Lecture 3 Chapter 2 Wave Motion Phase and phase velocity The superposition principle Complex representation 3D waves: plane waves Harmonic waves: summary Functional shape: Wave parameters: Asin k(x vt) “-” for wave moving right k - propagation number “+” for wave moving left - wavelength Alternative forms: - period - frequency - angular temporal frequency x t Asin 2 - wave number mostly k 2 used Asin2 x t v Asinkx t 1 1 x v 2 2 Asin2 t v single frequency These eq-ns describe an infinite monochromatic (monoenergetic) wave. Real waves are not infinite and can be described by superposition of harmonic waves. If frequencies of these waves cluster closely to a single frequency (form narrow band) the wave is called quasimonochromatic Periodic waves Waveform produced by saxophone: profile-elements - when repeated can reproduce the whole waveform Can use the same parameters to describe: - wavelength - the length of one profile-element - period - the duration in time for one profile-element - wave number - number of profile-elements per unit length -etc… Harmonic waves: example 1. Write an equation of a “red” light wave that propagates along x axis (at speed of light c) and has a wavelength 600 nm. Solution: Asin k(x vt) k 2 2 Asin (x ct) 6 107 m 2. What is the frequency of this light? Hz1/s Solution: v 8 v c 310 m/s 14 510 Hz 600 109 m Harmonic wave: Initial phase Consider wave x,t Asinkx t phase: kx t When written like that it implies that x,t x0 0 t0 With a single wave we can always chose x axis so that above is true But in general case x,t x0 0 t0 This is equivalent to the shift of coordinate x by some value a a x x,t Asinkx at x,t Asinkx t ka x,t Asinkx t - initial phase phase: kx t Harmonic wave: Phase Asinkx t Can use cos(): x,t Asin kx t x x,t Acoskx t 2 equivalent equations Special case: = = 180o phase shift x,t Asin kx t x,t Asin t kx x,t Acos t kx / 2 x Note: sin(kx-t) and sin(t-kx) both describe wave moving right, but phase-shifted by 180 degrees (). Harmonic wave: Phase derivatives x,t Asin kx t Phase: kx t Partial derivatives: rate of change of phase with time is equal t x to angular frequency (=2) rate of change of phase with distance is k x t equal to propagation number Harmonic wave: Phase velocity Phase: kx t What is the speed of motion of a point with constant phase? from the theory of partial derivatives x t x v t x t k sign gives direction phase velocity of a wave In general case, for any wave we can find the phase velocity: t v x x t Phase (red) vs. group (green) velocity always >0 (to be discussed later) by definition Add sign to give direction: + in positive x direction - in negative x direction The superposition principle 2 1 2 Consider differential wave equation: x2 v2 t2 If 1 and 2 are both solutions to that eq-n, then their superposition (1+2 ) is also a solution: 2 1 2 Proof: 1 2 1 2 x2 v2 t2 = 2 2 1 2 1 2 1 2 1 2 x2 x2 v2 t2 v2 t2 = The superposition principle Superposition principle: the resulting disturbance at each point in the region of overlap of two or more waves is the algebraic sum of the individual constituent waves at that location. Note: once waves pass the intersecting region they will move away unaffected by encounter Superposition of traveling waves The superposition principle: example Note: the resulting wave is still a harmonic wave (the same k) The superposition principle: special cases Two waves are ‘in-phase’: Two waves are ‘out-of-phase’: (=180o=) A sinkx t 1 A1 sinkx t 1 1 A sinkx t 2 A2 sinkx t 2 2 2 A2 sinkx t A Akxtsin A12Akxtsin 12 Amplitude of the resulting wave Amplitude of the resulting wave increases: constructive interference decreases: destructive interference The complex representation Complex numbers: ~z x iy, where i 1 In polar coordinates: x r cos , y r sin ~z rcos i sin i Euler formula: e cos i sin Argand diagram Any complex number: ~z rei Wave: x,t Asinkx t ' Acoskx t can use sin or cos to describe a wave Convention - use cos: x,t ReAeikxt Acoskx t Usually omit ‘Re’: x,t Aeikxt Aei wave equation using complex numbers The complex number math i e cos i sin ei ei ei ei cos , sin ei cos i sin 2 2i Magnitude (modulus, absolute value): ~z x2 y2 r ~z~z Complex conjugate: ~z x iy x iy rei rei ~ ~ ~ ~ ~ ~ z1 z2 z1 z2 Math: z1 z2 x1 x2 iy1 y2 e e e ~z x ~ ~ i1 2 e e z1z2 r1r2e ~ i2 z r i e 1 1 1 e 1 2 ~ i z2 r2 e 1 ei / 2 i ~ 1 ~ ~ 1 Rez z z Imzzz ~ ~ 2 2i ez i2 ez Phasor moving left x,t Asinkx t Lets rotate the arrow in Argand diagram at angular frequency : t This rotating arrow is called phasor A CCW rotation - wave moves left CW rotation - wave moves right Phasor: superposition Phasor addition Adding two waves can be done using phasors i1 i2 1 2 A1e A2e Aei Complex numbers can be added as vectors Phasor: superposition Phasor addition Adding two waves can be done using phasors i1 i2 1 2 A1e A2e Aei Complex numbers can be added as vectors Example: out-of-phase waves Amplitudes subtract A A sin kx t 1 1 2 Phase does not change 3-D waves Surfaces joining all points of equal phase are called wavefronts. Example: Wavefronts of 2-D circular waves on water surface (superposition where waves overlap) http://www.falstad.com/wavebox/ 3-D waves: plane waves (simplest 3-D waves) All the surfaces of constant phase of disturbance form parallel planes that are perpendicular to the propagation direction 3-D waves: plane waves (simplest 3-D waves) All the surfaces of constant phase of disturbance form parallel planes that are perpendicular to the propagation direction An equation of plane that is ˆ ˆ ˆ perpendicular to k kx i k y j kzk Unit vectors k r const a All possible coordinates of vector r are on a plane k Can construct a set of planes over which varies in space harmonically: r Asink r or r Acosk r or r Aeikr.