical fibres is perhaps the easiest to see: light, instead of propagating in all directions, follows the fibre. In the case of electromagnetic waves, waveguides are used both to guide the power of the wave and, in telecommunications, the signals it carries.

In the acoustic case too, waveguides concentrate the energy of the sound wave in a particular direction in space: the wave is not amplified, but its energy does not spread in all directions, so the intensity does not decrease with distance as in free propagation (see the section on wind instruments).

How they work

  • The waves enter at one end and propagate in the guide.
  • Propagation along the guide is free (travelling wave), while in the transverse direction standing waves are formed
  • This mode of propagation, which "mixes" travelling waves and standing waves, produces some interesting properties, the most important of which are the following:
  1. The guide is in general a dispersive medium, even if the medium it contains (the one in which the wave actually propagates) is not dispersive. That is to say, waves of different frequencies in general propagate in the guide with different velocities.
  2. In general there is a cut-off frequency, that is, below a certain frequency the waves cannot propagate along the guide at all. They enter it at one end, but no signal comes out at the other. The cut-off frequency typically depends on the width of the guide. The narrower the guide, the higher the cut-off frequency. (In acoustic guides and in optical fibres, however, the fundamental mode propagates at any frequency: the cut-off concerns the higher modes.)
  3. The phase velocity of the wave in the guide can be greater than the propagation speed in the medium (see the page dispersion). For electromagnetic waves this means that the phase velocity in the guide can be greater than the speed of light in the medium.

Check your understanding

Problem. A speaking tube has a square cross-section with a 5 cm side (c = 343 m/s). The first higher mode can propagate only if half a wavelength fits across the width of the tube. Below what frequency does only the plane wave propagate? Solution. The first higher mode propagates if λ/2 is less than 5 cm, that is if the frequency exceeds c/(2a) = 343/0.1 Hz ≈ 3430 Hz. Below this frequency, where most of the energy of the voice lies, only the plane wave travels in the tube: the sound reaches the other end without being distorted by the different speeds of the modes.

Problem. In an optical fibre the core has a refractive index of 1.48 and the cladding 1.46. What is the critical angle? Within what angle to the axis of the fibre must the light travel to remain trapped? Solution. The critical angle (measured from the normal to the interface) is θlim=arcsin⁡(n2/n1) = arcsin(1.46/1.48) ≈ 80.6°. The light remains trapped if it strikes the wall at a larger angle, that is if it travels within about 9.4° of the axis of the fibre.

Problem. The ear canal is about 2.5 cm long, open to the outside and closed by the eardrum. As a half-open pipe, at what frequency does it resonate? Solution. f0=c/(4L) = 343/0.1 Hz ≈ 3430 Hz. This resonance partly explains why the ear is most sensitive to sounds between 2000 and 5000 Hz (see the page anatomy of the auditory system).

Links

  1. ↑ Many images of speaking tubes at http://www.douglas-self.com/MUSEUM/COMMS/voicepipe/voicepipe.htm
"Physics, Waves, Music": a website about the physics of waves, acoustics of musical instruments, musical scales, harmony and music.