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Suppose a string is fixed at both ends. What will happen if I give the string a shape that is not the shape of a normal mode, and let it go? Exemple: What happens after I give it the shape of a semicircle and let it go?
Why do the other modes die out completely (and so rapidly!), instead of their amplitude just diminishing uniformly?Meir Achuz said:With energy loss (caused mostly by air resistance), the string will eventually vibrate in only the lowest mode.
quasar987 said:Ok, I'm not very familiar with Fourier transforms, but as I understand you, given a wave y(x,t) with fixed ends, we can apply Fourier tranforms to find that it can be decomposed as a sum of normal modes?
Meir Achuz said:You may just have to wait til you learn more math to really understand this.
Could you give a hint as to what math leads to this conclusion?Meir Achuz said:The higher modes die out faster because they are moving faster, and so have faster energy loss to the air. It's just like the energy loss due to windage increase as a car goes faster. The string is lilght, so the loss of all the energy in higher modes is rapid.
You may just have to wait til you learn more math to really understand this.
is that it doesn't account for twang at all, right?Meir Achuz said:In Math Physics books, the dissipation is often described by a term
\gamma(dy/dt) in the wave equation. That is a gross simplification, but leads to simple equations for the "twang".
The shape of a string when it's released is determined by several factors, including the tension in the string, the material and thickness of the string, and the force applied to the string.
The length of a string plays a significant role in its shape when it's released. Longer strings tend to have more pronounced curves and loops, while shorter strings may have smaller, tighter shapes.
Yes, the shape of a string can be controlled to some extent when it's released. This can be achieved by adjusting the tension in the string or changing the angle at which it's released.
The shape of a string when it's released is directly related to its frequency. The more pronounced the curves and loops, the higher the frequency of the string's vibrations. This is because shorter wavelengths correspond to higher frequencies.
Gravity plays a significant role in determining the shape of a string when it's released. Gravity pulls the string downward, causing it to sag and form curves and loops. The strength of gravity can also affect the tension in the string, which in turn can affect its shape.