{"id":188,"date":"2017-02-19T03:12:05","date_gmt":"2017-02-19T03:12:05","guid":{"rendered":"http:\/\/www.maxwellmckinnon.com\/blog\/?p=188"},"modified":"2022-03-05T20:34:33","modified_gmt":"2022-03-05T20:34:33","slug":"fun-with-waves-in-matlab-wave-equation-modeling","status":"publish","type":"post","link":"http:\/\/maxwellmckinnon.com\/blog\/2017\/02\/19\/fun-with-waves-in-matlab-wave-equation-modeling\/","title":{"rendered":"Fun with Waves in Matlab – Wave Equation Modeling"},"content":{"rendered":"
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Setup<\/h1>\n
Matlab is cool. Math is cool. Enough said.
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Amazing how all this beauty is held inside such a seemingly simple formula, the wave equation.<\/div>\n
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This is the basics of the setup. It is a rectangular pool that can simulate perturbations based on the above wave equation. The model implements finite differences for its solution on the 2D mesh. e.g. (f_x = (f_i – f_[i-1]) \/ da)<\/div>\n
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Details of Model<\/h1>\n
The inner oscillations are a result of model error. I wasn’t sure if this error was from the size of the impulse or the size of the mesh. The variables needed to be isolated. I doubled the size of the mesh points in each direction. To keep the impulse size the same, the input was now a 2×2 impulse instead of a 1×1. As seen, the inner oscillations have a higher frequency compared to before. This means the model error of inner oscillation frequency is dependent on the mesh size.<\/div>\n
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I wondered what happens in real life when something gets dropped into a big bowl of water. It turns out that there is one big oscillation followed by medium sized ripples. Running the simulation with a bigger impulse and a very fine mesh, I obtain similar results. As shown, you can see one big oscillation followed by medium-sized ripples.<\/div>\n
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In addition to the medium-sized ripples, you can still see little oscillations of high frequency. As shown previously, these depend on the step size.<\/div>\n
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In real life, there are a discrete number of water molecules to move up and down. There aren’t infinitely many points in the mesh.<\/div>\n
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Maybe these frequencies that depend on the step size show up in real life too. Or maybe not.<\/em><\/div>\n
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Boundary Value Manipulation<\/h1>\n

Any Dirichlet boundary conditions* can be imposed by reassigning the boundary to its fixed value after every iteration. I wanted to see what happened on the shape of a guitar. It is interesting how there is sort of a mirror of the source in the front of the guitar. There was only one impulse applied to this shape, even though it looks like there is a smaller impulse on the top of the guitar.<\/p>\n

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*A Dirichlet boundary condition is a fancy word for a boundary that doesn’t move, like a wall that has a jump rope tied to it – or a guitar bridge with guitar strings tied to it.<\/div>\n
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Fun Experiment DOUBLE SLIT!! xD<\/h1>\n

Applying more Dirichlet boundary values to form two slits for waves to come out. Look at that beautiful diffraction pattern!! (:<\/p>\n

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Until next time<\/div>\n
\u2208<\/div>\n","protected":false},"excerpt":{"rendered":"

Setup Matlab is cool. Math is cool. Enough said. Amazing how all this beauty is held inside such a seemingly simple formula, the wave equation. This is the basics of the setup. It is a rectangular pool that can simulate perturbations based on the above wave equation. The model implements finite differences for its solution on…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"class_list":["post-188","post","type-post","status-publish","format-standard","hentry","category-dev"],"_links":{"self":[{"href":"http:\/\/maxwellmckinnon.com\/blog\/wp-json\/wp\/v2\/posts\/188","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/maxwellmckinnon.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/maxwellmckinnon.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/maxwellmckinnon.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/maxwellmckinnon.com\/blog\/wp-json\/wp\/v2\/comments?post=188"}],"version-history":[{"count":5,"href":"http:\/\/maxwellmckinnon.com\/blog\/wp-json\/wp\/v2\/posts\/188\/revisions"}],"predecessor-version":[{"id":203,"href":"http:\/\/maxwellmckinnon.com\/blog\/wp-json\/wp\/v2\/posts\/188\/revisions\/203"}],"wp:attachment":[{"href":"http:\/\/maxwellmckinnon.com\/blog\/wp-json\/wp\/v2\/media?parent=188"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/maxwellmckinnon.com\/blog\/wp-json\/wp\/v2\/categories?post=188"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/maxwellmckinnon.com\/blog\/wp-json\/wp\/v2\/tags?post=188"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}