Lect_32_post

# Lect_32_post - Lecture 32 Standing waves Doppler effect...

This preview shows pages 1–9. Sign up to view the full content.

Lecture 32 Standing waves. Doppler effect.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Reflected waves: fixed end A pulse travels through a rope towards the end that is tied to a hook in the wall (ie, fixed end ) The pulse is inverted (because of Newton s 3 rd law!) F on wall by string F on string by wall The force by the wall always pulls in the direction opposite to the pulse. Another way: Consider one wave going into the wall and another coming out of the wall. The superposition must give 0 at the wall. Virtual wave must be inverted:
Reflected waves: free end A pulse travels through a rope towards the end that is tied to a ring that can slide up and down without friction along a vertical pole (ie, free end ) No force exerted on the free end, it just keeps going Fixed boundary condition Free boundary condition

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Standing waves A harmonic wave traveling along the + x direction is reflected at a fixed point. What is the result of its superposition with the reflected wave? No motion for these points ( nodes ) 2 k π λ = These points oscillate with the maximum possible amplitude ( antinodes ) Standing wave cos b ( ) cos a ( ) = 2sin a + b 2 " # \$ \$ % & ' ' sin a b 2 " # \$ \$ % & ' ' y 1 x , t ( ) = A cos kx ω t ( ) y 2 x , t ( ) = A cos kx + ω t ( ) y x , t ( ) = A cos kx ω t ( ) cos kx + ω t ( ) # \$ % & ' ( y x , t ( ) = 2 A sin kx ( ) sin ω t ( ) If kx = 0, π ,2 π ,... x = 0, λ 2 , λ , 3 λ 2 ... y x , t ( ) = 0 If kx = π 2 , 3 π 2 ,... x = λ 4 , 3 λ 4 ... y x , t ( ) = ± 2 A sin ω t ( )
+x -x Standing wave Node Antinode

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Standing waves and boundary conditions We obtained Nodes 0, , ,... 2 x λ λ = 3 Antinodes , ... 4 4 x λ λ = We need fixed ends to be nodes and free ends to be antinodes! Strong restriction on the waves that can survive with a given set of boundary conditions. y x , t ( ) = 2 A sin kx ( ) sin ω t ( )
Normal modes Which standing waves can I have for a string of length L fixed at both ends? I need nodes at x = 0 and x = L Nodes 0, , ,... 2 x λ λ = , ,... for 1,2,... 2 2 L n n λ λ λ = = = 2 for 1,2,... n L n n λ = = Allowed standing waves ( normal modes ) between two fixed ends Mode n = n-th harmonic f n = v λ n = n v 2 L for n = 1,2,...

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 free ends 1 fixed, 1 free 2 fixed ends 1 2 L λ = 2 L λ = 3 2 3 L λ = 4 2 L λ = Normal modes for fixed ends (lower row)
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern