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What is the physical explanation for how a travelling wave sent down an open-open end pipe reflect from the ends (even though they are open) to form a standing wave?

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A wave propagating through a wave guide of a constant wave impedance does not reflect, but does partially reflect from the place where the wave impedance of the wave guide changes. For example, if you connect a 75-Ohm cable to a 50-Ohm input of an oscilloscope, then a high frequency signal would partially reflect from the connection. Similarly, the wave impedance of a pipe is different from the wave impedance of the open space, so the wave partially reflects and forms a standing wave in the pipe.

This is valid for all types of waves, electromagnetic, sound, etc., as a result of the Huygens-Fresnel principle in an odd number of dimensions. In contrast, waves in an even number of dimensions, such as on water, reflect back, even if the wave impedance is constant. For example, if you drop a rock in a lake, the wave does not just circle out leaving the center undisturbed (like a light flash would). Instead, the wave also reflects back and forms a standing wave in the center.

It is possible to use a transformer to match impedances of two wave guides. Any TV antenna has a small 4:1 transformer matching the 300-Ohm antenna to a 75-Ohm coaxial cable. Another example is the top of the acoustic guitar that acts as a matching transformer between the strings and the air. This explains why an electric guitar is so quiet without amplification, but has a longer sustain (the wave reflects back to the string). Different shapes and geometries at the ends of the pipe would act as transformers and change how much of the wave is reflected. This is why musical instruments like the tuba have the expanding end (to minimize the sound reflections and increase the volume).

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  • $\begingroup$ You haven’t actually given an explanation for the phenomena; what’s the definition of impedance and how does the impedance change at the boundary? $\endgroup$ – cms Aug 1 '18 at 13:06
  • $\begingroup$ @cms Have you checked the Huygens principle linked in my answer? Each point of the wavefront is a source of a new wave that goes in all directions, including backward. However, the backward wave is canceled out on two conditions (1) the space has an odd number of dimensions and (2) there is no change in the medium. When the geometry of the medium changes, the back wave does not cancel. So the fact that the wave reflects is natural, as it does not reflect only in special circumstances. See: en.wikipedia.org/wiki/Acoustic_impedance - and: en.wikipedia.org/wiki/Wave_impedance $\endgroup$ – safesphere Aug 1 '18 at 15:42
  • $\begingroup$ You just concisely described the phenomena. i would exchange your comment and answer. Just providing a link to an answer is generally frowned upon on SE; a small paragraph (like your comment) would be sufficient. Perhaps: “Huygens principle explains this because ... you can find more about it by searching for wave impedance (link here) or specifically for sound (here)”. </constructive criticism> $\endgroup$ – cms Aug 1 '18 at 16:37
  • $\begingroup$ @cms I see the Huygens principle as the theoretical foundation for the answer, but not the answer itself. I see the wave impedance as the answer. For example, when you match the cable impedance to the impedance of the high frequency input, you don't ponder the Huygens principle, you solder matching resistors. In any case, I hope there is enough information here for anyone with any background to understand the phenomenon. $\endgroup$ – safesphere Aug 1 '18 at 17:02
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    $\begingroup$ @RobertoValente Perhaps you would have a better luck if you posted your own question with a specific problem. $\endgroup$ – safesphere Aug 8 '18 at 18:14

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