1. Wave Motion & EM Waves (II)
Chih-Chieh Kang
Electrooptical Eng.Dept. STUT

2. Sinusoidal Traveling Waves

3. Sinusoidal Waves
Snapshot of a traveling sinusoidal wave (at a fixed time, t = 0), and 0=0
Vertical displacement of the traveling wave:
Wavelength  ：the distance between two successive crests or troughs.
Amplitude A：one half the wave height or the distance from either the crest or the trough to the equilibrium points
Phase  = 2z/

4. Sinusoidal Traveling Waves
A wave does not change its shape as it travels through space. For a traveling sinusoidal wave moving at a speed v, the wave function  at some later time t :

5. Phase Lead & Phase Lag
(Ulaby)

6. Sinusoidal Traveling Waves
For the time a wave traveling a distance of one wavelength is called period T
The frequency of a sinusoidal wave f

7. Sinusoidal Traveling Waves
The angular wave number (or propagation number) of a sinusoidal wave k
 Wave function

8. Harmonic Traveling Waves
For a traveling sinusoidal wave (at a fixed point z = 0)
angular frequency=2/T=2f
 wave function

9. Speed of a Wave
For a traveling wave, its waveform retains the same phase
Phase velocity v : the velocity of the waveform as it moves across the medium

10. Mathematical Description of a Wave
Waves are solutions to the wave equation：
1-D waves
：wave function, v：phase velocity
- Where does wave equation come from?
- What do solutions look like?
- How much energy do they carry?

11. Wave Equation for a String
Each small piece of string obeys Newton’s Law：
Small displacement, so
Net force is proportional to curvature：

12. Wave Equation for a String
Newton’s 2nd Law…
>>(mass density
…leads to the wave equation with
- wave function=transverse displacement
- phase velocity
<-restoring force
<-inertia

13. Solutions of 1-D Wave Equation
Consider

14. Solutions of 1-D Wave Equation
 is a solution
the same reason, is a solution
is a solution too
any linear combination of solutions is also a solution
: superposition

15. Description of Traveling Waves
waves traveling in the +z direction
no change in shape
point P moving with time

16. 1-D Harmonic Traveling Waves
1-D time-harmonic traveling waves propagating in the +z direction
/v = 2f / f = 2/= k : Angular wave number

17. 1-D Harmonic Traveling Waves
Complex representation of harmonic traveling waves propagating in +z direction
 If

18. References
F. T. Ulaby, Fundamentals of Applied Electromagnetics, Prentice Hall.
J. D. Cutnell, and K. W. Johnson, Physics, Wiley.