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By Fleisch, Daniel A.; Kinnaman, Laura

"Waves are a major subject within the fields of mechanics, electromagnetism, and quantum conception, yet many scholars fight with the mathematical points. Written to counterpoint direction textbooks, this ebook specializes in the subjects that scholars locate so much tough. conserving the hugely well known strategy utilized in Fleisch's different Student's courses, the publication makes use of simple language to give an explanation for basic rules in an easy and  Read more...

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2 you know that λ = 2π/k and T = 2π/ω. Putting these three expressions together gives λ 2π /k 2π ω = = T 2π /ω k 2π v= or v= ω . 36) This is a very useful result. If you know both the angular frequency (ω) and the wavenumber (k), you can divide ω by k to find the wave phase speed. And, if you’re given a function such as that shown in Eq. 32) and you need the wave speed, just take whatever multiplies the time term (t) and divide it by whatever multiplies the position term (x). As always, it’s useful to ask yourself whether the units work.

28. At θ = 0, both of these phasors point along the horizontal (real) axis of the complex plane, but the phasor −e−iθ points along the negative real axis while the phasor eiθ points along the positive real axis. Adding these two opposite-direction phasors produces a zero resultant phasor, which is shown as a dot labeled eiθ + (−e−iθ ). Hence the expression (eiθ − e−iθ )/(2i) has a magnitude of 0, which is the value of sin θ for θ = 0. Now look at the phasors eiθ and −e−iθ for θ = 30◦ . The phasor eiθ points above the positive real axis by 30◦ , and the phasor −e−iθ points above the negative real axis by 30◦ .

If you hope to understand the wave equation (and all other partial differential equations), a good place to begin is to make sure you have a solid understanding of partial derivatives. 2. 1 Partial derivatives If you’ve taken any course in calculus or college-level physics, you almost certainly encountered ordinary derivatives when you learned how to find the slope of a line (m = dy/dx) or how to determine the speed of an object given its position as a function of time (vx = dx/dt). As you probably learned, there are many functions in mathematics and physics that depend on only one independent variable, and ordinary derivatives are all you’ll ever need to analyze the changes in such functions.

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