By Fleisch, Daniel A.; Kinnaman, Laura
"Waves are an incredible subject within the fields of mechanics, electromagnetism, and quantum concept, yet many scholars fight with the mathematical features. Written to enrich path textbooks, this publication makes a speciality of the themes that scholars locate such a lot tricky. keeping the hugely well known strategy utilized in Fleisch's different Student's courses, the publication makes use of undeniable language to provide an explanation for basic rules in an easy and transparent approach. routines and fully-worked examples aid readers attempt their realizing of the options, making this an excellent booklet for undergraduates in physics and engineering attempting to become familiar with this difficult topic. The ebook is supported by means of a set of on-line assets to be had at www.cambridge.org/9781107643260. those comprise interactive suggestions for each workout and challenge within the textual content and a sequence of video podcasts within which the authors clarify the $64000 options of each part of the book"-- Read more...
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Extra resources for A student’s guide to waves
In the case of the cosine function, the same analysis applies, as long as you recall that cos φ = sin(φ + π/2) and draw the phasor representing the cosine function along the vertical axis at time t = 0. Using this simplified approach, the two wavefunctions (y1 and y2 ) shown in Figs. 24 can be represented by the two phasors shown in Fig. 30. Notice that both of these phasors rotate anti-clockwise at the same rate (since they have the same ω), maintaining the phase difference between them (π/2 in this case).
26. As θ increases from 180◦ to 360◦ , the eiθ and e−iθ phasors continue rotating, and the resultant eiθ + e−iθ returns to zero and then to 1, as expected for the cosine function at these angles. So the cosine function can be represented by the two counter-rotating phasors eiθ and e−iθ , and at any angle the addition of those phasors and division by 2 yields the value of the cosine of that angle. Can the sine function also be represented in this way? Yes it can. To see how that works, observe what happens if you subtract e−iθ from eiθ : eiθ − e−iθ = (cos θ + i sin θ ) − (cos θ − i sin θ ) = cos θ − cos θ + i sin θ − (−i sin θ ) = 2i sin θ or sin θ = eiθ − e−iθ .
26. At θ = 0, both of these phasors point along the positive horizontal (real) axis of the complex plane (for clarity the phasors eiθ and e−iθ are drawn using dashed lines and slightly offset from one another in Fig. 26). Adding these two same-direction phasors produces a resultant phasor (which we label eiθ + e−iθ ) that also points along the positive real axis and has a magnitude (length) of 2 (since eiθ and e−iθ both have a magnitude of 1). Hence the expression (eiθ + e−iθ )/2 has a magnitude of 1, which is the value of cos θ for θ = 0.
A student’s guide to waves by Fleisch, Daniel A.; Kinnaman, Laura