Welcome to Physics 7C! Lecture 3 -- Winter Quarter Professor Robin Erbacher 343 Phy/Geo

Announcements Course policy and regrade forms on the web: Quiz today! ~20 minutes long on Block 11. I will not be here next week! Prof. Daniel Cebra will lecture in my place on February 1 st. Block 12 continues: DLMs 5, 6, and 7 this week. Turn off cell phones and pagers during lecture.

The Wave Representation Because there is both a time-dependence and a translation of the wave in space, we need to represent the wave using both t and x. Think of the sin argument as one big phase (or angle)  The most general solution is of the form: Note: I swapped x and t term. Block notes differ from DL expression. Both ok. Use DL version So, the total displacement of a wave is determined by A and 

Wave Interference What happens when there is more than one wave? When two or more waves meet, they interfere with each other. Combining waves by adding them is known as superposition. Consider two waves on a string. What’s the maximum displacement of the string from equilibrium?  y(wave 1 +wave 2 ) = A 1 +A 2 In Phase:  1 -  2 = n  2  (n = integer) Or (as in DL):  n i (n i = integer) (constructive interference) Out of Phase:  1 -  2 = [ (2n-1)/2]  2  (n=integer) Or (as in DL):  n h (n h = half-integer) (destructive interference)

Superposition of Waves Adding 1D Waves Together: Using the Full Expressions: What determines the total excursion of the medium at arbitrary time and position? Phase angles and amplitudes!

Equal Amplitude Waves If A 1 =A 2 =A, then we can factor out A and use our trig identity: Wave part (avg)Degree of constructive interference Waves of Same Frequency: Period and wavelength the same, so total phase difference is constant in time. Constructive interference for  =2  n. Waves of Different Frequency: Wavelength not the same, so graph of superposed waves shows variations in amplitude as waves go in and out of phase.

Interference: Different Frequencies If we break this into pieces: We observe sound from a fixed position x, so path lengths to our ears for each wave are constant, x 1 and x 2 : Frequency difference Path-length, wavelength difference Phase difference Time-independent constant Frequency difference

What We Hear… So we have sound waves at different frequencies, which means the pressure displacements add as before: Frequency we hear is tonal average of waves. Amplitude (instensity) of pressure fluctuations goes from loud, to soft, to loud again: difference between f b =|f 1 -f 2 | At a fixed location, it’s a function of time only:

Pitch versus Beats When you hear sound waves at different frequencies, you experience beats as they interfere. Carrier frequency: responsible for pitch, or overall frequency. F carrier = (f 1 +f 2 )/2 Beat frequency: Interference modulates the amplitude F beat = |f 1 -f 2 |

Interference: Reflections ReflectionsReflections of transverse waves: Slow medium to high speed, or off hard boundary, wave shift =  Fast medium to slow, or off soft boundary, wave shift = 0 ReflectionsReflections of longitudinal waves are the opposite! (like sound). Sound waves travel faster through a dense medium (water v. air). Light waves travel slower through a dense medium.

Standing Waves A wave on a rope tied at two ends behaves like two waves interfering: add the original wave and the reflected wave. A standing wave is not a real wave, but is the superposition of the wave on itself. Time dependence gone from sine, spatial dependence dropped from cosine. Amplitude will always be zero for certain points in space (x = n /2)  Nodes!!

Resonances For a standing wave: Two waves w/ same wavelength Waves have same amplitude Traveling in opposite directions Nodes: where opposite waves destructively interfere. Antinodes: where the two waves constructively interfere. /2 A standing wave that resonates has a node or antinode at either end, determined by the medium. An open-ended tube has antinode at the end, for example. Only certain resonant wavelengths are allowed Only certain resonant frequencies are allowed.

Fundamental and Harmonics /2 Node-node fundamental: Node-antinode fundamental: Harmonics: multiples of the fundamental frequency Node-node harmonics: Node-antinode harmonics: