Class 38 - Waves I Chapter 16 - Wednesday November 24th

1. Class 38 - Waves I Chapter 16 - Wednesday November 24th • Exam 3: Friday December 3rd, 8:20pm to 10:20pm • You must go to the following locations based on the 1st letter of your last name: • Review sessions: Tues. Nov. 30 and Thurs. Dec. 2, 6:15 to 8:10pm • Traveling waves • Waves on a string • The wave equation • Sample problems Reading: pages 413 to 423 (chapter 16) in HRW Read and understand the sample problems Assigned problems from chapter 16 (due Dec. 2nd): 6, 20, 22, 24, 30, 34, 42, 44, 66, 70, 78, 82

2. Waves I - types of waves • These are the most familiar. We encounter them every day. The common feature of all mechanical waves is that they are governed entirely by Newton's laws, and can exist only within a material medium. • All electromagnetic waves travel through vacuum at the same speed c, the speed of light, where c = 299 792 458 m/s. Electromagnetic waves are governed by Maxwell's equations (PHY 2049). • Although one thinks of matter as being made up from particles, it is in fact made up from fundamental matter waves that travel in vacuum. Matter waves are governed by the laws of quantum mechanics, or the Schrödinger and Dirac equations. • Mechanical waves: water waves, sound waves, seismic waves. • Electromagnetic waves: radio waves, visible light, ultraviolet light, x-rays, gamma rays. • Matter waves: electrons, protons, neutrons, anti-protons, etc..

3. Wave interference Matter-wave interference

4. Waves I - types of waves Longitudinal waves Transverse waves (2 polarizations)

5. Waves I - wavelength and frequency Wavelength (consider wave at t = 0): Transverse sinusoidal wave You can always add 2p to the phase of a wave without changing its displacement, i.e. phase shift

6. Waves I - wavelength and frequency Transverse sinusoidal wave The SI unit is radian per meter, or meter-1. This k is NOT the same as spring constant. phase shift Wavelength (consider wave at t = 0): We call k the angular wavenumber.

7. Waves I - wavelength and frequency Transverse sinusoidal wave Again, we can add 2p to the phase, phase shift Period and frequency (consider wave at x = 0):

8. Waves I - wavelength and frequency Transverse sinusoidal wave We call w the angular frequency. The SI unit is radian per second. The frequency f is defined as 1/T. phase shift Period and frequency (consider wave at x = 0):

9. The speed of a traveling wave Or Transverse sinusoidal wave • A fixed point on a wave has a constant value of the phase, i.e.

10. The speed of a traveling wave Transverse sinusoidal wave • So, general sinusoidal solution is: • In fact, any function of the form is a solution. • For a wave traveling in the opposite direction, we simply set time to run backwards, i.e. replace t with -t.

11. Traveling waves on a stretched string mis the string's linear density, or force per unit length. • Tension t provides the restoring force (kg.m.s-2) in the string. Without tension, the wave could not propagate. • The mass per unit length m (kg.m-1) determines the response of the string to the restoring force (tension), through Newtorn's 2nd law. • Look for combinations of t and m that give dimensions of speed (m.s-1). Dimensional analysis

12. Traveling waves on a string • The tension in the string is t. • The mass of the element dm is mdl, where m is the mass per unit length of the string. y x

13. Traveling waves on a string y x • In the small q limit...

14. The wave equation • General solution: