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   Asin2 x t    v   Asinkx  t   1    1    x  v    2   2   Asin2   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 107 m

2. What is the frequency of this light? Hz1/s Solution: v   8 v c 310 m/s 14      510 Hz  600 109 m Harmonic wave: Initial phase

Consider wave  x,t Asinkx t phase:   kx  t

When written like that it implies that  x,t x0  0 t0 With a single wave we can always chose x axis so that above is true

But in general case  x,tx0  0  t0 This is equivalent to the shift of coordinate x by some value a a x  x,t  Asinkx  at  x,t  Asinkx t  ka

 x,t  Asinkx t    - initial phase phase:   kx  t   Harmonic wave: Phase   Asinkx t Can use cos():

 x,t  Asin kx  t  x  x,t  Acoskx 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  and  are both solutions to that eq-n, then their superposition 1 2  (1+2 ) is also a solution:  2    1 2    Proof: 1 2 1 2 2   2 2 x v t =  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 sinkx  t 1  A1 sinkx  t 1 1     A sinkx  t    2  A2 sin kx  t 2 2

 2  A2 sinkx  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  rcos  i sin  i Euler formula: e  cos  i sin Argand diagram Any complex number: ~z  rei

Wave:  x,t Asinkx  t  ' Acoskx  t   can use sin or cos to describe a wave   Convention - use cos:  x,t ReAeikx t  Acoskx  t     Usually omit ‘Re’:  x,t Aeikx t   Aei wave equation using complex numbers The complex number math     i e  cos  i sin ei  ei ei  ei  cos  , sin  ei  cos  i sin 2 2i

~ 2 2 ~~ Magnitude (modulus, absolute value): z  x  y  r  zz  Complex conjugate: ~z   x  iy   x  iy  rei   rei

~ ~ ~ ~ ~ ~ z1 z2 z1 z2 Math: z1  z2  x1  x2  iy1  y2  e  e e  ~z x ~ ~ i 1 2  e  e z1z2  r1r2e

~  i2 z r i e  1 1  1 e 1 2 ~ i z2 r2 e  1 ei / 2  i ~ 1 ~ ~ 1  Rez  z  z Imzzz~ ~ 2 2i ez i2  ez Phasor moving left  x,t Asinkx  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 

 i 1 i2   1  2  A1e  A2e

  Aei Complex numbers can be added as vectors Phasor: superposition Phasor addition

Adding two waves can be done using phasors 

 i 1 i2   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  Asink  r     or  r  Acosk  r     or  r  Aeikr