Wave Motion - University of Louisville

1Prof. Sergio B. MendesSummer 2018

Chapter 14 of Essential University Physics, Richard Wolfson, 3rd Edition

Wave Motion


Waves: propagation of energy, not particles


Longitudinal Waves:disturbance is along the direction of

wave propagation


Transverse Waves:disturbance is perpendicular to the

direction of wave propagation


Waves with Longitudinal Transverse Motions


Amplitude of a Wave

height

pressure

longitudinal displacement

transverse displacement


a pulse

a wave train

a continuous wave

Waveforms


PhET

https://phet.colorado.edu/sims/html/wave-on-a-string/latest/wave-on-a-string_en.html


Wavelength in a continuous wave


Wave Speed

𝑣𝑣 =𝜆𝜆𝑇𝑇


PhET



Two Snapshots of a Wave Pulse

𝑡𝑡 = 0

𝑦𝑦 = 𝑓𝑓 𝑥𝑥

𝑡𝑡 ≥ 0

𝑦𝑦 = 𝑓𝑓 𝑥𝑥 − 𝑣𝑣 𝑡𝑡


PhET



𝜓𝜓(𝑥𝑥, 𝑡𝑡) = 𝑓𝑓 𝑥𝑥 − 𝑣𝑣 𝑡𝑡

Fingerprint of a Wave:


A Harmonic Wave

𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 0 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 2 𝜋𝜋𝑥𝑥𝜆𝜆

𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 2 𝜋𝜋𝑥𝑥 − 𝑣𝑣 𝑡𝑡𝜆𝜆

𝑡𝑡 = 0

𝑡𝑡 ≥ 0


PhET



A Couple of Definitions

𝑘𝑘 ≡2 𝜋𝜋𝜆𝜆

Wave Number

𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 2 𝜋𝜋𝑥𝑥 − 𝑣𝑣 𝑡𝑡𝜆𝜆

𝜔𝜔 ≡2 𝜋𝜋𝑇𝑇

Angular Frequency

= 2 𝜋𝜋 𝑓𝑓

𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡

𝑣𝑣 =𝜔𝜔𝑘𝑘


Propagation towards Positive x-direction𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡

𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥 + 𝜔𝜔 𝑡𝑡

Propagation towards Negative x-direction


Got It ? 14.1


The Wave Equation𝜓𝜓(𝑥𝑥, 𝑡𝑡) = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡

𝜕𝜕𝜓𝜓𝜕𝜕𝑥𝑥

= − 𝑘𝑘 𝐴𝐴 𝑐𝑐𝑠𝑠𝑠𝑠 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡

𝜕𝜕2𝜓𝜓𝜕𝜕𝑥𝑥2

= − 𝑘𝑘2 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡

𝜕𝜕𝜓𝜓𝜕𝜕𝑡𝑡

= 𝜔𝜔 𝐴𝐴 𝑐𝑐𝑠𝑠𝑠𝑠 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡

𝜕𝜕2𝜓𝜓𝜕𝜕𝑡𝑡2

= − 𝜔𝜔2 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡

1𝑘𝑘2


=1𝜔𝜔2



=1𝑣𝑣2



Waves on a String


An example on how the properties of the carrying medium determines

the wave speed:


𝐹𝐹𝑛𝑛𝑛𝑛𝑛𝑛 ≅ 2 𝐹𝐹 𝜃𝜃

𝑚𝑚 ≅ 2 𝜃𝜃 𝑅𝑅 𝜇𝜇

𝑎𝑎 =𝑣𝑣2

𝑅𝑅2 𝐹𝐹 𝜃𝜃 ≅ 2 𝜃𝜃 𝑅𝑅 𝜇𝜇

𝑣𝑣2

𝑅𝑅

𝑣𝑣 =𝐹𝐹𝜇𝜇

𝐹𝐹 = 𝑚𝑚 𝑎𝑎

Wave Speed


Example 14.2

𝑚𝑚 = 5.0 𝑘𝑘𝑘𝑘

∆𝑥𝑥 = 43 𝑚𝑚

∆𝑡𝑡 = 1.4 𝑐𝑐

𝑣𝑣 =𝐹𝐹𝜇𝜇

𝐹𝐹 = ? ?


Got It ? 14.2


Wave Power

𝑃𝑃 = 𝑭𝑭.𝒗𝒗

= −𝐹𝐹 𝑣𝑣 𝑐𝑐𝑠𝑠𝑠𝑠 𝜃𝜃

≅ −𝐹𝐹 𝑣𝑣 𝑡𝑡𝑎𝑎𝑠𝑠 𝜃𝜃

𝑦𝑦 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡

𝑣𝑣 =𝜕𝜕𝑦𝑦𝜕𝜕𝑡𝑡

𝑡𝑡𝑎𝑎𝑠𝑠 𝜃𝜃 =𝜕𝜕𝑦𝑦𝜕𝜕𝑥𝑥

= 𝜔𝜔 𝐴𝐴 𝑐𝑐𝑠𝑠𝑠𝑠 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡

= −𝑘𝑘 𝐴𝐴 𝑐𝑐𝑠𝑠𝑠𝑠 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡

= 𝐹𝐹 𝜔𝜔 𝑘𝑘 𝐴𝐴2 𝑐𝑐𝑠𝑠𝑠𝑠2 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡

= 𝜇𝜇 𝑣𝑣 𝜔𝜔2 𝐴𝐴2 𝑐𝑐𝑠𝑠𝑠𝑠2 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡

𝑃𝑃 = 𝜇𝜇 𝑣𝑣 𝜔𝜔2 𝐴𝐴2 𝑐𝑐𝑠𝑠𝑠𝑠2 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡�𝑃𝑃 =12𝜇𝜇 𝑣𝑣 𝜔𝜔2 𝐴𝐴2


Wave Intensity

𝐼𝐼 ≡𝑃𝑃𝑐𝑐𝑃𝑃𝑃𝑃𝑃𝑃𝐴𝐴𝑃𝑃𝑃𝑃𝑎𝑎

𝐼𝐼 =𝑃𝑃𝑐𝑐𝑃𝑃𝑃𝑃𝑃𝑃4 𝜋𝜋 𝑃𝑃2


Example of Wave Intensities


Example 14.3𝑃𝑃1 = 9.2 𝑊𝑊

𝑥𝑥1 = 1.9 𝑚𝑚 𝑥𝑥2 = ? ?

𝑃𝑃2 = 4.9 𝑊𝑊

𝐼𝐼 =𝑃𝑃𝑐𝑐𝑃𝑃𝑃𝑃𝑃𝑃4 𝜋𝜋 𝑃𝑃2

𝐼𝐼1 =𝑃𝑃1

4 𝜋𝜋 𝑥𝑥12

𝐼𝐼2 =𝑃𝑃2

4 𝜋𝜋 𝑥𝑥22𝐼𝐼1 = 𝐼𝐼2

𝑃𝑃14 𝜋𝜋 𝑥𝑥12

=𝑃𝑃2

4 𝜋𝜋 𝑥𝑥22𝑥𝑥2 = 𝑥𝑥1

𝑃𝑃2𝑃𝑃1


Got It ? 14.3


Sound Waves

𝑣𝑣 =𝛾𝛾 𝑃𝑃𝜌𝜌

𝛾𝛾 is a constant characteristic of the gas


𝛽𝛽 𝑑𝑑𝑑𝑑 ≡ 10 log10𝐼𝐼𝐼𝐼𝑜𝑜

𝐼𝐼𝑜𝑜 ≡ 1 × 10−12 𝑊𝑊𝑚𝑚2

Audible Frequencies for Human Ears


Example 14.4𝛽𝛽1 = 75 𝑑𝑑𝑑𝑑

𝑃𝑃2𝑃𝑃1

= ? ?𝛽𝛽2 = 60 𝑑𝑑𝑑𝑑

𝑃𝑃2𝑃𝑃1

=𝐼𝐼2𝐼𝐼1

𝐼𝐼2𝐼𝐼1

=𝐼𝐼𝑜𝑜 10

𝛽𝛽210

𝐼𝐼𝑜𝑜 10𝛽𝛽110

= 10𝛽𝛽2−𝛽𝛽110

𝐼𝐼 = 𝐼𝐼𝑜𝑜 10𝛽𝛽10


Interferenceor what happened when two waves are present in the same region of space at a particular

time ?

Just add them up !!

• When wave crests coincide with crests, the interference is constructive.

• When crests coincide with troughs, the interference is destructive.

© 2016 Pearson Education, Inc.

Co-Propagating Waves

Constructive

Destructive


An application of destructive interference: getting waves

to cancel each other:


Adding Multiple Harmonic Waves: Fourier Analysis


Time and Frequency Descriptions


Dispersion: when the speed 𝑣𝑣 𝜔𝜔depends on the frequency

No dispersion

With dispersion

𝑣𝑣 𝜔𝜔 = 𝑐𝑐𝑐𝑐𝑠𝑠𝑐𝑐𝑡𝑡𝑎𝑎𝑠𝑠𝑡𝑡

𝑣𝑣 𝜔𝜔


Beats:

𝑦𝑦1 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝜔𝜔1 𝑡𝑡

𝑦𝑦2 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝜔𝜔2 𝑡𝑡

𝑦𝑦1 𝑡𝑡 + 𝑦𝑦2 𝑡𝑡 = 2 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐12𝜔𝜔1 − 𝜔𝜔2 𝑡𝑡 𝑐𝑐𝑐𝑐𝑐𝑐

12𝜔𝜔1 + 𝜔𝜔2 𝑡𝑡

two co-propagating waves of slightly different frequencies

𝑡𝑡



Interference in 2D

𝑦𝑦1 𝑃𝑃1 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑃𝑃1

𝑦𝑦2 𝑃𝑃2 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑃𝑃2

𝑘𝑘 𝑃𝑃1 − 𝑃𝑃2 = 𝜋𝜋 2 𝑚𝑚

𝑘𝑘 𝑃𝑃1 − 𝑃𝑃2 = 𝜋𝜋 2 𝑚𝑚 + 1

Constructive Interference:

Destructive Interference:


Total Reflection at an interface

PhET



Refraction at an interfaceand partial reflection and transmission


Partial Reflection and Transmission


Standing Waves: interference between

two counter-propagating waves of the same frequency

𝑦𝑦1 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡

𝑦𝑦2 𝑥𝑥, 𝑡𝑡 = −𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥 + 𝜔𝜔 𝑡𝑡

𝑦𝑦1 𝑡𝑡 + 𝑦𝑦2 𝑡𝑡 = 2 𝐴𝐴 𝑐𝑐𝑠𝑠𝑠𝑠 𝜔𝜔 𝑡𝑡 𝑐𝑐𝑠𝑠𝑠𝑠 𝑘𝑘 𝑥𝑥


𝑘𝑘 𝐿𝐿 = 𝑚𝑚 𝜋𝜋

𝑦𝑦1 𝑡𝑡 + 𝑦𝑦2 𝑡𝑡 = 2 𝐴𝐴𝑐𝑐𝑠𝑠 𝑠𝑠 𝜔𝜔 𝑡𝑡 𝑐𝑐𝑠𝑠𝑠𝑠 𝑘𝑘 𝑥𝑥

𝐿𝐿 = 𝑚𝑚𝜆𝜆2

𝑚𝑚 = 1, 2, 3, 4, …

2 𝐿𝐿1 ,

2 𝐿𝐿2 ,

2 𝐿𝐿3 ,

2 𝐿𝐿4 ,𝜆𝜆 =

𝑓𝑓 = 1𝑣𝑣

2 𝐿𝐿 , 2𝑣𝑣



2 𝐿𝐿 ,

…

…

fundamental harmonics

𝑥𝑥 = 𝐿𝐿 𝑐𝑐𝑠𝑠 𝑠𝑠 𝑘𝑘 𝐿𝐿 = 0

𝑥𝑥 = 0 𝑐𝑐𝑠𝑠 𝑠𝑠 𝑘𝑘 0 = 0


PhET

https://phet.colorado.edu/en/simulation/legacy/normal-modes


𝑘𝑘 𝐿𝐿 = 𝑚𝑚𝜆𝜆2

𝑦𝑦1 𝑡𝑡 + 𝑦𝑦2 𝑡𝑡 = 2 𝐴𝐴 𝑐𝑐𝑐𝑐 𝑐𝑐 𝜔𝜔 𝑡𝑡 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥

𝑘𝑘 𝐿𝐿 = 2 𝑚𝑚 + 1𝜆𝜆4

= 2 𝑚𝑚𝜆𝜆4


Doppler Effect

𝜆𝜆′ = 𝜆𝜆 − 𝑢𝑢 𝑇𝑇

= 𝜆𝜆 − 𝑢𝑢𝜆𝜆𝑣𝑣

from a moving sourcewith respect to the wave carrier medium,

observer at rest w.r.t to the carrier medium

= 𝜆𝜆 1 −𝑢𝑢𝑣𝑣

𝑓𝑓′ =𝑓𝑓

1 − 𝑢𝑢𝑣𝑣

𝜆𝜆′ = 𝜆𝜆 1 −𝑢𝑢𝑣𝑣


Doppler Effectfrom a moving source

with respect to the carrier medium,observer at rest w.r.t to the carrier medium

𝜆𝜆𝐴𝐴,𝐵𝐵′ = 𝜆𝜆 1 ∓

𝑢𝑢𝑣𝑣

𝑓𝑓𝐴𝐴,𝐵𝐵′ =

𝑓𝑓

1 ∓ 𝑢𝑢𝑣𝑣

𝑢𝑢


Doppler Effect

𝑥𝑥𝑝𝑝𝑛𝑛𝑝𝑝𝑝𝑝 = −𝜆𝜆 + 𝑣𝑣 𝑡𝑡

from a moving observerwith respect to the wave carrier medium,source at rest w.r.t to carrier medium

𝑓𝑓′ = 𝑓𝑓 1 +𝑢𝑢𝑣𝑣

𝜆𝜆′ =𝜆𝜆

1 + 𝑢𝑢𝑣𝑣

𝐴𝐴𝑢𝑢

𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 = 0 − 𝑢𝑢 𝑡𝑡

−𝑢𝑢 𝑇𝑇′ = −𝜆𝜆 + 𝑣𝑣 𝑇𝑇𝑇

−𝑢𝑢 𝑇𝑇′ = −𝑣𝑣 𝑇𝑇 + 𝑣𝑣 𝑇𝑇𝑇

𝑇𝑇′ =𝑇𝑇

1 + 𝑢𝑢𝑣𝑣

𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑥𝑥𝑝𝑝𝑛𝑛𝑝𝑝𝑝𝑝


Velocity of a Wave

𝑣𝑣𝑤𝑤𝑝𝑝𝑤𝑤𝑛𝑛, 𝑜𝑜𝑜𝑜𝑜𝑜𝑛𝑛𝑜𝑜𝑤𝑤𝑛𝑛𝑜𝑜 = 𝑣𝑣𝑤𝑤𝑝𝑝𝑤𝑤𝑛𝑛, 𝑐𝑐𝑝𝑝𝑜𝑜𝑜𝑜𝑐𝑐𝑛𝑛𝑜𝑜

𝑣𝑣

𝑢𝑢

𝑣𝑣𝑤𝑤𝑝𝑝𝑤𝑤𝑛𝑛, 𝑜𝑜𝑜𝑜𝑜𝑜𝑛𝑛𝑜𝑜𝑤𝑤𝑛𝑛𝑜𝑜 = 𝑣𝑣 + 𝑢𝑢

𝑣𝑣𝑢𝑢

+ 𝑣𝑣𝑐𝑐𝑝𝑝𝑜𝑜𝑜𝑜𝑐𝑐𝑛𝑛𝑜𝑜, 𝑜𝑜𝑜𝑜𝑜𝑜𝑛𝑛𝑜𝑜𝑤𝑤𝑛𝑛𝑜𝑜

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