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Chapter 2: Q12E (page 498)

Speed of Propagation vs. Particle Speed. (a) Show that Eq. (15.3) may be written as

\(y\left( {x,t} \right) = Acos\left[ {\frac{{2\pi }}{\lambda }\left( {x - vt} \right)} \right]\)

(b) Use \(y\left( {x,t} \right)\) to find an expression for the transverse velocity \({v_y}\)of a particle in the string on which the wave travels. (c) Find the maximum speed of a particle of the string. Under what circumstances is this equal to the propagation speed \(v\) ? Less than\(v\)? Greater than\(v\)?

Short Answer

Expert verified

(a) \(y\left( {x,t} \right) = Acos\left[ {\frac{{2\pi }}{\lambda }\left( {x - vt} \right)} \right]\)

Step by step solution

01

Given data

Wave function for a sinusoidal wave propagation in +x-direction is

\(y\left( {x,t} \right) = A\cos \left[ {\omega \left( {\frac{x}{v} - t} \right)} \right]\,{\rm{ }}{\rm{. }}{\rm{. }}{\rm{. (15}}{\rm{.3)}}\)

02

Concept/ Formula used

\(\frac{\lambda }{T} = \lambda f = v\)

Where,\(\lambda \)is wavelength

\(f\)Is frequency and\(v\)is wave speed.

03

Derive equation \(y\left( {x,t} \right) = Acos\left[ {\frac{{2\pi }}{\lambda }\left( {x - vt} \right)} \right]\)

(a)

\(\begin{aligned}{l}Acos2\pi \left( {\frac{x}{\lambda } - \frac{t}{T}} \right)\\ = + Acos\frac{{2\pi }}{\lambda }\left( {x - \frac{\lambda }{T}t} \right)\\ = + Acos\frac{{2\pi }}{\lambda }\left( {x - vt} \right)\end{aligned}\)

Where, \(\frac{\lambda }{T} = \lambda f = v\)

\(\therefore y\left( {x,t} \right) = Acos\left[ {\frac{{2\pi }}{\lambda }\left( {x - vt} \right)} \right]\)

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