Review: Waves - I
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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
yx 0 0
yx L 0
sinkL 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:
yx,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 =10s-1
z(y, t)=(3.0mm)sin[(60 cm-1)y -(10s-1)t]
(b) Speed
z(y,t)
uz
w zm cosky 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
2f
x,t
x 2ft
v
2 f
x
v
350m/s
x
0.117 m
2f
2 500Hz 3
v
(b)
2
v lf
l
w
k
w 2f
2ft 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˚.