{"id":409,"date":"2016-11-04T02:07:32","date_gmt":"2016-11-04T02:07:32","guid":{"rendered":"https:\/\/jhubisz.expressions.syr.edu\/phy360\/?p=409"},"modified":"2016-11-04T02:44:11","modified_gmt":"2016-11-04T02:44:11","slug":"week-10","status":"publish","type":"post","link":"https:\/\/jhubisz.expressions.syr.edu\/phy360\/2016\/11\/04\/week-10\/","title":{"rendered":"Week 10"},"content":{"rendered":"

This week we began our study of Fourier Analysis<\/strong> and Fourier Series<\/strong>. \u00a0Our system of study for this discussion will be a string on which the wave velocity is v, and which has fixed endpoints at x=0 and x=L where x is a position along the length of the string.<\/p>\n

Recall that the wave equation is<\/p>\n\\frac{\\partial^2 y(x,t)}{\\partial x^2} = \\frac{1}{v^2} \\frac{\\partial^2 y(x,t)}{\\partial t^2}\n

and that we can write any solution to this wave equation as a superposition of normal modes:<\/p>\ny(x,t) = \\sum_{n=1}^{\\infty} A_n \\cos \\left[ \\omega_n t - \\delta_n \\right] \\sin \\left[ \\frac{n \\pi x}{L} \\right]\n

where \\omega_n = \\frac{n \\pi v}{L}.<\/p>\n

We would like to go from a set of initial conditions (position and velocity of every segment of the string) to the values of each A_n and \\delta_n. \u00a0We were able to do this for systems with a finite number of degrees of freedom, so we should, in principle, be able to do it in the case of systems with a very large number of degrees of freedom.<\/p>\n

Let us first consider just a set of positions at time $t =0$. \u00a0Let’s say we are given these, and they take the form y(x,t=0) = f(x), where f(x) is any function with f(0)=f(L)=0. \u00a0These are the boundary conditions obeyed by our segment of string.<\/p>\n

If we now apply this to the expression for y in terms of the normal modes, we find<\/p>\ny(x,t=0) = f(x) = \\sum_{n=1}^{\\infty} A_n \\cos (\\delta_n) \\sin \\left[ \\frac{n \\pi x}{L} \\right]\n

Now for the moment, let us combine the coefficients of each \\sin into a set of coefficients B_n = A_n \\cos \\delta_n, in which case we then have<\/p>\nf(x) = \\sum_{n=1}^{\\infty} B_n \\sin \\left[ \\frac{n \\pi x}{L} \\right]\n

Let us now try to get the B coefficients from the function f(x).<\/p>\n

To do this, we use a trick involving integrals of sin functions:<\/p>\n\\frac{2}{L} \\int_0^L \\sin \\left[ \\frac{n \\pi x}{L} \\right] \\sin \\left[ \\frac{m \\pi x}{L} \\right] dx = \\left\\{ \\begin{array}{ll} 1 & n = m \\\\ 0 & n \\ne m \\end{array} \\right.\n

multiplying both sides of the equation for f(x) by \\frac{2}{L} \\sin \\left[ \\frac{m \\pi x}{L} \\right] and integrating over $x$ from $0$ to $L$, we find<\/p>\n\\frac{2}{L} \\int_0^L f(x) \\sin \\left[ \\frac{m \\pi x}{L} \\right] = \\sum_{n=1}^{\\infty} B_n \\cdot \\frac{2}{L} \\sin\\left[ \\frac{n \\pi x}{L} \\right] \\sin \\left[ \\frac{m \\pi x}{L} \\right] dx = B_m\n

This means that if we know the function f(x), and perform some integrals, then we can get the B_m coefficients!<\/p>\n

See this weeks Mathematica code to begin to get an idea of how this works:<\/p>\n

https:\/\/jhubisz.expressions.syr.edu\/phy360\/wp-content\/uploads\/sites\/5\/2016\/11\/Week-10.nb<\/a><\/p>\n

We can then go a bit further. \u00a0Again, the goal is to get both the A_n and \\delta_n. \u00a0In order to do this, we need not just initial position information, but also initial velocity information. \u00a0Right now, we have B_n = A_n \\cos \\delta_n, so we only have fixed a combination of A_n and \\delta_n. \u00a0Let us say we have that velocity information as a given:<\/p>\n

\\left. \\frac{\\partial y(x,t)}{\\partial t} \\right|_{t=0} = g(x) = \\sum_{n=1}^{\\infty} A_n \\omega_n \\sin \\delta_n \\sin \\left[ \\frac{n \\pi x}{L} \\right].<\/p>\n

In this case, we can write C_n = A_n \\omega_n \\sin \\delta_n, and thus we can also express the velocity of each point of the string in terms of a sum over sin functions:<\/p>\ng(x) = \\sum_{n=1}^\\infty C_n \\sin \\left[ \\frac{n \\pi x}{L} \\right]\n

We can then use the same trick to get:<\/p>\nC_n = \\frac{2}{L} \\int_0^L g(x) \\sin \\left[ \\frac{n \\pi x}{L} \\right]\n

So now you have all of the B’s, and all of the C’s (since we presumed we know both initial position, f(x), and initial velocity, g(x)). \u00a0We then have<\/p>\nB_n = A_n \\cos \\delta_n \\text{ and } C_n = A_n \\omega_n \\sin \\delta_n\n

From this, we can finally get both A_n and \\delta_n:<\/p>\n

A_n = \\left[ B_n^2 + \\left( \\frac{C_n}{\\omega_n} \\right)^2 \\right]^{1\/2} \\text{ and } \\delta_n = \\arctan \\left[ \\frac{ C_n\/\\omega_n }{B_n} \\right].<\/p>\n

I will soon be uploading a mathematica notebook that goes through an example of a particular set of initial positions and velocities, and go through how to get the A’s and \\delta‘s for a specific example.<\/p>\n","protected":false},"excerpt":{"rendered":"

This week we began our study of Fourier Analysis and Fourier Series. \u00a0Our system of study for this discussion will be a string on which the wave velocity is , and which has fixed endpoints at x=0 and x=L where … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/jhubisz.expressions.syr.edu\/phy360\/wp-json\/wp\/v2\/posts\/409"}],"collection":[{"href":"https:\/\/jhubisz.expressions.syr.edu\/phy360\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/jhubisz.expressions.syr.edu\/phy360\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/jhubisz.expressions.syr.edu\/phy360\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/jhubisz.expressions.syr.edu\/phy360\/wp-json\/wp\/v2\/comments?post=409"}],"version-history":[{"count":10,"href":"https:\/\/jhubisz.expressions.syr.edu\/phy360\/wp-json\/wp\/v2\/posts\/409\/revisions"}],"predecessor-version":[{"id":420,"href":"https:\/\/jhubisz.expressions.syr.edu\/phy360\/wp-json\/wp\/v2\/posts\/409\/revisions\/420"}],"wp:attachment":[{"href":"https:\/\/jhubisz.expressions.syr.edu\/phy360\/wp-json\/wp\/v2\/media?parent=409"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/jhubisz.expressions.syr.edu\/phy360\/wp-json\/wp\/v2\/categories?post=409"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/jhubisz.expressions.syr.edu\/phy360\/wp-json\/wp\/v2\/tags?post=409"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}