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 session: Thursday Dec. 2 (Woodard), 6:15 to 8:10pm in NPB 1001 (HERE!) Final Exam (cumulative): Tuesday December 14th, 12:30pm to 2:30pm. Room assignments: A to K in NPB1001 (in here); L to Z in Norman Hall 137. Two more review sessions: Dec. 7 (Hill) and Dec. 9 (Woodard), 6:15 to 8:10pm in NPB1001 (HERE!)

Chapter 16 - Wednesday December 1st Class 40 - Waves I Chapter 16 - Wednesday December 1st QUICK review Wave interference Standing waves and resonance Sample exam problems HiTT (if time permits, otherwise Friday) Reading: pages 413 to 437 (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

Review - traveling waves on a string Velocity 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. Energy transfer rates

The principle of superposition for waves It often happens that waves travel simultaneously through the same region, e.g. Radio waves from many broadcasters Sound waves from many musical instruments Different colored light from many locations from your TV Nature is such that all of these waves can exist without altering each others' motion Their effects simply add And have solutions: This is a result of the principle of superposition, which applies to all harmonic waves, i.e. waves that obey the linear wave equation

The principle of superposition for waves If two waves travel simultaneously along the same stretched string, the resultant displacement y' of the string is simply given by the summation where y1 and y2 would have been the displacements had the waves traveled alone. Overlapping waves algebraically add to produce a resultant wave (or net wave). Overlapping waves do not in any way alter the travel of each other This is the principle of superposition. Link

Interference of waves Suppose two sinusoidal waves with the same frequency and amplitude travel in the same direction along a string, such that The waves will add. If they are in phase (i.e. f = 0), they combine to double the displacement of either wave acting alone. If they are out of phase (i.e. f = p), they combine to cancel everywhere, since sin(a) = -sin(a + p). This phenomenon is called interference.

Interference of waves

Interference of waves Phase shift Wave part Amplitude Mathematical proof: Then: But: Phase shift So: Wave part Amplitude

Interference of waves If two sinusoidal waves of the same amplitude and frequency travel in the same direction along a stretched string, they interfere to produce a resultant sinusoidal wave traveling in the same direction. If f = 0, the waves interfere constructively, cos½f = 1 and the wave amplitude is 2ym. If f = p, the waves interfere destructively, cos(p/2) = 0 and the wave amplitude is 0, i.e. no wave at all. All other cases are intermediate between an amplitude of 0 and 2ym. Note that the phase of the resultant wave also depends on the phase difference. Link

Standing waves If two sinusoidal waves of the same amplitude and wavelength travel in opposite directions along a stretched string, their interference with each other produces a standing wave. x dependence t dependence This is clearly not a traveling wave, because it does not have the form f(kx - wt). In fact, it is a stationary wave, with a sinusoidal varying amplitude 2ymcos(wt). Link

Reflections at a boundary Waves reflect from boundaries. This is the reason for echoes - you hear sound reflecting back to you. However, the nature of the reflection depends on the boundary condition. For the two examples on the left, the nature of the reflection depends on whether the end of the string is fixed or loose. Movies

Standing waves and resonance At ordinary frequencies, waves travel backwards and forwards along the string. Each new reflected wave has a new phase. The interference is basically a mess, and no significant oscillations build up.

Standing waves and resonance However, at certain special frequencies, the interference produces strong standing wave patterns. Such a standing wave is said to be produced at resonance. These certain frequencies are called resonant frequencies.

Standing waves and resonance Standing waves occur whenever the phase of the wave returning to the oscillating end of the string is precisely in phase with the forced oscillations. Thus, the trip along the string and back should be equal to an integral number of wavelengths, i.e. Each of the frequencies f1, f1, f1, etc, are called harmonics, or a harmonic series; n is the harmonic number. l determined by geometry

Standing waves and resonance Here is an example of a two-dimensional vibrating diaphragm. The dark powder shows the positions of the nodes in the vibration.