Wave Motion & EM Waves (II) Chih-Chieh Kang Electrooptical Eng.Dept. STUT email:kangc@mail.stut.edu.tw

Sinusoidal Traveling Waves

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/

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 :

Phase Lead & Phase Lag (Ulaby)

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

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

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

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

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?

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

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

Solutions of 1-D Wave Equation Consider

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

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

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

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

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