Fourier - [Please have my notes by your side. I will refer...

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Unformatted text preview: [Please have my notes by your side. I will refer to many things there.] Lets start from the vertical analogue of N=2 system. As we have defined horizontal MT mode as that in which both masses move to the same side (right or left) of the equilibrium, we can define MT mode of vertical vibration as that in which both masses are either above or below the center equilibrium line. Similarly SM is where the vertical displacements have opposite sign all the time. And lets also specify t i t i e a t y e a t y 2 2 1 1 ) ( , ) ( = = . These have the same mathematical structure as the horizontal case, except for the variable y in the place of x. Forget about N=2 case for a moment, and take N to infinity. Then we have continuous rope, or spring. And recollect the second demonstration in the class, in which the professor used a long black spring with one end attached on the wall. He tried to show diverse standing wave patterns, with different frequencies and wavelength. The result was : 1. In the lowest mode there was a single bump with no node, and the entire spring moved up and down altogether. See the figure below. And the spring vibrated relatively slow. 2. In the next higher mode, there were one node in the middle and two bumps, and the left / right half moved opposite way. When one side is above the center, the other side is always below. So it looks like the figure below. Peter had to move his hand faster to establish this mode. (higher frequency) 3. In the second higher mode, we can predict how it will look like two nodes, three bumps, with adjacent bumps moving oppsite direction. And the vibration was quite fast. These are the first three (lowest) normal modes. But in what respect can we identify them as normal modes? Recall the definition of normal mode. It is a pattern of vibration in which all the constituent particles vibrate with the same frequency, like MT and SM for N=2 case. This holds for all the above three modes; every part of the rope or spring vibrated at the same frequency. In fact, the professor was applying a special initial displacement to excite only one normal mode, exactly the same way as he displaced the two carts either to the same direction or opposite direction in order to excite either MT...
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Fourier - [Please have my notes by your side. I will refer...

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