(i) Use group theory to determine the representation for the mode motion. (ii) Reduce the representation, showing your work, and identify th translational, rotational and vibrational modes.
Added by Mary H.
Close
Your feedback will help us improve your experience
Surendra Kumar and 60 other Chemistry 102 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Let the system of Problem $4.7$ be set in motion with the initial conditions $x=A, y=0, \dot{x}=\dot{y}=0$ at $t=0 .$ Show that the normal mode amplitudes are $X_{0}=\left(m_{1} / M\right) A$ and $Y_{0}=A$ to yield $$ x=\frac{A}{M}\left(m_{1} \cos \omega_{1} t+m_{2} \cos \omega_{2} t\right) $$ and $$ y=A \frac{m_{1}}{M}\left(\cos \omega_{1} t-\cos \omega_{2} t\right) $$ where $M=m_{1}+m_{2}$ Express these displacements as $$ x=2 A \cos \omega_{m} t \cos \omega_{a} t+\frac{2 A}{M}\left(m_{1}-m_{2}\right) \sin \omega_{m} t \sin \omega_{a} t $$ and $$ y=2 A \frac{m_{1}}{M} \sin \omega_{m} t \sin \omega_{a} t $$ where $\omega_{m}=\left(\omega_{2}-\omega_{1}\right) / 2$ and $\omega_{a}=\left(\omega_{1}+\omega_{2}\right) / 2$
Sketch some of the normal modes of vibration for a semicircular drumhead and find the characteristic vibration frequencies as multiples of the fundamental for the corresponding circular drumhead.
PARTIAL DIFFERENTIAL EQUATIONS
Miscellaneous problems
Consider a symmetric step-index waveguide [see Eq. (29.1)] with $n_{1}=1.50, n_{2}=1.46, d=4 \mu \mathrm{m}$ operating at $\lambda_{0}=0.6328 \mu \mathrm{m} .$ Calculate the number of $\mathrm{TE}$ and $\mathrm{TM}$ modes.
Recommended Textbooks
Chemistry: Structure and Properties
Chemistry The Central Science
Chemistry
Watch the video solution with this free unlock.
EMAIL
PASSWORD