A student’s guide to waves by Fleisch, Daniel A.; Kinnaman, Laura

By Fleisch, Daniel A.; Kinnaman, Laura

"Waves are a huge subject within the fields of mechanics, electromagnetism, and quantum concept, yet many scholars fight with the mathematical elements. Written to enrich direction textbooks, this ebook makes a speciality of the subjects that scholars locate such a lot tough. conserving the hugely renowned method utilized in Fleisch's different Student's publications, the e-book makes use of undeniable language to provide an explanation for primary principles in an easy and  Read more...

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But you can always convert the distances in the argument (x ± vt) to phase (with units of radians) by multiplying by the wavenumber (k). 7 Phasor representation of wavefunctions Putting together the concepts of the complex plane, the Euler relations, and phasors provides a very powerful tool for the analysis of wavefunctions. To understand how this works, consider two waves represented by the following wavefunctions: y1 (x, t) = A1 sin(k1 x + ω1 t + 1 ), y2 (x, t) = A2 sin(k2 x + ω2 t + 2 ). 39) If the amplitudes of these waves are equal (so A1 = A2 = A), and if the waves also have the same wavelength (hence the same wavenumber, so k1 = k2 = k) and the same frequency (hence the the same angular frequency, so ω1 = ω2 = ω), then the only difference between the waves must be due to their phase constants ( 1 and 2 ).

1. That’s a limitation of showing functions of space and time in a static environment, such as this book. To see the difference between a wave and a wave profile, imagine drawing a few periods of a sine wave, as in Fig. 18(a). If you repeat the drawing on separate pieces of paper with the wave profile shifted to the left or right by a small amount each time, you can put the pieces together to create a “flipbook”. Flipping through the book shows the wave itself as it moves over time, while each page shows the wave profile at a certain instant in time.

29 A simplified representation of sinusoidal functions. It cancels out the horizontal (real) component of the resultant; and it doubles the length of the vertical (imaginary) component. So the process of adding the counter-rotating phasors and then dividing by 2i gives exactly the same result as finding the projection of a single phasor onto the vertical axis. 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.

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