Q12E - University Physics with Modern Physics

Question

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\(v\)?

Step-by-Step Solution

Answer

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

Step 1: Given data

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

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

Step 2: Concept/ Formula used

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

Where, \(\lambda \) is wavelength 

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

Step 3: 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, \(\frac{\lambda }{T} = \lambda f = v\)

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