- Interference occurs when two waves meet and can be constructive or destructive. Constructive interference happens when peaks and troughs of waves align, while destructive interference happens when peaks meet troughs. - Plane waves traveling in different directions can interfere. Solid points on a diagram represent constructive interference where wave peaks meet, while open points represent destructive interference where a peak meets a trough. - Path difference is the difference in distances traveled by two waves, and determines whether interference is constructive or destructive based on whether the difference is an integer or half-integer multiple of the wavelength. - Beats occur when two waves with nearly identical but not quite the same frequencies are combined, causing the amplitude of the resulting wave to

1. Interference and Beats
— Shiyue Ma

2. Interference of
one-dimensional waves
• Left Side Picture: When we put these two waves together, the
peak of the first wave will meet the peak of the second wave.
The trough will meet the trough. So, these two waves are
perfectly in phase and we will observe constructive
interference.
• Right Side Picture: The peak will meet the trough. So they are
perfectly out of phase. So we will observe destructive
interference.

3. Interference of
two plane waves
• The picture on the right
shows two plane waves,
which have the same
wavelength,but travels
different direction. The
lines represent the peaks
of each wave.
• QUESTION: The solid
points represent
constructive interference
or the open points.

4. Interference of
two plane waves
(Answer)
• As we can see, the solid
points are the intersection of
two lines. So we know, at
these points, the two peaks of
two plane waves meet with
each other. Therefore, the
solid points are constructive
interference.
• For a open points, we can
know there are one peak of
the red wave, and one trough
of the blue wave. So these
points represent
deconstructive interference.

5. Practise Question
• The thicker lines represent the crests of waves.
The thin lines represent the troughs.

6. Answer
• (D) Because only points A and B are the
interceptions of two thick lines.

7. Path Difference
**We only path difference when the two waves have the same wavelength.
• Definition: Two waves from two different point source travel different distance. The
difference between two distance is Path Difference. (Path difference = d1 - d2)
An Awesome Video:
https://www.youtube.com/watch?v=S9U_KWP5gB8
The condition for
constructive interference:
The path difference
between two waves
must be the integer
multiples of
wavelength.
(Path difference = n *
lambda)
The condition for
destructive interference:
When one path is an
integer number of
wavelength but the
other one is half-
integer multiples of
wavelength.
(Path difference = (n +
0.5) * lambda)

8. Question
• The thicker lines represent the crests of waves. The thin lines represent the troughs.
• What are the path difference for point
C and point D?
• Are they constructive interference or
destructive interference?

9. Answer
• Point C: 6 λ — 4 λ = 2 λ
• Point D: 5 λ — 4.5 λ = 0.5 λ
• Because the path difference for point C is a integer
multiple of wavelength. So it is a constructive
interference.
• Because the path difference for point D is a half-
integer multiple of wavelength. So it is a destructive
interference.

10. Beats
• When there are two waves and the frequencies of
these two waves are slightly different from each
other. We can detect the increasing and decreasing
amplitude as a function of time. The variation of
amplitude is Beats.
• Demonstration Video:
https://www.youtube.com/watch?v=IYeV2Wq82fw

11. Formulas about beats
• If we have two waves:
• And we add them together in order to get the
resultant wave: • The red
line is
Wave1
• The blue
line is
Wave 2
• The
green line
is Wave 3

12. Tune
• Sometimes beats will make instruments out of tune.
In order to provide better musics to audience,
musicians need to adjust the frequency of its notes
until the beats disappear.
• This is a video that musicians are tuning their
instruments:
https://www.youtube.com/watch?v=KfSH1ezevjM

13. Thank you!!!!