Welcome to Physics 7C!Welcome to Physics 7C!

Lecture 3 -- Winter Quarter -- 2005

Professor Robin Erbacher

343 Phy/Geo

erbacher@physics.ucdavis.edu

AnnouncementsAnnouncements

• Course policy and regrade forms on the web: http://physics7.ucdavis.edu

• 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 1st.

• Block 12 continues: DLMs 5, 6, and 7 this week.

• Turn off cell phones and pagers during lecture.

The Wave RepresentationThe Wave RepresentationBecause 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.

€

€

Δy(x,t) = Asin[Φ(x,t)]

Think of the sin argument as one big phase (or angle) €

y(x,t) - y0 = A sin(2π

Tt ±

2π

λx +φ) + BThe 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 Wave InterferenceWhat 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?

Δy(wave1+wave2) = A1+A2

In Phase: 1 - 2 = n2 (n = integer)

Or (as in DL): Δ ni (ni = integer)

(constructive interference)

Out of Phase: 1 - 2 = [(2n-1)/2]2 (n=integer)

Or (as in DL): Δ nh (nh = half-integer)

(destructive interference)

Superposition of WavesSuperposition of Waves

Adding 1D Waves Together:

€

Δytotal(x,t) = Δy1(x,t) +Δy2(x,t) = A1 sinΦ1 + A2 sinΦ2

€

Δytotal(x,t) = A1 sin(2πt

T1

± 2πx

λ1

+ϕ1) + A2 sin(2πt

T2

± 2πx

λ 2

+ϕ 2)

Using the Full Expressions:

What determines the total excursion of the medium at arbitrary time and position?

Phase angles and amplitudes!

Equal Amplitude WavesEqual Amplitude WavesIf A1=A2=A, then we can factor out A and use our trig identity:

€

Δy(x, t) = A(sinΦ1 + sinΦ2) = 2AsinΦ1 + Φ2( )

2

⎡

⎣ ⎢

⎤

⎦ ⎥cos

Φ1 − Φ2( )2

⎡

⎣ ⎢

⎤

⎦ ⎥

Wave part (avg) Degree ofconstructiveinterference

Waves of Same Frequency:Period and wavelength the same, so total phase difference is constant in time. Constructive interference for Δ=2n.

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.

+

− A

+ A

0 t [ ]s

yy

total

= y

1

Interference: Different Frequencies

Interference: Different Frequencies

€

Δ(t) = (2πt

T1

−2πx1

λ1

+ϕ1) − (2πt

T2

−2πx2

λ 2

+ϕ 2)

€

Δ(x,t) = 2πt(1

T1

−1

T2

) − 2π (x1

λ1

−x2

λ 2

) + (ϕ1 −ϕ 2)

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, x1 and x2:

Frequency difference

Path-length,wavelength difference

Phase difference

Time-independent constantFrequency difference

What We Hear…What We Hear…So we have sound waves at different frequencies, which means the pressure displacements add as before:

€

ΔP(x, t) = 2AsinΦ1 + Φ2( )

2

⎡

⎣ ⎢

⎤

⎦ ⎥cos

Φ1 − Φ2( )2

⎡

⎣ ⎢

⎤

⎦ ⎥

• Frequency we hear is tonal average of waves.

• Amplitude (instensity) of pressure fluctuations goes from loud, to soft, to loud again: difference between fb=|f1-f2|

At a fixed location, it’s a function of time only:

€

ΔP(x, t) = 2Asinf1 + f2( )t

2

⎡

⎣ ⎢

⎤

⎦ ⎥cos

f1 − f2( )t

2

⎡

⎣ ⎢

⎤

⎦ ⎥

Pitch versus BeatsPitch versus BeatsWhen you hear sound waves at different frequencies, you experience beats as they interfere.

Carrier frequency: responsible for pitch, or overall frequency. Fcarrier = (f1+f2)/2

Beat frequency: Interference modulates the amplitude Fbeat = |f1-f2|

Interference: ReflectionsInterference: 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

Reflections 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 WavesStanding 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.

€

y(x, t) = 2Asin 2πx

λ

⎡ ⎣ ⎢

⎤ ⎦ ⎥cos 2π

t

T

⎡ ⎣ ⎢

⎤ ⎦ ⎥

Time dependence gone from sine, spatial dependence dropped from cosine. Amplitude will always be zero for certain points in space (x = n/2) Nodes!!

Nodes

Where two oppositely

traveling waves

always destructively

intefere.

Antinodes

Where two oppositely

traveling waves

always constructively

intefere.

+

=

ResonancesResonances

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 HarmonicsFundamental and Harmonics

/2Node-node fundamental:

Node-antinode fundamental:€

f1 =vwave

2L

€

f1 =vwave

4L

Harmonics: multiples of the fundamental frequency

€

fn = nf1 = nvwave

2L

⎡ ⎣ ⎢

⎤ ⎦ ⎥

Node-nodeharmonics:

€

fn = nf1 = nodd

vwave

2L

⎡ ⎣ ⎢

⎤ ⎦ ⎥

Node-antinodeharmonics: