The app controls are labelled in Italian: below, each label is given as it appears in the app, followed by its English meaning.

In this series of experiments a sound wave is sent through pipes that are open, closed, or "flared" at one end. In musical instruments the role of the wave source is "played" by the musician who, with lips and breath, sets the air column inside the bore of the instrument oscillating. For this reason, in what follows we will also use the term "mouthpiece" for the physical location of the source.

Reflection of sound at the ends of an open cylindrical tube

1. Instructions
   • Click here to start the 2D Waves app and select from the menu at the top left Strumenti: canna cilindrica aperta (Instruments: open cylindrical pipe).
   • Try increasing the Frequenza sorgente 1 (source 1 frequency) with the corresponding slider
2. Remarks
   • A reflected wave (in antiphase) appears at the open end of the pipe. At the open end of the pipe there is a pressure node (the pressure is constrained to equal atmospheric pressure). As far as pressure is concerned, the open end behaves like a fixed end, and the pressure wave reverses its phase on reflection (from green to red).
   • Besides the reflected wave there is a transmitted wave, which radiates into the surroundings and whose wavefronts soon become spherical.
   • The "default" frequency is chosen so low that the reflected wave does not interfere with the wave generated by the source. By increasing the frequency these waves can be made to "interfere": by adjusting the frequency with the corresponding slider, you can "find" a stationary configuration in which the reflected wave and the wave generated by the source meet in antiphase, forming nodes.
3. Related topics: see reflection in strings, interference between reflected waves, reflection of sound, musical instruments, refraction in one dimension (in Italian)
4. To find out more: see impedance, impedance matching, refraction of sound, trumpet, flute, oboe, clarinet

Reflection of sound in an open cylindrical tube with holes

1. Instructions
   • Click here to start the 2D Waves app and select from the menu at the top left Strumenti: canna cilindrica aperta (Instruments: open cylindrical pipe).
   • Select the commands Stop, Cancella Onde (clear waves) and Modifica pareti (edit walls)
   • Using the mouse (dragging it over a wall), make a hole in the cylindrical pipe (about halfway along the pipe)
   • Deselect Stop
2. Remarks
   • The reflected wave (in antiphase) already appears at the hole. It is as if the hole, connecting the tube with the outside, created a new pressure node (now the pressure is constrained to equal atmospheric pressure at the hole as well);
   • The net effect is a reduction of the effective length of the cylindrical pipe. This effect is fundamental for changing the standing wavelengths of the pipe and therefore the pitch of the sound it emits.
   • Near the holes there is a non-negligible radiation of the wave (sound also comes out of the holes!!). In the trumpet the emitted frequencies are changed mostly by means of valves that lengthen the path of the air through the tube. The sound of the trumpet is radiated almost entirely from the open (and flared) end.
3. Related topics: see reflection in strings, interference between reflected waves, reflection of sound, musical instruments, refraction in one dimension (in Italian)
4. To find out more: see impedance, impedance matching, refraction of sound, trumpet, flute, oboe, clarinet
5. Other experiments
   • Try changing the size of the hole: the "early" reflection effect is negligible if the hole is too small compared with the transverse size of the pipe. In this case the effective length of the pipe lies between the real length and the hole-to-mouthpiece distance.

Reflection of sound at the ends of a half-open cylindrical tube

1. Instructions
   • Click here to start the 2D Waves app and select from the menu at the top left Strumenti: canna cilindrica semiaperta (Instruments: half-open cylindrical pipe).
   • Try increasing the Frequenza sorgente 1 (source 1 frequency) with the corresponding slider
2. Remarks
   • A reflected wave (in phase) appears at the closed end of the pipe. The closed end of the pipe prevents the longitudinal oscillation of the molecules (the displacement has a node); this favours the variation of pressure between a maximum value (corresponding to the highest possible density) and a minimum value (corresponding to the greatest rarefaction). That is, we have a pressure antinode (as if, for pressure, the closed end were a free end). The pressure wave is not inverted on reflection (green stays green).
   • The closed end of course prevents the wave from radiating into the surroundings.
   • The "default" frequency is chosen so low that the reflected wave does not interfere with the wave generated by the source. By increasing the frequency these waves can be made to "interfere": by adjusting the frequency with the corresponding slider, you can "find" a stationary configuration in which the reflected wave and the wave generated by the source meet in antiphase, forming nodes.
3. Related topics: see reflection in strings, interference between reflected waves, reflection of sound, musical instruments, refraction in one dimension (in Italian)
4. To find out more: see impedance, impedance matching, refraction of sound, trumpet, flute, oboe, clarinet

Pipe with a widening and a narrowing

1. Instructions
   • Click here to start the 2D Waves app and select from the menu at the top left Strumenti: canna con allargamento (Instruments: pipe with widening) or Strumenti: canna con restringimento (Instruments: pipe with narrowing).
2. Remarks
   • Observe how the waves behave as the frequency changes and, in particular, try to work out whether the pipe radiates waves into space better at high or at low frequencies. Can we say that it behaves like a filter?
   • You will find the complete theory for this configuration on the pages refraction in pipes, impedance, impedance matching, and refraction in strings
   • Other related examples on the page modified cylindrical pipe

Propagation in a curved duct

1. Instructions
   • Click here to start the 2D Waves app and select from the menu at the top left Strumenti: canna cilindrica curva (Instruments: curved cylindrical pipe).
2. Remarks
   • The simulation shows that the bend, although at a right angle, does not substantially change the propagation of sound waves in the duct, which behaves like a single straight cylinder whose total length is the sum of the two arms
   • Try changing the frequency of the wave
   • Draw other shapes of curved pipe yourself, and check whether the propagation is modified.

Reflection of sound at the ends of a conical and an exponential horn

1. Instructions
   • Click here to start the 2D Waves app and select from the menu at the top left Strumenti: corno conico 1 (Instruments: conical horn 1).
   • To change the opening of the conical horn select from the menu at the top left Strumenti: corno conico 2 (Instruments: conical horn 2).
   • To change the "flare" of the horn select from the menu at the top left Strumenti: corno esponenziale (Instruments: exponential horn).
2. Remarks
   • the reflected wave seems almost absent; in reality it has a very low intensity. To see it better, increase the Intensità (intensity) using the corresponding slider.
   • the wave reflected at the end of the horn also hits the walls on its way back and is reflected again by them. It gives rise to interference patterns more similar to those of a two-dimensional membrane than to those of a string. The problem loses its one-dimensional character.
   • widening the opening of the conical horn, the reflected wave has an even lower intensity. Clearly the impedance matching with the outside is even more gradual and reduces the reflection coefficient.
   • in the case of the exponential horn the reflected wave is practically absent; clearly the impedance matching between adjacent cross-sections of the horn, and between the last section and the outside air, is optimal.
3. Related topics: see reflection in strings, interference between reflected waves, reflection of sound, musical instruments, refraction in one dimension (in Italian)
4. To find out more: see impedance, impedance matching, refraction of sound, trumpet, clarinet, acoustics of model wind instruments, conical pipe, mixed pipe