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Modes are an interesting, secret playground of engineers. It seems like the idea of modes could be explained to first graders on a playground with a jump rope, but teachers in the US are probably too busy trying to make ends meet. Simply put, modes are naturally occurring motion or wave patterns (often sinusoidal) that are constrained to boundary conditions. A drummer can make at least two distinct tones out of a drum based on where he strikes it. In the first image below, the drum is struck in the center, and the center vibrates up and down at the fundamental frequency. The perimeter of the drum head is constrained and thus is a zero crossing on any sine wave you can envision across the drum head...
Sometimes complex behavior can be interpreted in three seconds if you are familiar with our Rules of Thumb. A microwave Rules of Thumb could be an inexact but notable relationship of one or more design parameters with performance, or it could just be an easy way to remember something that other, lesser people often mix up.
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MTT-S is an international organization with more than 11,000 members and 150 chapters worldwide. The society promotes the advancement of microwave theory and its applications, including RF, microwave, millimeter-wave, and terahertz technologies.