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Waves and Sound

- Ch. 15-16

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

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

Mathematical Description

wave function

- Wave equation
- Model for any wave (periodic or non-periodic,

electromagnetic, sound, on string, in water)

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)

Reflection

Boundary conditions whether the end is fixed

(right) or free (left)

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

Free Response

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)

Standing Waves

- Operate based on the principle of superposition

(see right) - Nodes occur at x0, ?/2, ?, 3?/2,
- Where do antinodes occur?

Normal Modes

- n1,2,3, (standing wave, fixed at

both ends) - L is length of string, n is of antinodes
- n1,2,3,
- 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

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?

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?

Sound what? Sound!

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

Speed of Sound

- In a fluid
- In a solid rod
- In an ideal gas
- At 20C, speed of sound in air is 344 m/s
- Does sound travel fastest in a gas, a liquid, or

a solid? Slowest?

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.

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?

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

Open Resonators

- Open pipe open at both ends
- Displacement nodes are at each end
- Ex organ, flute, recorder, ocarina?

Stopped Resonators

- Stopped pipe one open end, one closed
- Antinode at the open end and node at closed end
- Ex Oboe, clarinet

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

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

Interference

- Constructive/ in phase waves differ by an

integer multiple of ? (?, 2?, 3?, )

- Destructive/ out of phase waves differ by an

integer multiple of ?/2 (?/2, 3?/2, 5?/2,)

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?

Beats

- When two sounds destructively interfere slightly

out of phase, the resulting superposition (wave)

has a different frequency, the beat frequency

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

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?