1. (Road to discuss harmony)
Combination of tones
(Road to discuss harmony)
Linear superposition
If two driving forces are applied simultaneously, the response will
be the sum of the responses to the driving forces individually.
For instance: doubling the driving force doubles the response.
In linear systems independent signals do not influence each other.
Linear systems (examples):
Loudspeakers, microphones and amplifiers should be linear to some extent.
Is ear a linear system?
Linear addition of two sound waves:

2. 2. Interference (review)
Interference – combination of waves
(an interaction of two or more waves arriving at the same place)
Important: principle of linear superposition:
No shift or shift by
Shift by
Waves source
(a) If the interfering waves add up so that they reinforce each other, the total
wave is larger; this is called “constructive interference”.
(b) If the interfering waves add up so that they cancel each other, the total
wave is smaller (or even zero); this is called “destructive interference”.

3. Question: Two speakers S1 and S2 are driven by the same signal generator and are different distances from a microphone P as shown.
The minimum frequency for constructive interference to occur at P is __ Hz. (The speed of sound is v = 340 m/s.)
A B C D. 800
Question: Two speakers S1 and S2 are separated by 2.0 m and are driven by the same signal generator. Each speaker radiates sound waves isotropically with wavelength  = 1.5 m. The sound waves from one speaker do not reflect off the other speaker. The number of standing wave antinodes on the line between the two speakers is __. 1 2 3 5

4. 3. Relation between circular motion and simple harmonic oscillations (Review)y
R=A x
linear equation
Waves (Review):

5. 4. Interference between two waves moving in the same direction, the same frequency and wavelengthIf
The sum of two sinusoidal traveling waves with the same frequency and wavelength moving in the same direction is another sinusoidal wave with the same frequency and wavelength moving in the same direction.

6. 5. Beats
Calculations (optional)

7. 6. Slightly mismatched frequencies cause audible “beats”
f1 = 16 Hz
f2 = 18 Hz

8. There is not enough information to choose
Question: The beat frequency between tones with frequencies f1 and f2 is 2.0 Hz. In order to increase the beat frequency, one must __.
increase f1
increase f2
decrease f1
decrease f2
There is not enough information to choose
Amplitude modulation
Nonlinear effects