Transcript Electric Potential - McMaster University

Superposition of Waves
1) Identical waves in opposite directions:
“standing waves”
2) 2 waves at slightly different frequencies:
“beats”
3) 2 identical waves, but not in phase:
“interference”
Physics 1B03summer-Lecture
Principle of Superposition
Two Waves In The Same Medium:
The observed displacement y(x,t) is the sum of
the individual displacements:
y1(x,t) + y2(x,t) = y(x,t)
(for a “linear medium”)
Physics 1B03summer-Lecture
Quiz
What do you get if you add two
identical (but out-of-phase)
square or triangular waves?
+
+
=?
=?
Physics 1B03summer-Lecture
What’s Special about Sine Waves?
Two waves, of the same frequency, arrive out of phase:
Eg.
y1  A sin(t )
y2  A sin(t   )
From Trig:
sin a + sin b = 2 cos [(a-b)/2] sin [(a+b)/2]
Result:
y  y1  y2

 

 2 A cos  sin  t  
2
 2

amplitude
Physics 1B03summer-Lecture
Asin t
Asin (t+)
+
ARsin (t+R)
=
Resultant: Sine wave, AR depends on phase difference
Physics 1B03summer-Lecture
“Constructive interference:”
phase difference = 0, 2p, 4p, ...
AR =A1 + A2
“Destructive interference:”
phase difference = p, 3p, 5p,...
AR =|A1 - A2|
Physics 1B03summer-Lecture
Standing Waves
A standing wave is an oscillation pattern with a
stationary outline that results from the superposition
of two identical waves traveling in opposite directions.
Physics 1B03summer-Lecture
Sine Waves In Opposite Directions:
y1 = Aosin(kx – ωt)
y2 = Aosin(kx + ωt)
Total displacement, y(x,t) = y1 + y2
a b
a -b


Trigonometry : sin a  sin b  2 sin 
 cos

 2 
 2 
kx  t  " a"
kx - t  " b"
Then:
y ( x, t )  2 A0 sin kx cos t
Physics 1B03summer-Lecture
y( x, t )  ( A sin kx) cos t
(where A = 2Ao )
The particle motions are simple harmonic oscillations which are
all in phase (or ½ cycle out of phase) with each other, but with
different amplitudes.
y A
t=0
t = T/8
x
t = 3T/8
-A
node
t = T/2
node
node
Physics 1B03summer-Lecture
node
antinode
Antinodes form where the waves always arrive in phase
(“constructive interference”); nodes form at locations where the
waves are 180o (½ cycle) out of phase (“destructive interference”).
Physics 1B03summer-Lecture
Nodes are positions where the amplitude is zero:
at
kx = 0
(x = 0)
kx = π
(x = λ/2)
kx = 2π
(x = λ),
kx = 3π
(x=3λ/2)
etc.
i.e., Nodes are ½ wavelength apart.
Antinodes (maximum amplitude) are halfway
between nodes.
Physics 1B03summer-Lecture