Standing Waves and Harmonics

Professor Dave here, let's discuss standing waves. Waves in the ocean can travel great distance, just like sound waves. But some are confined to specific region, like if you shake taut rope with one end fixed in space. Waves will travel down this rope and then back again, reflected at the boundary. Some waves are confined between two fixed boundaries, like on plucked string, where they will experience repeated reflections at both ends of the string, resulting in multitude of wave cycles traveling in both directions. If this vibration is of particular frequency, it will produce an interference pattern that appears to be completely stationary wave. These are called transverse standing waves. These kinds of waves can only have particular frequencies for strings of given length, because they can only have integer numbers of half wavelengths, since the waves must return to zero amplitude at both boundaries. If the number of half wavelengths was not an integer, the wave couldn't exist. This means that the number of half wavelengths in any standing wave must be quantized, meaning it can only exhibit certain discrete values, such as the set of integers, rather than any value from continuous spectrum. And this idea of quantization will be an important one when we get to the modern physics course. Standing waves contain nodes, where there is destructive interference and an amplitude of zero, as well as antinodes where the amplitude is at maximum. The string will be completely stationary at the nodes, and the other sections, if vibrating rapidly enough, will appear to the human eye to create loops, and more nodes means more energy. Two dimensional standing waves also exist, which manifest on flat two-dimensional surface like drum head, rather than one-dimensional string. On one-dimensional string the nodes are points, but on two-dimensional surface the nodes are lines and curves. These can be radial nodes, which are circles of particular radius, and angular nodes, which are lines at particular angles. These nodes can be visualized if we place sand on vibrating surface. The sand gets pushed over to the nodes by the vibrations and this helps us visualize the two-dimensional standing wave. three-dimensional standing wave would have two dimensional planes as nodes, but we will have to use our imaginations for that. One interesting thing about standing waves is that they combine to produce all of the consonant intervals in music that sound good to our ears. If we consider this half wavelength to be the first harmonic with frequency of f1, then doubling this frequency to 2f1 will produce standing wave with one full wavelength and, will give us the second harmonic, which in music would be perceived as an octave. Tripling the original frequency to 3f1 will give us the third harmonic, which in music is called fifth. This will continue to give the rest of the consonant intervals. So the reason that certain combinations of notes sound pleasant is no random matter, it is rooted in the harmonics of standing waves. This concept of standing waves also represents our first encounter with quantization, given that each harmonic has an integer number of half wavelengths, and quantization will be very important concept in the modern physics course. As we will later see, standing waves are at the heart of number of quantum phenomena that seem to completely defy logic, such as the quantization of energy of an electron. But we aren't quite there yet, so let's check comprehension. Thanks for watching, guys. Subscribe to my channel for more tutorials, support me on patreon so can keep making content, and as always feel free to email me: