1. Lecture 38 Review of waves Interference Standing waves on a string

2. What is a wave ? Examples: –Sound waves (air moves back & forth) –Stadium waves (people move up & down) –Water waves (water moves up & down) –Light waves (what moves??) A wave is a traveling disturbance that transports energy but not matter. Waves exist as excitations of a (more or less) elastic medium.

3. Amplitude: The maximum displacement A of a point on the wave. Amplitude A A Period: The timeT for a point on the wave to undergo one complete oscillation. x y A few parameters Frequency: Number of oscillations f for a point on the wave in one unit of time. Angular frequency: radians ω for a point on the wave in one unit of time.

4. y Wavelength: The distance λ between identical points on the wave. Speed: The wave moves one wavelength λ in one period T, so its speed is Connecting all these simple harmonic motions Wave number k : x λ

5. Propagation Moves in the +x direction with v Moves in the − x direction with v Examples: Harmonic waves

6. The wave equation Wave equation

7. Wave energy Work is clearly being done: F. dr > 0 as hand moves up and down. This energy must be moving away from your hand (to the right) since the kinetic energy (motion) of the end of the string grabbed by the hand stays the same. P

8. Power Energy for a particle in SHM (attached to a spring k) This energy propagates at speed v. the average energy per unit time that flows in the direction of propagation should be proportional to v Average power

9. Intensity Example: A siren emits a sound of power 2W at 100 m from you. How much power reaches your ear (eardrum area = 0.7 cm 2 ) Intensity at distance r from source: Average power (over time) in wave Area of the surface where this power is distributed Power absorbed by eardrum: r

10. Interference, superposition Q: What happens when two waves “collide?” A: They ADD together! We say the waves are superposed. Constructive Destructive

11. Why superposition works The wave equation is linear: If f 1 and f 2 are solution, then Bf 1 + Cf 2 is also a solution! These points are now displaced by both waves (It has no terms where variables are squared.)

12. Superposition of two identical harmonic waves out of phase Two identical waves out of phase: intermediate constructive destructive Wave 2 is little ahead or behind wave 1

13. Superposition of two identical harmonic waves out of phase: the math It’s all about the phase difference

14. Interference for 2D and 3D waves Let S 1, S 2 be two sources that emit spherical sound waves in phase. S 1 S 2 P d1d1 d2d2 At point P: Destructive interference This is what matters… At some points the resulting wave can be very faint!

15. These points are always nodes!

16. 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) 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 (more mathematical): 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: The pulse is inverted (simply because of Newton’s 3 rd law!)

17. 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

18. Reflection on fixed end – inversion, or change of phase of π

19. Reflection on free end – no inversion

20. Standing waves A wave traveling along the +x direction is reflected at a fixed point. What is the result of the its superposition with the reflected wave? Standing wave

21. Nodes No motion for these points (nodes)

22. Antinodes These points oscillate with the maximum possible amplitude (antinodes)

23. Standing waves and boundary conditions We obtained 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.

24. 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 Allowed standing waves (normal modes) between two fixed ends Mode n = n-th harmonic DEMO: Normal modes on string Ruben’s tube