Transcript Review: Waves - I

Review: Waves - I
Waves
Particle: a tiny concentration of matter, can
transmit energy.
Wave: broad distribution of energy, filling the
space through which it travels.
Quantum Mechanics:
Wave
Particle
Types of Waves
Types of waves: Mechanical Waves,
Electromagnetic Waves, Matter Waves,
Electron, Neutron, People, etc ……
Transverse Waves:
Displacement of medium
 Wave travel direction
Longitudinal Waves:
Displacement of medium
|| Wave travel direction
Parameters of a Periodic Wave
l: Wavelength, length of one complete wave
form
T: Period, time taken for one wavelength of
wave to pass a fixed point
v: Wave speed, with which the wave moves
f: Frequency, number of periods per second
l = vT
v = l/T = l f
Wave Function of Sinusoidal Waves
y(x,t) = ymsin(kx-wt)
ym: amplitude
When ∆x=l, 2 is
added to the phase
kx-wt : phase
k: wave number k 
w: angular frequency
2
l
When ∆t=T, 2 is
added to the phase
2
w
 2 f
T
Wave Speed
How fast does the wave form travel?
Wave Speed
How fast does the wave form travel?
Pick a fixed displacement   a fixed phase
kx-wt = constant
dx w
v

dt k
y(x,t) = ymsin(kx-wt)
v>0
y(x,t) = ymsin(kx+wt)
v<0
Transverse Waves (String):
v


Principle of Superposition
Overlapping waves add to produce a
resultant wave
y’(x,t) = y1 (x,t) + y2 (x,t)
Overlapping waves do not alter the
travel of each other
Interference
y1 t   ym sin kx  wt 
y2 t   ym sin kx  wt   
1  
1 

y t   y1  y 2  2ym cos  sin kx  w t  

2  
2 
Constructive:   n 2 
Destructive:
1 

  n  2 
2
k
2
l
n=0,1,2, ...
Phasor Addition
PHASOR: a vector with the amplitude ym of the
wave and rotates around origin with w of the
wave
When the interfering waves have the
same w
INTERFERENCE
PHASOR ADDITION
Can deal with waves with different
amplitudes
sin   sin   2 sin
1
1
   cos    
2
2
Standing Waves
Two sinusoidal waves with same AMPLITUDE
and WAVELENGTH traveling in OPPOSITE
DIRECTIONS interfere to produce a standing
wave
y1  x, t   ym sin  kx  wt 
y2  x, t   ym sin  kx  wt 
y x,t   y1  y2  2ym sin kx cos wt
Amplitude depends
on position
The wave does
not travel
y x,t   2ym sin kx cos wt
NODES: points of zero amplitude
kx  n ,
or
nl
x
2
n  0,1,2,...
ANTINODES: points of maximum (2ym)
amplitude
 1 
 1 l
kx  n   , or x  n 
 2 
 2  2
k
2
l
sin n   0
n  0,1,2,...
 1  

sin n    1
 2  
Standing Waves in a String
The BOUNDARY CONDITIONS determines
how the wave is reflected.
Fixed End: y = 0, a node at the end
The reflected wave has an opposite sign
Free End: an antinode at the end
The reflected wave has the same sign
Case: Both Ends Fixed
yx  0  0
yx  L  0
sinkL  0
OR
2L
l
n
OR
nv
f
2L
n
k
,
L
n  1,2,3,....
k can only take
these values
where

v

RESONANT FREQUENCIES:
yx,t   2ym sin kx cos wt
n 
f
2L 
f
v
l
k
2
l
HRW 11E (5th ed.). (a) Write an expression describing a sinusoidal transverse wave
traveling on a cord in the y direction with an angular wave number of 60 cm-1, a
period of 0.20 s, and an amplitude of 3.0 mm. Take the transverse direction to be the z
direction. (b) What is the maximum transverse speed of a point on the cord?
(a)
k = 60 cm-1, T=0.2 s, zm=3.0 mm
z(y,t)=zmsin(ky-wt)
w = 2/T = 2/0.2 s =10s-1
z(y, t)=(3.0mm)sin[(60 cm-1)y -(10s-1)t]
(b) Speed
z(y,t)
uz 
 w zm cosky  wt 
t

 w zm sin
 (ky  wt)
2
uz,min= wzm = 94 mm/s

HRW 16P (5th ed.). A sinusoidal wave of frequency 500 Hz has a velocity of 350
m/s. (a) How far apart are two points that differ in phase by /3 rad? (b) What is
the phase difference between two displacements at a certain point at times 1.00 ms
apart?
f = 500Hz, v=350 mm/s
(a) Phase
y(x,t) = ymsin(kx-wt)
k
 x,t   kx  wt
2f
  x,t  
x  2ft
v
2 f
 
x
v
350m/s  
x 
 
 0.117 m
2f
2 500Hz 3 
v
(b)
2
v  lf 
l
w
k
w  2f
  2ft  2 500 Hz (1.00  10 3 )   rad.
HRW 36E (5th ed.). Two identical traveling waves, moving in the
same direction, are out of phase by /2 rad. What is the amplitude of
the resultant wave in terms of the common amplitude ym of the two
combining waves?
y1 t   ym sin kx  wt 
y2 t   ym sin kx  wt   
1  
1 

y t   2ym cos  sin kx  wt  

2  
2 
For


2
1

A  2 ym cos   2 ym cos  1.4ym
2
4
HRW 41E (5th ed.). Two sinusoidal waves of the same wavelength
travel in the same direction along a stretched string with amplitudes
of 4.0 and 7.0 mm and phase constant of 0 and 0.8 rad,
respectively. What are (a) the amplitude and (b) the phase constant
of the resultant wave?
(a)
2
ym  ym1
 y2m2  2 ym1 ym2 cos 
2
 ym1
 y2m2  2 ym1 ym2 cos 
 4.4mm
(b)
h
ym2
 sin   sin 
ym2=7.0 mm
h
h
 sin 
ym
ym2 sin 
sin  
 0.935
ym
ym


0.8
ym1=4.0 mm
The angle  is either 68˚ or 112˚. Choose 112˚, since >90˚.