Mathematical description of waves

[ldarko29]

I'm looking for a book or some kind of course on "wave mathematics".

 

I'm familiar with basics, like wavelength, frequency, amplitude, interferrence and all that, but I want to further improve my understanding of this subject.

[ajb]

I think you need to give us a better understanding of your existing level of understanding.

The mathematical definition of a wave is a solution to the wave equation. There is a lot of hard-core analysis behind this in different dimensions and proving existence of solutions etc. This is the kind of thing one meets in a graduate course in analysis.

 

I think, this is not what you seek.

 

In physics and engineering one could also mean a "periodic function" by a wave. There is a whole load of mathematics devoted to periodic functions, principally the ideas of a Fourier transform and a Fourier series.

 

In physics we also have a wave function which describes the quantum behaviour of atoms and molecules. Strictly, they are not waves, they are not solution of the wave equation, but the wave functions have the same qualitative behaviour as waves. They are solutions to the Schrödinger equation.

[ldarko29]

I'm interested in quantum mechanics, so I thought I'd get really familiar with all aspects of 'classical mechanics' first.

Since I'm graduating high school this year, and I intend to study physics at university, it will help for both.

That's why I'm currently reading "Classical Mechanics" by John R. Taylor, which seems like a really good book.

 

I am trying to find some kind of book that goes from basics (what's a wave, types, diffraction, polarization, etc.), towards describing them with more complex mathematical tools (like partial derivatives, laplacean, etc.).

 

I'm not familiar with wave equation (where I'm from, we still didn't learn partial differentiation in high school, so we didn't learn any concepts involving them), but I AM familiar with partial derivatives (I taught myself), integreals, and such tools that I see on the wave equation Wikipedia page.

 

My 'final goal' is to really understand quantum mechanical wavefunction.

[ajb]

I'm interested in quantum mechanics, so I thought I'd get really familiar with all aspects of 'classical mechanics' first.

Since I'm graduating high school this year, and I intend to study physics at university, it will help for both.

That's why I'm currently reading "Classical Mechanics" by John R. Taylor, which seems like a really good book.

That book looks a little more in depth than you may need at this level.

I am trying to find some kind of book that goes from basics (what's a wave, types, diffraction, polarization, etc.), towards describing them with more complex mathematical tools (like partial derivatives, laplacean, etc.).

I would recommend a book intended for first year students in physics, something like Halliday, Resnik and Walker. They have a good discussion about oscillations and waves. The book is not heavily mathematical, the study of waves could take you into the depths of analysis is you wish.

I'm not familiar with wave equation (where I'm from, we still didn't learn partial differentiation in high school, so we didn't learn any concepts involving them), but I AM familiar with partial derivatives (I taught myself), integreals, and such tools that I see on the wave equation Wikipedia page.

Then I suggest as an exercise looking into how to solve simple first order differential equations.

My 'final goal' is to really understand quantum mechanical wavefunction.

You will do when you get to university.

[ldarko29]

That book looks a little more in depth than you may need at this level.

Well, I just started reading it, and I can still follow it. However, I'm still at the very basics (Newton's laws derivation etc.), so it'll probably get tougher.

I may just read what the author calls "essentials" -- first chapter of the book.

 

I would recommend a book intended for first year students in physics, something like Halliday, Resnik and Walker. They have a good discussion about oscillations and waves. The book is not heavily mathematical, the study of waves could take you into the depths of analysis is you wish.

Thanks, I looked it up, and it seems like a great one.

Many illustrations and problems explained, althought it is 1300 pages long!

 

 

I will definitelly need to learn more about differential equations, as I'm only vaguely familiar with them..

Edited February 10, 2013 by ldarko29

[elfmotat]

Taylor's book is very good. It's probably my favorite classical mechanics text. Halliday and Resnik is good as well.

[ajb]

Taylor's book is very good. It's probably my favourite classical mechanics text.

My favourite is R. Abraham and J. E. Marsden, Foundations of Mechanics. I have the first edition from 1967. It is really the first book on modern geometric mechanics and contains many modern things. But I would not say it is the place to start your adventure in classical mechanics.

 

Arnold's Mathematical Methods of Classical Mechanics also comes highly recommended. As does Goldstein's Classical Mechanics. However, these are graduate texts and probability not the best place to start.

[x(x-y)]

"University Physics" by Young and Freedman (International, 13th Ed.) is what I use as a first year physics student and I can say that it is a good text - explains concepts well, with well derived mathematical treatments where necessary.