Download Waves -- Revision Guide

Waves
yck2F6/2004
Kinds of waves
Pulse / Periodic waves
Transverse / Longitudinal
Progressive (traveling) / Stationary (standing)
Frequency (f): Number of oscillations per second
Period (T): Time for one oscillation (unit is s)
Amplitude (A):
The largest magnitude of oscillation (unit depends on type of wave)
Wavelength (): The distance traveled by wave in one period of time (m)
Wave velocity (v): The distance traveled by wave per second ( m s1)
Intensity (I): Energy transported through a unit perpendicular area per second
Usually,
(W m2)
Intensity  Amplitude2
Mathematical representation of wave:
y  A cost  kx  
where,
y is the magnitude of oscillation of wave from equilibrium state;
A is the amplitude of wave;
 is the angular frequency of wave;
 = 2 f
(unit is s1)
t is the time ;
(s)
x is the position on line along which the wave propagates; (m)
`
k is the wave number;
(m1)
 is the initial phase (starting phase)
For a certain moment, say t = 0 :
k corresponds to the phase of 2
k  2
so,
k
(x=)
2

For a certain position on the propagation line, say x = 0 :
T corresponds to the phase of 2
T  2
so,
(t=T)
T
2

Graphical representations of waves:
y-t graph (one point, all time)
y-x graph (one moment, all positions)
There are crests and troughs for transverse waves; compressions and rarefactions for
longitudinal waves.
Wave velocity depends on the medium, not on frequency of oscillations.
Doppler Effect
(for sound and light)
v  vr
f
f
v  vs o
where
fo is the original frequency of wave
f is the apparent frequency received by the receiver
v is the wave velocity
vr is the velocity of the receiver (+ approaching,  moving away)
vs is the velocity of wave source ( approaching, + moving away)
Examples of Doppler effect
Electromagnetic spectrum
Wave speed is speed of light.
The approximate ranges () of different e.m. waves:
Radio waves
104 m -- 102 m
Microwaves
102 m -- 103 m
Infra red
103 m -- 107 m
Visible light
7107 m -- 4107 m
Ultra-violet rays 107 m -- 109 m
X-rays
109 m -- 1011 m
Gamma rays
1011 m -- shorter
Examples and applications
Huygens' Principle
Every point on a wavefront behaves as a new source (sourcelet), sending out
secondary wavelets. A smooth line (tangent) joining the most forward part of the
secondary wavelets forms the new wavefront.
Reflection of waves
There is phase change of  when a wave is reflected from a denser medium.
Laws of reflection are obeyed.
Refraction of waves
Snell's Law is obeyed.
Refractive index of a medium depends on the speed of wave in that medium.
Selective Polarization of waves (only for transverse waves)
A polarizer and an analyser are media that only allow oscillations of a certain
direction to pass through.
When a unpolarized wave passes through a polarizer, its intensity drops to half.
When a polarized wave passes through an analyser, then its intensity I is
I  I o cos 2 
where
 is the angle between the directions of polarization of the polarizer
and analyser.
Polarization of waves by reflection
Reflected light is completely polarized when the transmitted and reflected
rays are perpendicular at Brewster's angle of incidence, b , where
n
tan  b  2
n1
(n1 and n2 are refractive indice of the two media, and light is incident from
medium 1 to medium 2)
Polarization by scattering
Scattered light is completely polarized when it is normal to the incident
direction.
Principle of Superposition
When two waves of the same nature meet at a point, the resultant magnitude of
oscillation at that point is the algebraic sum of oscillations caused by the
individual waves.
Interference of waves
When two waves of the same nature, same polarization direction (for transverse
waves), same frequency, constant phase difference and almost the same
amplitude meet, a steady interference pattern may be resulted.
For two coherent waves, (waves of same phase)
Constructive interference occurs at a point when
path difference from the two sources = n
(or phase difference = n (2) )
Destructive interference occurs at a point when

Path difference from the two sources = n 
(n = 0, 1, 2, ...)
1
2

(or phase difference = (2n + 1)  )
Young's double-slit experiment
produces a interference pattern of uniform separation
(detail description and calculations are needed)
Multiple-slits interference
Diffraction grating (for light)
Constructive interference occurs at angle  when
a sin   m
(a : grating spacing;
m : order of maximum)
Destructive interference occurs at angle ' when
1

a sin '   m   
( N : no. of slits involved )
N

Angular half-width,
  '
Examples and application of interference
Beat: occurs when two waves of slightly different frequencies superposition.
Beat frequency = f  f '
Diffraction of light
For wave passing through a single slit of width a ( a >  )
There will be a central maximum (which is wider and brighter)
Maximum occurs at  when
a sin  
2m  1

2
Minimum occurs at ' when
a sin '  m  1
(m = 0, 1, 2, ...)
Stationary waves
Stationary wave is a result of superposition of a progressive wave (of certain
wavelengths) and its reflected wave.
Nodes and antinodes are found in a stationary wave. At node, oscillation and
kinetic energy are the minimum; at antinode, oscillation and kinetic energy are
the maximum.
Resonance in air columns and strings
Harmonics and overtones
Finding velocity of sound by experiments:
1. tuning fork and air column
finding length of air column for resonance (error due to end correction)
2. Kundt's tube
measuring distance between two leaps of powder (foam chips)
Intensity Level of sound ( h , unit is dB)
Decibel : using 1012 W m2 as reference intensity.
 I 
h  10 log 10  
 Io 
Noise and musical note
Geometric Optics
Systems of lenses and mirrors
Optical instruments (magnifying glass and telescope)
Angular magnification and angle subtended by object and image
Optical spectrometer