Post on 23-Feb-2022
1Prof. Sergio B. MendesSummer 2018
Chapter 14 of Essential University Physics, Richard Wolfson, 3rd Edition
Wave Motion
2Prof. Sergio B. MendesSummer 2018
Waves: propagation of energy, not particles
3Prof. Sergio B. MendesSummer 2018
Longitudinal Waves:disturbance is along the direction of
wave propagation
4Prof. Sergio B. MendesSummer 2018
Transverse Waves:disturbance is perpendicular to the
direction of wave propagation
5Prof. Sergio B. MendesSummer 2018
Waves with Longitudinal Transverse Motions
6Prof. Sergio B. MendesSummer 2018
Amplitude of a Wave
height
pressure
longitudinal displacement
transverse displacement
7Prof. Sergio B. MendesSummer 2018
a pulse
a wave train
a continuous wave
Waveforms
8Prof. Sergio B. MendesSummer 2018
PhET
9Prof. Sergio B. MendesSummer 2018
Wavelength in a continuous wave
10Prof. Sergio B. MendesSummer 2018
Wave Speed
π£π£ =ππππ
11Prof. Sergio B. MendesSummer 2018
PhET
12Prof. Sergio B. MendesSummer 2018
Two Snapshots of a Wave Pulse
π‘π‘ = 0
π¦π¦ = ππ π₯π₯
π‘π‘ β₯ 0
π¦π¦ = ππ π₯π₯ β π£π£ π‘π‘
13Prof. Sergio B. MendesSummer 2018
PhET
14Prof. Sergio B. MendesSummer 2018
ππ(π₯π₯, π‘π‘) = ππ π₯π₯ β π£π£ π‘π‘
Fingerprint of a Wave:
15Prof. Sergio B. MendesSummer 2018
A Harmonic Wave
ππ π₯π₯, π‘π‘ = 0 = π΄π΄ ππππππ 2 πππ₯π₯ππ
ππ π₯π₯, π‘π‘ = π΄π΄ ππππππ 2 πππ₯π₯ β π£π£ π‘π‘ππ
π‘π‘ = 0
π‘π‘ β₯ 0
16Prof. Sergio B. MendesSummer 2018
PhET
17Prof. Sergio B. MendesSummer 2018
A Couple of Definitions
ππ β‘2 ππππ
Wave Number
ππ π₯π₯, π‘π‘ = π΄π΄ ππππππ 2 πππ₯π₯ β π£π£ π‘π‘ππ
ππ β‘2 ππππ
Angular Frequency
= 2 ππ ππ
ππ π₯π₯, π‘π‘ = π΄π΄ ππππππ ππ π₯π₯ β ππ π‘π‘
π£π£ =ππππ
18Prof. Sergio B. MendesSummer 2018
Propagation towards Positive x-directionππ π₯π₯, π‘π‘ = π΄π΄ ππππππ ππ π₯π₯ β ππ π‘π‘
ππ π₯π₯, π‘π‘ = π΄π΄ ππππππ ππ π₯π₯ + ππ π‘π‘
Propagation towards Negative x-direction
19Prof. Sergio B. MendesSummer 2018
Got It ? 14.1
20Prof. Sergio B. MendesSummer 2018
The Wave Equationππ(π₯π₯, π‘π‘) = π΄π΄ ππππππ ππ π₯π₯ β ππ π‘π‘
πππππππ₯π₯
= β ππ π΄π΄ πππ π π π ππ π₯π₯ β ππ π‘π‘
ππ2πππππ₯π₯2
= β ππ2 π΄π΄ ππππππ ππ π₯π₯ β ππ π‘π‘
πππππππ‘π‘
= ππ π΄π΄ πππ π π π ππ π₯π₯ β ππ π‘π‘
ππ2πππππ‘π‘2
= β ππ2 π΄π΄ ππππππ ππ π₯π₯ β ππ π‘π‘
1ππ2
ππ2πππππ₯π₯2
=1ππ2
ππ2πππππ‘π‘2
ππ2πππππ₯π₯2
=1π£π£2
ππ2πππππ‘π‘2
21Prof. Sergio B. MendesSummer 2018
Waves on a String
22Prof. Sergio B. MendesSummer 2018
An example on how the properties of the carrying medium determines
the wave speed:
23Prof. Sergio B. MendesSummer 2018
πΉπΉππππππ β 2 πΉπΉ ππ
ππ β 2 ππ π π ππ
ππ =π£π£2
π π 2 πΉπΉ ππ β 2 ππ π π ππ
π£π£2
π π
π£π£ =πΉπΉππ
πΉπΉ = ππ ππ
Wave Speed
24Prof. Sergio B. MendesSummer 2018
Example 14.2
ππ = 5.0 ππππ
βπ₯π₯ = 43 ππ
βπ‘π‘ = 1.4 ππ
π£π£ =πΉπΉππ
πΉπΉ = ? ?
25Prof. Sergio B. MendesSummer 2018
Got It ? 14.2
26Prof. Sergio B. MendesSummer 2018
Wave Power
ππ = ππ.ππ
= βπΉπΉ π£π£ πππ π π π ππ
β βπΉπΉ π£π£ π‘π‘πππ π ππ
π¦π¦ π₯π₯, π‘π‘ = π΄π΄ ππππππ ππ π₯π₯ β ππ π‘π‘
π£π£ =πππ¦π¦πππ‘π‘
π‘π‘πππ π ππ =πππ¦π¦πππ₯π₯
= ππ π΄π΄ πππ π π π ππ π₯π₯ β ππ π‘π‘
= βππ π΄π΄ πππ π π π ππ π₯π₯ β ππ π‘π‘
= πΉπΉ ππ ππ π΄π΄2 πππ π π π 2 ππ π₯π₯ β ππ π‘π‘
= ππ π£π£ ππ2 π΄π΄2 πππ π π π 2 ππ π₯π₯ β ππ π‘π‘
ππ = ππ π£π£ ππ2 π΄π΄2 πππ π π π 2 ππ π₯π₯ β ππ π‘π‘οΏ½ππ =12ππ π£π£ ππ2 π΄π΄2
27Prof. Sergio B. MendesSummer 2018
Wave Intensity
πΌπΌ β‘πππππππππππ΄π΄ππππππ
πΌπΌ =ππππππππππ4 ππ ππ2
28Prof. Sergio B. MendesSummer 2018
Example of Wave Intensities
29Prof. Sergio B. MendesSummer 2018
Example 14.3ππ1 = 9.2 ππ
π₯π₯1 = 1.9 ππ π₯π₯2 = ? ?
ππ2 = 4.9 ππ
πΌπΌ =ππππππππππ4 ππ ππ2
πΌπΌ1 =ππ1
4 ππ π₯π₯12
πΌπΌ2 =ππ2
4 ππ π₯π₯22πΌπΌ1 = πΌπΌ2
ππ14 ππ π₯π₯12
=ππ2
4 ππ π₯π₯22π₯π₯2 = π₯π₯1
ππ2ππ1
30Prof. Sergio B. MendesSummer 2018
Got It ? 14.3
31Prof. Sergio B. MendesSummer 2018
Sound Waves
π£π£ =πΎπΎ ππππ
πΎπΎ is a constant characteristic of the gas
32Prof. Sergio B. MendesSummer 2018
π½π½ ππππ β‘ 10 log10πΌπΌπΌπΌππ
πΌπΌππ β‘ 1 Γ 10β12 ππππ2
Audible Frequencies for Human Ears
33Prof. Sergio B. MendesSummer 2018
Example 14.4π½π½1 = 75 ππππ
ππ2ππ1
= ? ?π½π½2 = 60 ππππ
ππ2ππ1
=πΌπΌ2πΌπΌ1
πΌπΌ2πΌπΌ1
=πΌπΌππ 10
π½π½210
πΌπΌππ 10π½π½110
= 10π½π½2βπ½π½110
πΌπΌ = πΌπΌππ 10π½π½10
34Prof. Sergio B. MendesSummer 2018
Interferenceor what happened when two waves are present in the same region of space at a particular
time ?
Just add them up !!
β’ When wave crests coincide with crests, the interference is constructive.
β’ When crests coincide with troughs, the interference is destructive.
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Co-Propagating Waves
Constructive
Destructive
36Prof. Sergio B. MendesSummer 2018
An application of destructive interference: getting waves
to cancel each other:
Β© 2016 Pearson Education, Inc.
Adding Multiple Harmonic Waves: Fourier Analysis
Β© 2016 Pearson Education, Inc.
Time and Frequency Descriptions
39Prof. Sergio B. MendesSummer 2018
Dispersion: when the speed π£π£ ππdepends on the frequency
No dispersion
With dispersion
π£π£ ππ = πππππ π πππ‘π‘πππ π π‘π‘
π£π£ ππ
Β© 2016 Pearson Education, Inc.
Beats:
π¦π¦1 π‘π‘ = π΄π΄ ππππππ ππ1 π‘π‘
π¦π¦2 π‘π‘ = π΄π΄ ππππππ ππ2 π‘π‘
π¦π¦1 π‘π‘ + π¦π¦2 π‘π‘ = 2 π΄π΄ ππππππ12ππ1 β ππ2 π‘π‘ ππππππ
12ππ1 + ππ2 π‘π‘
two co-propagating waves of slightly different frequencies
π‘π‘
41Prof. Sergio B. MendesSummer 2018
42Prof. Sergio B. MendesSummer 2018
Interference in 2D
π¦π¦1 ππ1 = π΄π΄ ππππππ ππ ππ1
π¦π¦2 ππ2 = π΄π΄ ππππππ ππ ππ2
ππ ππ1 β ππ2 = ππ 2 ππ
ππ ππ1 β ππ2 = ππ 2 ππ + 1
Constructive Interference:
Destructive Interference:
43Prof. Sergio B. MendesSummer 2018
Total Reflection at an interface
PhET
44Prof. Sergio B. MendesSummer 2018
Refraction at an interfaceand partial reflection and transmission
45Prof. Sergio B. MendesSummer 2018
Partial Reflection and Transmission
Β© 2016 Pearson Education, Inc.
Standing Waves: interference between
two counter-propagating waves of the same frequency
π¦π¦1 π₯π₯, π‘π‘ = π΄π΄ ππππππ ππ π₯π₯ β ππ π‘π‘
π¦π¦2 π₯π₯, π‘π‘ = βπ΄π΄ ππππππ ππ π₯π₯ + ππ π‘π‘
π¦π¦1 π‘π‘ + π¦π¦2 π‘π‘ = 2 π΄π΄ πππ π π π ππ π‘π‘ πππ π π π ππ π₯π₯
47Prof. Sergio B. MendesSummer 2018
ππ πΏπΏ = ππ ππ
π¦π¦1 π‘π‘ + π¦π¦2 π‘π‘ = 2 π΄π΄πππ π π π ππ π‘π‘ πππ π π π ππ π₯π₯
πΏπΏ = ππππ2
ππ = 1, 2, 3, 4, β¦
2 πΏπΏ1 ,
2 πΏπΏ2 ,
2 πΏπΏ3 ,
2 πΏπΏ4 ,ππ =
ππ = 1π£π£
2 πΏπΏ , 2π£π£
2 πΏπΏ , 3π£π£
2 πΏπΏ , 4π£π£
2 πΏπΏ ,
β¦
β¦
fundamental harmonics
π₯π₯ = πΏπΏ πππ π π π ππ πΏπΏ = 0
π₯π₯ = 0 πππ π π π ππ 0 = 0
48Prof. Sergio B. MendesSummer 2018
PhET
49Prof. Sergio B. MendesSummer 2018
ππ πΏπΏ = ππππ2
π¦π¦1 π‘π‘ + π¦π¦2 π‘π‘ = 2 π΄π΄ ππππ ππ ππ π‘π‘ ππππππ ππ π₯π₯
ππ πΏπΏ = 2 ππ + 1ππ4
= 2 ππππ4
50Prof. Sergio B. MendesSummer 2018
Doppler Effect
ππβ² = ππ β π’π’ ππ
= ππ β π’π’πππ£π£
from a moving sourcewith respect to the wave carrier medium,
observer at rest w.r.t to the carrier medium
= ππ 1 βπ’π’π£π£
ππβ² =ππ
1 β π’π’π£π£
ππβ² = ππ 1 βπ’π’π£π£
51Prof. Sergio B. MendesSummer 2018
Doppler Effectfrom a moving source
with respect to the carrier medium,observer at rest w.r.t to the carrier medium
πππ΄π΄,π΅π΅β² = ππ 1 β
π’π’π£π£
πππ΄π΄,π΅π΅β² =
ππ
1 β π’π’π£π£
π’π’
52Prof. Sergio B. MendesSummer 2018
Doppler Effect
π₯π₯ππππππππ = βππ + π£π£ π‘π‘
from a moving observerwith respect to the wave carrier medium,source at rest w.r.t to carrier medium
ππβ² = ππ 1 +π’π’π£π£
ππβ² =ππ
1 + π’π’π£π£
π΄π΄π’π’
π₯π₯ππππππ = 0 β π’π’ π‘π‘
βπ’π’ ππβ² = βππ + π£π£ πππ
βπ’π’ ππβ² = βπ£π£ ππ + π£π£ πππ
ππβ² =ππ
1 + π’π’π£π£
π₯π₯ππππππ = π₯π₯ππππππππ
53Prof. Sergio B. MendesSummer 2018
Velocity of a Wave
π£π£π€π€πππ€π€ππ, πππππππππππ€π€ππππ = π£π£π€π€πππ€π€ππ, ππππππππππππππ
π£π£
π’π’
π£π£π€π€πππ€π€ππ, πππππππππππ€π€ππππ = π£π£ + π’π’
π£π£π’π’
+ π£π£ππππππππππππππ, πππππππππππ€π€ππππ