Phys 101 beats LO2

1. BEATS
Orchestrating a Solution

2. FIRSTYOU MUST
UNDERSTAND INTERFERENCE

3. EXPLAINING INTERFERENCE
The interaction between two or more
wave functions
For Example: waves with differing phase constants
+

4. INTERFERENCE
When mixing the two wave functions they interfere
constructively and destructively in different times and
at different spaces

5. RESULTANT WAVE
The amplitude heard by you is the resultant wave of the two or more
waves functions interacting
Red:Wave Function 1
Blue:Wave Function 2
Purple: Resultant Wave
Here you can
see wave functions
interacting

6. RESULTANT WAVE II
As you can see the resultant is the sum of the two amplitudes

7. BEATS
Beats:
!
Are the resultant variation in amplitude of
two or more waves that are interacting
constructively or destructively.
!
!
However, beats are produced by
waves out of frequency

8. RELATEDTO FREQUENCY
The rate at which the amplitude varies is
proportional to the frequency difference
*However, if frequency difference is too large than you hear two distinct tones
rather than one tone which varies in intensity

9. EFFECT OF FREQUENCY
Functions out of phase, same frequency
Functions with different frequencies

10. RESULTANT WAVES
Resultant wave of functions with same frequency, but different phase
constant
Resultant wave of functions with different frequency

11. SEETHE DIFFERENCE?
Resultant
wave of
differing
phase
constant
functions
Resultant
wave of
differing
frequency
functions

12. SINUSOIDAL EFFECT
Amplitude vs time graph for the sum of the waves

13. EQUATION FOR RESULTANT
AMPLITUDE
Variables:
!
ω* = (ω1 + ω2)/2 Mean angular frequency
!
Δω = (ω1 - ω2)/2 Angular frequency difference
!
t = time
!
*set X0 to 0
STotal(0,t) = 2smcos(ω*t)cos(Δωt)

14. EXAMPLE QUESTION

15. Exactly 5 minutes into a musical play you
hear a peculiar tone that sounds really
good.The sound is made from the
combined tones of a guitar and violin.
Being a rockstar, you want to know what
tone that was to be able to incorporate
it into your next album.Also being a
Physics Major you decide to use your
combined knowledge of music and
physics to find the tone’s amplitude.
What is the tone’s amplitude if you know that:
The guitar tone has a wave function of :
2cos(4x-3t)
and the violin has a wave function of
2cos(6x-9t)
Sm = nm

16. SOLUTION
You want to put the equation in the form
STotal(0,t) = 2smcos(ω*t)cos(Δωt)
You know that
S(x,t)= 2cos(4x-3t)
and
S’(x,t)= 2cos(6x-9t)
So therefore
ω = 3 Hz
and
ω’ = 9 Hz

17. SOLUTION
Therefore find
!
ω* = (ω1 + ω2)/2 (Mean angular frequency)
&
Δω = (ω1 - ω2)/2 (Angular frequency difference)
ω* = (3+9)/2
= 6 Hz
!
Δω = (3-9)/2
= -3 Hz

18. SOLUTION
At t = 5 minutes
t = 5minutes * 60seconds/1minute
t = 300s
STotal(0,t) = 2smcos(ω*t)cos(Δωt)
STotal(0,t) = 2(2)cos(6(300))cos(-3(300))
The tone you heard had an amplitude of
-0.263 nm

19. END OF BEATS LO
Graphing done at Desmos Online Graphing Calculator