1. Physics 1B03summer-Lecture 10
Superposition of Waves
Identical waves in opposite directions:
“standing waves”
2 waves at slightly different frequencies:
“beats”
2 identical waves, but not in phase:
“interference”
Physics 1B03summer-Lecture 10

2. 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 10

3. Physics 1B03summer-Lecture 10
Quiz
What do you get if you add two identical (but out-of-phase) square or triangular waves? +
= ? +
= ?
Physics 1B03summer-Lecture 10

4. Physics 1B03summer-Lecture 10
What’s Special about Sine Waves?
Two waves, of the same frequency, arrive out of phase:
Eg.
From Trig:
sin a + sin b = 2 cos [(a-b)/2] sin [(a+b)/2]
Result:
amplitude
Physics 1B03summer-Lecture 10

5. Physics 1B03summer-Lecture 10
Asin wt
Asin (wt+f) +
ARsin (wt+fR) =
Resultant: Sine wave, AR depends on phase difference
Physics 1B03summer-Lecture 10

6. Physics 1B03summer-Lecture 10
“Constructive interference:”
phase difference = 0, 2p, 4p, ...
AR =A1 + A2
“Destructive interference:”
phase difference = p, 3p, 5p,...
AR =|A1 - A2|
Physics 1B03summer-Lecture 10

7. Physics 1B03summer-Lecture 10
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 10

8. Sine Waves In Opposite Directions:
y2 = Aosin(kx + ωt)
y1 = Aosin(kx – ωt)
Total displacement, y(x,t) = y1 + y2
Trigonometry :
Then:
Physics 1B03summer-Lecture 10

9. Physics 1B03summer-Lecture 10
(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
t = T/2
node
node
node
Physics 1B03summer-Lecture 10

10. Physics 1B03summer-Lecture 10
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 10

11. Physics 1B03summer-Lecture 10
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 10