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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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You live on a busy street, but as a music lover, you want to reduce the traffic noise. (a) If you install special sound reflecting windows that reduce the sound intensity level (in dB) by 30 dB, by what fraction have you lowered the sound intensity(W/m2)?(b) If, instead, you reduce the intensity by half, what change (in dB) do you make in the sound intensity level?

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