PHYS 422 Lecture 23 Continuous Systems and Fourier Analysis III

# PHYS 422 Lecture 23 Continuous Systems and Fourier Analysis III

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Continuous Systems and Fourier Analysis III

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Fourier Analysis a stretched string of length L 0 L () ,s i n c o s nn n n nx yx t A t L π ω δ ⎛⎞ =− ⎜⎟ ⎝⎠ 1/2 1 n nT n L μ == in general all the modes of the string are present so the motion of the string can be completely described by now imagine we capture an image of the string at some time t 0 so that the cosine terms are essentially fixed numbers. 1 i n c o s n n t A t L = 1 sin n n B L = ⇒=
Fourier Analysis the coefficient B n is now just we can now make the following statement about the string that holds as long as it is constrained to have y = 0 at x = 0 and x = L We can represent any profile of the string between x = 0 and x = L and resolve it into an infinite series of sine functions given by this is just a consequence of extending our discussion of N coupled oscillators to the limit where N this is a Fourier analysis in space ( ) 0 cos nn n n BA t ω δ =− () 1 sin n n nx yx B L π = ⎛⎞ = ⎜⎟ ⎝⎠ 1 sin n n B L = =

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Fourier Analysis now let’s focus on a particular value of x along the string where the coefficient C n is given by this is just a Fourier analysis in time the spatial representation depends on a series of sines while the temporal representation depends on a series of cosines () 1 ,s i n c o s nn n n nx yxt A
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## This note was uploaded on 03/23/2011 for the course PHYS 422 taught by Professor Staff during the Spring '08 term at Purdue University-West Lafayette.

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PHYS 422 Lecture 23 Continuous Systems and Fourier Analysis III

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