PHYS 422 Lecture 23 Continuous Systems and Fourier Analysis III

PHYS 422 Lecture 23 Continuous Systems and Fourier Analysis III

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Continuous Systems and Fourier Analysis III
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 = ⇒=
Background image of page 2
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 = =
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

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.

Page1 / 14

PHYS 422 Lecture 23 Continuous Systems and Fourier Analysis III

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online