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First, consult the figure at the end of the first page (Problem 3):

http://www.iro.umontreal.ca/~montanom/cours/phy/1620_tp_8.pdf

The problem reads: "A rope of length L and mass M is stretched with a tension T between the 2 rings of negligible mass allowed to slide w/o friction along the lines parallel to the y axis. Initially, the rings are maintained at y = 0, while we give the rope a shape of the form y(x, t=0) = Asin²(pi x/L) as illustrated. We then let go of the rope and the rings at time t = 0.

a) Draw the periodical function corresponding to the definition of the problem and give its spatial period.

b) Give the complete expression of the motion of the rope as a function of time in terms of its Fourier components."

The answer to b) is

[tex]y(x,t) = \left( \frac{A}{2} - \frac{A}{2}cos\left(\frac{2\pi x}{L}\right)\right) cos(\omega t)[/tex]

Where the part in parenthesis is just the Fourier expansion of Asin²(pi x/L). From this answer I read 2 surprising things

1) Judging from the term cos(xt), the rope oscillate in a normal mode.

2) At x = 0 and x = L, the displacement y is 0 at all times. So the resulting motion is exactly the same as if the rope was glued to a wall.

My question is:

How am I suppose to know what will happen to this rope when I release it? I.e. how was I supposed to know that I all I needed to do was multiply the Fourier term by cos(wt)?! Stated otherwise: given an initial configuration of the rope, how can I predict how it will behave when released? Does any deformation on a rope whatesoever vibrate in a normal mode when released?!

http://www.iro.umontreal.ca/~montanom/cours/phy/1620_tp_8.pdf

The problem reads: "A rope of length L and mass M is stretched with a tension T between the 2 rings of negligible mass allowed to slide w/o friction along the lines parallel to the y axis. Initially, the rings are maintained at y = 0, while we give the rope a shape of the form y(x, t=0) = Asin²(pi x/L) as illustrated. We then let go of the rope and the rings at time t = 0.

a) Draw the periodical function corresponding to the definition of the problem and give its spatial period.

b) Give the complete expression of the motion of the rope as a function of time in terms of its Fourier components."

The answer to b) is

[tex]y(x,t) = \left( \frac{A}{2} - \frac{A}{2}cos\left(\frac{2\pi x}{L}\right)\right) cos(\omega t)[/tex]

Where the part in parenthesis is just the Fourier expansion of Asin²(pi x/L). From this answer I read 2 surprising things

1) Judging from the term cos(xt), the rope oscillate in a normal mode.

2) At x = 0 and x = L, the displacement y is 0 at all times. So the resulting motion is exactly the same as if the rope was glued to a wall.

My question is:

How am I suppose to know what will happen to this rope when I release it? I.e. how was I supposed to know that I all I needed to do was multiply the Fourier term by cos(wt)?! Stated otherwise: given an initial configuration of the rope, how can I predict how it will behave when released? Does any deformation on a rope whatesoever vibrate in a normal mode when released?!

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