By Fleisch, Daniel A.; Kinnaman, Laura
"Waves are a massive subject within the fields of mechanics, electromagnetism, and quantum idea, yet many scholars fight with the mathematical features. Written to counterpoint path textbooks, this publication makes a speciality of the subjects that scholars locate so much tough. maintaining the hugely renowned procedure utilized in Fleisch's different Student's courses, the e-book makes use of simple language to provide an explanation for primary rules in an easy and transparent approach. workouts and fully-worked examples support readers attempt their knowing of the thoughts, making this an incredible ebook for undergraduates in physics and engineering attempting to familiarize yourself with this demanding topic. The ebook is supported via a set of on-line assets to be had at www.cambridge.org/9781107643260. those contain interactive ideas for each workout and challenge within the textual content and a sequence of video podcasts during which the authors clarify the $64000 thoughts of each element of the book"-- Read more...
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Additional resources for A student’s guide to waves
4. 5 μs at a fixed location? (b) What is the difference in phase of a mechanical wave with period of 2 seconds and speed of 15 m/s at two locations separated by 4 meters at some instant? 5. If vector D = −5ˆı − 2jˆ and vector E = 4jˆ, find the magnitude and direction of the vector F = D + E both graphically and algebraically. 6. Verify that each of the complex numbers in Fig. 10 has the polar form shown in Fig. 12. 7. Solve the differential equation dz/dθ = iz for z. 8. Use the power-series representation of sin θ , cos θ , and eiθ to prove the Euler relation eiθ = cos θ + i sin θ .
To distinguish between ordinary and partial derivatives, ordinary derivatives are written as d/dx or d/dt and partial derivatives are written as ∂/∂x or ∂/∂t. As you may recall, ordinary derivatives allow you to determine the change of one variable with respect to another. For example, if you have a function y = f (x), the ordinary derivative of y with respect to x (that is, dy/dx) tells you how much the value of y changes for a small change in the variable x. If you make a graph with y on the vertical axis and x on the horizontal axis, as in Fig.
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 . 44) This result shows that the sine function can also be represented by two counter-rotating phasors, since as θ increases eiθ rotates anti-clockwise and −e−iθ rotates clockwise, as illustrated in Fig. 27. Notice that in this figure, in addition to the phasors eiθ and e−iθ , the phasor −e−iθ is also shown.