wave motion - uofl department of physics & astronomy 355 fall 18/notes/chapter … · wave...
TRANSCRIPT
1Prof. Sergio B. MendesFall 2018
Wave MotionChapter 2 of Optics
Eugene Hecht, 5th Edition
2Prof. Sergio B. MendesFall 2018
Waves: propagation of energy, not particles
3Prof. Sergio B. MendesFall 2018
Longitudinal Waves:disturbance is along the direction of
wave propagation
4Prof. Sergio B. MendesFall 2018
Transverse Waves:disturbance is perpendicular to the
direction of wave propagation
5Prof. Sergio B. MendesFall 2018
Waves with Longitudinal Transverse Motions
6Prof. Sergio B. MendesFall 2018
Amplitude of a Wave
height
pressure
longitudinal displacement
transverse displacement
7Prof. Sergio B. MendesFall 2018
a pulse
a wave train
a continuous harmonic wave
Different Waveforms
8Prof. Sergio B. MendesFall 2018
PhET
9Prof. Sergio B. MendesFall 2018
Two Snapshots of a Wave Pulse
𝑡𝑡 = 0
𝑦𝑦 𝑥𝑥, 𝑡𝑡 = 0 = 𝑓𝑓 𝑥𝑥
𝑡𝑡 ≥ 0
𝑦𝑦 𝑥𝑥, 𝑡𝑡 = 𝑓𝑓 𝑥𝑥 − 𝑣𝑣 𝑡𝑡
propagating with velocity v along the x-axis
10Prof. Sergio B. MendesFall 2018
Propagation towards Positive x-direction𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝑓𝑓 𝑥𝑥 − 𝑣𝑣 𝑡𝑡
𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝑓𝑓 𝑥𝑥 + 𝑣𝑣 𝑡𝑡
Propagation towards Negative x-direction
𝑣𝑣 > 0
𝑣𝑣 > 0
11Prof. Sergio B. MendesFall 2018
𝜓𝜓(𝑥𝑥, 𝑡𝑡) = 𝑓𝑓 𝑥𝑥 − 𝑣𝑣 𝑡𝑡
Fingerprint of the Wave Phenomena: 𝑥𝑥 − 𝑣𝑣 𝑡𝑡
𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴 𝑒𝑒−𝑥𝑥 − 𝑣𝑣 𝑡𝑡𝜎𝜎2
2
𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 0 = 𝐴𝐴 𝑒𝑒−𝑥𝑥2𝜎𝜎2
𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 0 = 𝐴𝐴 𝑒𝑒−𝑥𝑥𝜎𝜎 𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴 𝑒𝑒−
𝑥𝑥 − 𝑣𝑣 𝑡𝑡𝜎𝜎
𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 0 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥 𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥 − 𝑣𝑣 𝑡𝑡
12Prof. Sergio B. MendesFall 2018
𝜓𝜓(𝑥𝑥, 𝑡𝑡) = 𝑓𝑓 𝑥𝑥 − 𝑣𝑣 𝑡𝑡
Therefore:
𝑣𝑣 = −�𝜕𝜕𝜓𝜓𝜕𝜕𝑡𝑡�𝜕𝜕𝜓𝜓𝜕𝜕𝑥𝑥
𝜕𝜕2𝜓𝜓𝜕𝜕𝑥𝑥2
=1𝑣𝑣2𝜕𝜕2𝜓𝜓𝜕𝜕𝑡𝑡2
13Prof. Sergio B. MendesFall 2018
𝜕𝜕2𝜓𝜓𝜕𝜕𝑥𝑥2
=1𝑣𝑣2𝜕𝜕2𝜓𝜓𝜕𝜕𝑡𝑡2
Wave Equation
• 2nd order partial differential equation
• Linear equation in 𝝍𝝍 : 𝝍𝝍, 𝝏𝝏𝜶𝜶𝝍𝝍, 𝝏𝝏𝜶𝜶𝟐𝟐𝝍𝝍,
• Homogeneous equation: no term involving independent variables
If 𝝍𝝍𝟏𝟏 and 𝝍𝝍𝟐𝟐 are solutions then 𝝍𝝍𝟏𝟏 + 𝝍𝝍𝟐𝟐 is also a solution equation.
If 𝝍𝝍 is a solution then 𝒂𝒂 𝝍𝝍 is also a solution equation.
14Prof. Sergio B. MendesFall 2018
Principle of Superposition
If 𝝍𝝍𝟏𝟏 and 𝝍𝝍𝟐𝟐 are solutions to the wave equation then 𝒂𝒂 𝝍𝝍𝟏𝟏 + 𝒃𝒃 𝝍𝝍𝟐𝟐 is also a solution equation.
Wave superposition is the foundation of wave phenomena such as interference and diffraction
15Prof. Sergio B. MendesFall 2018
A Harmonic Wave Solution:
𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥 − 𝑣𝑣 𝑡𝑡 + 𝜀𝜀
𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝐵𝐵 𝑐𝑐𝑠𝑠𝑠𝑠 𝑘𝑘 𝑥𝑥 − 𝑣𝑣 𝑡𝑡 + 𝜀𝜀′
𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥 − 𝑣𝑣 𝑡𝑡 + 𝜀𝜀 + 𝐵𝐵 𝑐𝑐𝑠𝑠𝑠𝑠 𝑘𝑘 𝑥𝑥 − 𝑣𝑣 𝑡𝑡 + 𝜀𝜀′
or
or
16Prof. Sergio B. MendesFall 2018
Let’s consider:
𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥 − 𝑣𝑣 𝑡𝑡 + 𝜀𝜀
17Prof. Sergio B. MendesFall 2018
Wavelength ina Harmonic Wave
𝑡𝑡 = 𝑓𝑓𝑠𝑠𝑥𝑥𝑒𝑒𝑓𝑓
𝑘𝑘 ≡2 𝜋𝜋𝜆𝜆
𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥 − 𝑣𝑣 𝑡𝑡 + 𝜀𝜀
𝑥𝑥
18Prof. Sergio B. MendesFall 2018
Period ina Harmonic Wave
𝑥𝑥 = 𝑓𝑓𝑠𝑠𝑥𝑥𝑒𝑒𝑓𝑓
𝜏𝜏 ≡𝜆𝜆𝑣𝑣
𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥 − 𝑣𝑣 𝑡𝑡 + 𝜀𝜀
𝜏𝜏
𝜏𝜏
𝜏𝜏
𝑡𝑡
19Prof. Sergio B. MendesFall 2018
Wave Speed
𝑣𝑣 =𝜆𝜆𝜏𝜏𝜏𝜏
𝜏𝜏
𝜏𝜏
20Prof. Sergio B. MendesFall 2018
A Few Definitions:
𝑘𝑘 ≡2 𝜋𝜋𝜆𝜆
Wave Number
𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥 − 𝑣𝑣 𝑡𝑡 + 𝜀𝜀
𝑓𝑓 ≡1𝜏𝜏
Angular Frequency
= 2 𝜋𝜋 𝑓𝑓
𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀
𝜔𝜔 ≡2 𝜋𝜋𝜏𝜏
Frequency
𝑣𝑣 =𝜆𝜆𝜏𝜏
𝑣𝑣 =𝜔𝜔𝑘𝑘
21Prof. Sergio B. MendesFall 2018
𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀
= 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑 𝑥𝑥, 𝑡𝑡
𝜑𝜑 𝑥𝑥, 𝑡𝑡 ≡ 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀Phase:
Amplitude: 𝐴𝐴
22Prof. Sergio B. MendesFall 2018
Constant Phase
𝜑𝜑 𝑥𝑥, 𝑡𝑡 ≡ 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀
𝑓𝑓𝜑𝜑 𝑥𝑥, 𝑡𝑡 = 𝑘𝑘 𝑓𝑓𝑥𝑥 − 𝜔𝜔 𝑓𝑓𝑡𝑡
𝑓𝑓𝜑𝜑 𝑥𝑥, 𝑡𝑡 = 0
𝑓𝑓𝑥𝑥𝑓𝑓𝑡𝑡
=𝜔𝜔𝑘𝑘 =
𝜆𝜆𝜏𝜏
𝑘𝑘 𝑓𝑓𝑥𝑥 − 𝜔𝜔 𝑓𝑓𝑡𝑡 = 0
𝑣𝑣𝑝𝑝𝑝 =
23Prof. Sergio B. MendesFall 2018
The Wave Equation𝜓𝜓(𝑥𝑥, 𝑡𝑡) = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀
𝜕𝜕𝜓𝜓𝜕𝜕𝑥𝑥
= − 𝑘𝑘 𝐴𝐴 𝑐𝑐𝑠𝑠𝑠𝑠 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀
𝜕𝜕2𝜓𝜓𝜕𝜕𝑥𝑥2
= − 𝑘𝑘2 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀
𝜕𝜕𝜓𝜓𝜕𝜕𝑡𝑡
= 𝜔𝜔 𝐴𝐴 𝑐𝑐𝑠𝑠𝑠𝑠 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀
𝜕𝜕2𝜓𝜓𝜕𝜕𝑡𝑡2
= − 𝜔𝜔2 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀
1𝑘𝑘2
𝜕𝜕2𝜓𝜓𝜕𝜕𝑥𝑥2
=1𝜔𝜔2
𝜕𝜕2𝜓𝜓𝜕𝜕𝑡𝑡2
𝜕𝜕2𝜓𝜓𝜕𝜕𝑥𝑥2
=1𝑣𝑣2
𝜕𝜕2𝜓𝜓𝜕𝜕𝑡𝑡2
24Prof. Sergio B. MendesFall 2018
Complex Numbers
�̃�𝑧 ≡ 𝑥𝑥 + 𝑠𝑠 𝑦𝑦𝑥𝑥,𝑦𝑦 ∈ ℛℯ
�̃�𝑧1 = 𝑥𝑥1 + 𝑠𝑠 𝑦𝑦1�̃�𝑧2 = 𝑥𝑥2 + 𝑠𝑠 𝑦𝑦2
�̃�𝑧1 + �̃�𝑧2 = 𝑥𝑥1 + 𝑥𝑥2 + 𝑠𝑠 𝑦𝑦1 + 𝑦𝑦2
𝑠𝑠 ≡ −1
25Prof. Sergio B. MendesFall 2018
Euler Formula
𝑒𝑒𝑖𝑖 𝜃𝜃 = 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 + 𝑠𝑠 𝑐𝑐𝑠𝑠𝑠𝑠 𝜃𝜃
𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 =12𝑒𝑒𝑖𝑖 𝜃𝜃 + 𝑒𝑒− 𝑖𝑖 𝜃𝜃
𝑒𝑒− 𝑖𝑖 𝜃𝜃 = 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 − 𝑠𝑠 𝑐𝑐𝑠𝑠𝑠𝑠 𝜃𝜃
𝑐𝑐𝑠𝑠𝑠𝑠 𝜃𝜃 =1
2 𝑠𝑠𝑒𝑒𝑖𝑖 𝜃𝜃 − 𝑒𝑒− 𝑖𝑖 𝜃𝜃
26Prof. Sergio B. MendesFall 2018
�̃�𝑧 ≡ 𝑥𝑥 + 𝑠𝑠 𝑦𝑦
𝑥𝑥 = 𝑟𝑟 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃
𝑦𝑦 = 𝑟𝑟 𝑐𝑐𝑠𝑠𝑠𝑠 𝜃𝜃
𝜃𝜃 = 𝑡𝑡𝑡𝑡𝑠𝑠−1𝑦𝑦𝑥𝑥
𝑟𝑟 = 𝑥𝑥2 + 𝑦𝑦2
Graphical Representation
𝑦𝑦
𝑥𝑥
𝑟𝑟
𝜃𝜃
= 𝑟𝑟 𝑒𝑒𝑖𝑖 𝜃𝜃
𝑒𝑒𝑖𝑖 𝜃𝜃 = 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 + 𝑠𝑠 𝑐𝑐𝑠𝑠𝑠𝑠 𝜃𝜃𝑟𝑟 𝑟𝑟 𝑟𝑟
27Prof. Sergio B. MendesFall 2018
�̃�𝑧∗ ≡ 𝑥𝑥 − 𝑠𝑠 𝑦𝑦
�̃�𝑧 �̃�𝑧∗ = 𝑥𝑥2 + 𝑦𝑦2 = 𝑟𝑟2
�̃�𝑧 = 𝑥𝑥 + 𝑠𝑠 𝑦𝑦
Complex Conjugate
= 𝑟𝑟 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 + 𝑠𝑠 𝑟𝑟 𝑐𝑐𝑠𝑠𝑠𝑠 𝜃𝜃 = 𝑟𝑟 𝑒𝑒𝑖𝑖 𝜃𝜃
= 𝑟𝑟 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 − 𝑠𝑠 𝑟𝑟 𝑐𝑐𝑠𝑠𝑠𝑠 𝜃𝜃 = 𝑟𝑟 𝑒𝑒− 𝑖𝑖 𝜃𝜃
28Prof. Sergio B. MendesFall 2018
𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑 𝑥𝑥, 𝑡𝑡
= ℛℯ 𝐴𝐴 𝑒𝑒𝑖𝑖 𝜑𝜑 𝑥𝑥,𝑡𝑡
Complex Notation for Waves
𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑 𝑥𝑥, 𝑡𝑡
𝐴𝐴
𝜑𝜑 𝑥𝑥, 𝑡𝑡
𝐴𝐴 𝑐𝑐𝑠𝑠𝑠𝑠 𝜑𝜑 𝑥𝑥, 𝑡𝑡
𝜓𝜓1 = 𝐴𝐴1 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑1 𝜓𝜓2 = 𝐴𝐴2 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑2
𝜓𝜓 = 𝜓𝜓1 + 𝜓𝜓2
&
Adding Waves
29
𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑 = 𝐴𝐴1 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑1 + 𝐴𝐴2 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑2𝐴𝐴 = ? ?𝜑𝜑 = ? ?
to be determined
Graphical Representation & Phasor:
𝐴𝐴1 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑1
𝐴𝐴1𝐴𝐴1 𝑐𝑐𝑠𝑠𝑠𝑠 𝜑𝜑1
𝐴𝐴2 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑2
𝐴𝐴2𝐴𝐴2 𝑐𝑐𝑠𝑠𝑠𝑠 𝜑𝜑2𝜑𝜑2
𝜑𝜑1
𝜑𝜑
𝐴𝐴𝜑𝜑2 − 𝜑𝜑1
30
𝜓𝜓1 = 𝐴𝐴1 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑1 𝜓𝜓2 = 𝐴𝐴2 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑2𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑 = 𝐴𝐴1 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑1 + 𝐴𝐴2 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑2
𝐴𝐴 𝑐𝑐𝑠𝑠𝑠𝑠 𝜑𝜑 = 𝐴𝐴1 𝑐𝑐𝑠𝑠𝑠𝑠 𝜑𝜑1 + 𝐴𝐴2 𝑐𝑐𝑠𝑠𝑠𝑠 𝜑𝜑2𝑡𝑡𝑡𝑡𝑠𝑠 𝛼𝛼 =
𝐴𝐴1 𝑐𝑐𝑠𝑠𝑠𝑠 𝜑𝜑1 + 𝐴𝐴2 𝑐𝑐𝑠𝑠𝑠𝑠 𝜑𝜑2𝐴𝐴1 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑1 + 𝐴𝐴2 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑2
𝐴𝐴2 = 𝐴𝐴12 + 𝐴𝐴22 + 2 𝐴𝐴1 𝐴𝐴2 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑2 − 𝜑𝜑1
31Prof. Sergio B. MendesFall 2018
𝑡𝑡𝑡𝑡𝑠𝑠 𝛼𝛼 =𝐴𝐴1 𝑐𝑐𝑠𝑠𝑠𝑠 𝜑𝜑1 + 𝐴𝐴2 𝑐𝑐𝑠𝑠𝑠𝑠 𝜑𝜑2𝐴𝐴1 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑1 + 𝐴𝐴2 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑2
𝐴𝐴2 = 𝐴𝐴12 + 𝐴𝐴22 + 2 𝐴𝐴1 𝐴𝐴2 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑2 − 𝜑𝜑1
Phase:
Amplitude:
32Prof. Sergio B. MendesFall 2018
Extending the Wave Equation from 1D to 3D:
𝜕𝜕2𝜓𝜓𝜕𝜕𝑥𝑥2
=1𝑣𝑣2
𝜕𝜕2𝜓𝜓𝜕𝜕𝑡𝑡2
𝜕𝜕2𝜓𝜓𝜕𝜕𝑥𝑥2
+𝜕𝜕2𝜓𝜓𝜕𝜕𝑦𝑦2
+𝜕𝜕2𝜓𝜓𝜕𝜕𝑧𝑧2
=1𝑣𝑣2
𝜕𝜕2𝜓𝜓𝜕𝜕𝑡𝑡2
𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀
3D1D
𝜓𝜓 𝒓𝒓, 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝒌𝒌 . 𝒓𝒓 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀
Harmonic Plane WaveHarmonic Wave
33Prof. Sergio B. MendesFall 2018
𝜕𝜕2𝜓𝜓𝜕𝜕𝑥𝑥2
+𝜕𝜕2𝜓𝜓𝜕𝜕𝑦𝑦2
+𝜕𝜕2𝜓𝜓𝜕𝜕𝑧𝑧2
=1𝑣𝑣2
𝜕𝜕2𝜓𝜓𝜕𝜕𝑡𝑡2
𝜓𝜓 𝒓𝒓, 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝒌𝒌 . 𝒓𝒓 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀
Harmonic Plane Wave:
𝒌𝒌 . 𝒓𝒓 = 𝑘𝑘𝑥𝑥 𝑥𝑥 + 𝑘𝑘𝑦𝑦 𝑦𝑦 + 𝑘𝑘𝑧𝑧 𝑧𝑧
𝑘𝑘𝑥𝑥 2 + 𝑘𝑘𝑦𝑦2 + 𝑘𝑘𝑧𝑧 2 =
𝜔𝜔2
𝑣𝑣2𝑣𝑣 =
𝜔𝜔𝑘𝑘
34Prof. Sergio B. MendesFall 2018
𝜓𝜓 𝒓𝒓, 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝒌𝒌. 𝒓𝒓 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀
Why the name: Plane Wave ?
𝒌𝒌
𝑆𝑆1𝑆𝑆2
𝑆𝑆3𝑆𝑆4
𝒓𝒓1 ∈ 𝑆𝑆1𝒓𝒓1
𝒌𝒌 . 𝒓𝒓1 = 𝒌𝒌 𝑏𝑏
𝜓𝜓 𝒓𝒓1, 𝑡𝑡 = 𝑐𝑐𝑐𝑐𝑠𝑠𝑐𝑐𝑡𝑡𝑡𝑡𝑠𝑠𝑡𝑡
𝑏𝑏
35Prof. Sergio B. MendesFall 2018
𝒌𝒌
𝑆𝑆1
𝑆𝑆2
𝒓𝒓1
�𝒌𝒌𝑐𝑐
𝜑𝜑 𝒓𝒓1, 𝑡𝑡 = 𝒌𝒌 . 𝒓𝒓1 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀
𝜑𝜑 𝒓𝒓2, 𝑡𝑡 = 𝜑𝜑 𝒓𝒓1 + 𝑐𝑐 �𝒌𝒌, 𝑡𝑡 = 𝒌𝒌 . 𝒓𝒓1 + 𝑐𝑐 �𝒌𝒌 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀
= 𝜑𝜑 𝒓𝒓1, 𝑡𝑡 + 𝑘𝑘 𝑐𝑐
𝑘𝑘 𝑐𝑐 = 2𝜋𝜋 𝑐𝑐 = 𝜆𝜆 =2𝜋𝜋𝑘𝑘
𝒓𝒓2 = 𝒓𝒓1 + 𝑐𝑐 �𝒌𝒌
36Prof. Sergio B. MendesFall 2018
𝜓𝜓 𝒓𝒓, 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝒌𝒌. 𝒓𝒓 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀
𝒌𝒌
Harmonic Plane Wave:
37Prof. Sergio B. MendesFall 2018
𝑟𝑟 = 𝑥𝑥2 + 𝑦𝑦2 + 𝑧𝑧2
𝜑𝜑 = 𝑡𝑡𝑡𝑡𝑠𝑠−1𝑦𝑦𝑥𝑥
𝜃𝜃 = 𝑐𝑐𝑐𝑐𝑐𝑐−1𝑧𝑧
𝑥𝑥2 + 𝑦𝑦2 + 𝑧𝑧2
𝑥𝑥 = 𝑟𝑟 𝑐𝑐𝑠𝑠𝑠𝑠 𝜃𝜃 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑
𝑦𝑦 = 𝑟𝑟 𝑐𝑐𝑠𝑠𝑠𝑠 𝜃𝜃 𝑐𝑐𝑠𝑠𝑠𝑠 𝜑𝜑
𝑧𝑧 = 𝑟𝑟 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃
Spherical Coordinates
38Prof. Sergio B. MendesFall 2018
𝛻𝛻2 =1𝑟𝑟2
𝜕𝜕𝜕𝜕𝑟𝑟
𝑟𝑟2𝜕𝜕𝜕𝜕𝑟𝑟
+1
𝑟𝑟2 𝑐𝑐𝑠𝑠𝑠𝑠 𝜃𝜃𝜕𝜕𝜕𝜕𝜃𝜃
𝑐𝑐𝑠𝑠𝑠𝑠 𝜃𝜃𝜕𝜕𝜕𝜕𝜃𝜃
+1
𝑟𝑟2 𝑐𝑐𝑠𝑠𝑠𝑠2 𝜃𝜃𝜕𝜕2
𝜕𝜕𝜑𝜑2
𝜕𝜕2
𝜕𝜕𝑥𝑥2+
𝜕𝜕2
𝜕𝜕𝑦𝑦2+𝜕𝜕2
𝜕𝜕𝑧𝑧2𝜓𝜓 𝑥𝑥,𝑦𝑦, 𝑧𝑧, 𝑡𝑡 =
1𝑣𝑣2
𝜕𝜕2𝜓𝜓 𝑥𝑥,𝑦𝑦, 𝑧𝑧, 𝑡𝑡𝜕𝜕𝑡𝑡2
𝛻𝛻2 =𝜕𝜕2
𝜕𝜕𝑥𝑥2+
𝜕𝜕2
𝜕𝜕𝑦𝑦2+𝜕𝜕2
𝜕𝜕𝑧𝑧2
1𝑟𝑟2
𝜕𝜕𝜕𝜕𝑟𝑟
𝑟𝑟2𝜕𝜕𝜕𝜕𝑟𝑟
+1
𝑟𝑟2 𝑐𝑐𝑠𝑠𝑠𝑠 𝜃𝜃𝜕𝜕𝜕𝜕𝜃𝜃
𝑐𝑐𝑠𝑠𝑠𝑠 𝜃𝜃𝜕𝜕𝜕𝜕𝜃𝜃
+1
𝑟𝑟2 𝑐𝑐𝑠𝑠𝑠𝑠2 𝜃𝜃𝜕𝜕2
𝜕𝜕𝜑𝜑2𝜓𝜓 𝑟𝑟,𝜑𝜑,𝜃𝜃, 𝑡𝑡 =
1𝑣𝑣2
𝜕𝜕2𝜓𝜓 𝑟𝑟,𝜑𝜑,𝜃𝜃, 𝑡𝑡𝜕𝜕𝑡𝑡2
39Prof. Sergio B. MendesFall 2018
𝜓𝜓 𝑟𝑟,𝜑𝜑, 𝜃𝜃, 𝑡𝑡 = 𝜓𝜓 𝑟𝑟, 𝑡𝑡
Spherical Symmetry
1𝑟𝑟2
𝜕𝜕𝜕𝜕𝑟𝑟
𝑟𝑟2𝜕𝜕𝜕𝜕𝑟𝑟
𝜓𝜓 𝑟𝑟, 𝑡𝑡 =1𝑣𝑣2
𝜕𝜕2𝜓𝜓 𝑟𝑟, 𝑡𝑡𝜕𝜕𝑡𝑡2
1𝑟𝑟2
𝜕𝜕𝜕𝜕𝑟𝑟
𝑟𝑟2𝜕𝜕𝜕𝜕𝑟𝑟
+1
𝑟𝑟2 𝑐𝑐𝑠𝑠𝑠𝑠 𝜃𝜃𝜕𝜕𝜕𝜕𝜃𝜃
𝑐𝑐𝑠𝑠𝑠𝑠 𝜃𝜃𝜕𝜕𝜕𝜕𝜃𝜃
+1
𝑟𝑟2 𝑐𝑐𝑠𝑠𝑠𝑠2 𝜃𝜃𝜕𝜕2
𝜕𝜕𝜑𝜑2𝜓𝜓 𝑟𝑟,𝜑𝜑, 𝜃𝜃, 𝑡𝑡 =
1𝑣𝑣2
𝜕𝜕2𝜓𝜓 𝑟𝑟,𝜑𝜑,𝜃𝜃, 𝑡𝑡𝜕𝜕𝑡𝑡2
40Prof. Sergio B. MendesFall 2018
1𝑟𝑟2
𝜕𝜕𝜕𝜕𝑟𝑟
𝑟𝑟2𝜕𝜕𝜕𝜕𝑟𝑟
𝜓𝜓 𝑟𝑟, 𝑡𝑡 =1𝑣𝑣2
𝜕𝜕2𝜓𝜓 𝑟𝑟, 𝑡𝑡𝜕𝜕𝑡𝑡2
𝜓𝜓 𝑟𝑟, 𝑡𝑡 ≡𝑓𝑓 𝑟𝑟, 𝑡𝑡𝑟𝑟
𝜕𝜕2𝑓𝑓 𝑟𝑟, 𝑡𝑡𝜕𝜕𝑟𝑟2
=1𝑣𝑣2
𝜕𝜕2𝑓𝑓 𝑟𝑟, 𝑡𝑡𝜕𝜕𝑡𝑡2
𝜓𝜓 𝑟𝑟, 𝑡𝑡 =𝑓𝑓 𝑟𝑟 − 𝑣𝑣 𝑡𝑡
𝑟𝑟
41Prof. Sergio B. MendesFall 2018
𝜓𝜓 𝑟𝑟, 𝑡𝑡 =𝑓𝑓 𝑟𝑟 − 𝑣𝑣 𝑡𝑡
𝑟𝑟
𝜓𝜓 𝑟𝑟, 𝑡𝑡 =𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑟𝑟 − 𝑣𝑣 𝑡𝑡 + 𝜀𝜀
𝑟𝑟
=𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑟𝑟 − 𝜔𝜔𝑡𝑡 + 𝜀𝜀
𝑟𝑟Spherical (diverging) Wave