1. Combining Waves
interference
§ 14.7 1

2. Principle of Superposition
Where waves meet, the displacement is the sum of the displacements from the individual waves. 3
Run at: W1 = 0.2; k1 = 0.2; ampl = 15
integral multiples (half, third, quarter) of lambda: multiply W1, k1 by 2, 3, 4
add same-lambda wave with negative amplitude
Beats: slightly vary w2 and k2 together from wave 1 values
Standing waves (use phase velocity of 0.5: k = 2*w)
result –3

3. Interference Constructive: Sum of waves has increased amplitude
Destructive: Sum of waves has decreased amplitude
Two-wave simulation

4. Interference Patterns
Interference of similar wavelengths 4

5. Patterns Positions of constructive and destructive interference
destructive: nodes
constructive: antinodes
Ripple tank simulator

6. waves that don’t actually travel
Standing Waves
waves that don’t actually travel
§ 14.8 6

7. Standing Waves
Sum of waves of equal amplitude and wavelength traveling in opposite directions
Half-wavelength divides exactly into the available space
Wave pattern has locations of minimum and maximum variation (nodes and antinodes)
(standing longitudinal waves)
Run at: w1 = 0.2; k1 = 0.2; ampl = 15
integral multiples (half, third, quarter) of lambda: multiply w1, k1 by 2, 3, 4
add same-lambda wave with negative amplitude
Beats: slightly vary w2 and k2 together from wave 1 values (0.22 and 0.22; 0.21 and 0.21, etc.)
Standing waves (use w of about 0.2; try w = 0.2, k = 0.1) 7

8. standing waves generalized
Normal modes
standing waves generalized 8

9. Modes
Objects have characteristic frequencies at which standing waves are sustained
Lowest frequency = fundamental
Higher frequencies = overtones
Sustained motion is a combination of normal modes 9

10. Vibrational Modes: Clamped String
Insert Figure 15.3 from class text
Source: Griffith, The Physics of Everyday Phenomena, Figure 15.13 10

11. Combinations of Harmonics
Characteristic sounds arise from combining particular harmonics in specific ratios
Fourier analysis suimulation
flute
oboe
saxophone
Simulation

12. “Closed” and “Open” Tube Modes
Source: Halliday, Resnick, and Walker, Fundamentals of Physics, 2003, p 419.

13. Sequence of Harmonics
Western musical scale and harmonies are based on overtone series
(sound files)
Sound files: overtones of open tube or clamped string

14. Circular membrane standing waves
2-D Standing Waves
Nodes are lines or curves
Circular membrane standing waves
edge node only
diameter node
circular node
Source: Dan Russel’s page
Higher frequency  more nodes

15. Aside Electron orbitals in atoms and molecules are 3-D standing waves
All particles have wave natures
Orbitals are interference patterns that persist (don’t cancel over time)
Stationary states are like harmonics

16. Resonance Boundary conditions determine nodal positions
For uniform media, resonant wavelengths and frequencies have simple relationships
Clamped strings
Air cylinders
More complex media are more interesting 16

17. coincidence of similar frequencies
Beats
coincidence of similar frequencies
§ 14.9 17

18. Beats
Waves of similar frequency combine to give alternating times of constructive and destructive interference
Distinctive “waa-waa” sound with beat frequency equal to the difference in frequency of the component waves
fbeat = |f1 – f2|
(Why?)

19. Beats
Sound files
Ripple tank simulator