1. Waves and Sound Ch

2. Types of Mechanical Waves
A mechanical wave is a wave that requires a medium
Transverse waves – the displacement of the medium is perpendicular to the direction the wave travels
Longitudinal waves – the motion of the medium is back and forth in the same direction as the wave
Wave speed ≠ particle speed
Waves are disturbances; transport energy (but not matter) from one region to another

3. Periodic Waves Periodic/Sinusoidal wave:
Wave has constant velocity, but every particle undergoes simple harmonic motion
Wavelength – distance from point on one wave shape to identical pt. on the next
Frequency - number of cycles per unit time:
Velocity:
rarefaction
compression λ

4. Mathematical Description
wave function
Wave equation:
Model for any wave (periodic or non-periodic, electromagnetic, sound, on string, in water…)

5. Speed of Transverse Waves and Wave Energy
(speed of wave on string)
F is tension and μ is mass per unit length
Waves transport energy from one place to another
(avg. power of wave on string)
Intensity – the average rate at which energy is transported by the wave, per unit of surface area
(inverse-square law for intensity)

6. Reflection Boundary conditions – whether
the end is fixed (right) or free (left)

7. Interference and Superposition
Interference – two or more waves passing through the same region
Principle of Superposition – when two waves overlap, the displacement at any pt is the sum of the individual waves’ displacements

8. Free Response

9. Standing Waves
Pattern resulting from the combination of the reflected wave and the original wave
Nodes – pts where amplitude never changes
Antinodes – pts where string fluctuates up to the amplitude and then back down with each passing wave body
Does not appear to move, so is called a standing wave (as opposed to a traveling wave – last slide)

10. Constructive
interference
destructive
Standing Waves
Operate based on the principle of superposition (see right)
Nodes occur at x=0, λ/2, λ, 3λ/2,…
Where do antinodes occur?

11. Normal Modes [n=1,2,3,…] (standing wave, fixed at both ends)
L is length of string, n is # of antinodes
[n=1,2,3,…]
f1 is the fundamental frequency
These frequencies are harmonics within the harmonic series
f2 is the 2nd harmonic and 1st overtone; f3 is the third harmonic and the second overtone

12. Free Response
If a string forms a standing wave with 3 antinodes and a wavelength of 4 cm, how long is the string?
A standing wave travels at 5 m/s on a 1 m long string. What is the frequency of the 2nd overtone?

13. Normal Modes (string fixed at both ends)
Normal mode – motion in which all particles in a system move sinusoidally with the same frequency
see previous diagram
not the case for musical instruments
What are timbre and harmonic content?

14. Sound what? Sound!

15. Sound Waves Sound is a longitudinal wave
20 – 20,000 Hz is the audible range
Displacement amplitude – the max. displacement of a particle from its equilibrium position
Useful to describe sound waves in terms of pressure differences
Pressure amplitude = max pressure fluctuation =

16. Speed of Sound In a fluid: In a solid rod: In an ideal gas:
At 20°C, speed of sound in air is 344 m/s
Does sound travel fastest in a gas, a liquid, or a solid? Slowest?

17. Sound Intensity
A low-frequency sound need a large amplitude to have the same intensity as a high-frequency sound (b/c )
Decibel scale
logarithmic scale
+10 dB = x 10 intensity; +20 dB = x 100 intens.

18. Free Response
A concert is recorded as having a decibel level of 170 dB. Since I0 is 10^-12 W/m², what is the intensity of the sound?
If one person speaks at 8x10^-4 W/m² and a second person yells at 4x10^-3 W/m², how many decibels louder is the second?

19. Standing Sound Waves
In same way that transverse standing waves form, longitudinal standing waves can form
Pressure nodes vs. Displacement nodes
Pressure node is a point in a standing sound wave at which pressure and density do not vary
Pressure antinode is a point at which pressure and density vary the greatest a n

20. Open Resonators Open pipe – open at both ends
Displacement nodes are at each end
Ex – organ, flute, recorder, ocarina?

21. Stopped Resonators Stopped pipe – one open end, one closed
Antinode at the open end and node at closed end
Ex – Oboe, clarinet

22. Resonance Similar to concept of driven oscillation
If the frequency of a speaker, voice, etc. matches one of the normal-mode frequencies of resonator (i.e. pipe), then the resonator vibrates with maximum amplitude.
Aretha Franklin example

23. Interference
Constructive interference – two or more waves meet so that the resulting wave is larger than either of the originals (i.e. crest to crest, trough to trough)
Waves are in phase
Destructive Interference – two or more waves seem to cancel each other out (i.e. crest meets trough)
Waves are out of phase

24. Interference
Constructive/ in phase: waves differ by an integer multiple of λ (λ, 2λ, 3λ, …)
Destructive/ out of phase: waves differ by an integer multiple of λ/2 (λ/2, 3λ/2, 5λ/2,…)

25. Free Response
Two loudspeakers are positioned as below. The both produce a frequency of 784 Hz. The speed of sound in air is 344 m/s. a) At what distances from B will there be destructive interference? b) What distances will produce constructive interference? c) If the frequency is made low enough, there will be no positions along the line BC at which destructive interference occurs. How low must this frequency be?

26. Beats
When two sounds destructively interfere slightly out of phase, the resulting superposition (wave) has a different frequency, the beat frequency

27. Doppler Effect
Simply – when a source of sound and a listener are in relative motion, the perceived frequency differs from the actual
Think of a boat moving towards and away from shore
Applies to light as well – red shifting

28. Free Response
A car alarm is emitting sound waves of frequency 520 Hz. You are on a motorcycle, traveling directly away from the car. How fast must you be traveling if you detect a frequency of 490 Hz?