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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