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HL Math IA: help me understand a few things


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Hey, I'm a senior in HL and need help with my IA.  Right now we are developing our ideas, and I am thinking about musical tones and timbre.  I found a great powerpoint that explains the math behind this (http://faculty.tru.ca/rtaylor/music_seminar.pdf) that involves second derivatives and trigonometry, but I have trouble figuring out the notation behind all of this (pg. 22 and beyond is what I'm talking about).  For example, I don't know what u(x,t) means.

 

If anyone can help me clarify things, that would be great.

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u(x, t) is explained, on page 22 actually, as a function that takes two values (position x and time elapsed t) and returns a single value (displacement, vector quantity, of that specific particle). The 2nd derivative of displacement is acceleration (with direction). Note the acceleration is not constant. 

 

The lower case delta means partial derivative that is we look at instantaneous change to displacement when only time is changing but position of the particle we are looking at is not. Vice versa if it were delta x in the denominator instead of delta t.

The partial differential equation is much beyond the IB curriculum so proceed with caution and never write down something you don't understand. Demonstrating a basic understanding of an advanced topic is better than a superficial grasp of more advanced topics. To continue with this IA topic you should learn about waves, standing waves, and simple harmonic motion in physics (ideally from a teacher), not to mention all the math involved.

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Thanks for your help, but I'm still having some difficulty with understanding my topic.  My main problem is with the Laplacian Operator.  This looks beyond what we will be learning, so I need to do some extra research.  I don't know how I would calculate that nor do I know what it represents.  If you could offer me an explanation or just a website I could look at, that would help a lot.  Also, what does c represent in the wave equation.  I assume that means speed of vibrating object (sound), but I'm not sure.

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i think c is a damping constant (energy loss due to sound).https://en.wikipedia.org/wiki/Vibration#Free_vibration_with_damping
and the Lapacian operator seems to be defined as a piece-wise function of 2nd derivatives (it's all on the pdf). 
that symbol though, you should check it out in a course/textbook on vector calculus. I haven't learn it yet.

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