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: Tf /1

fv rarefaction compression

λ

Mathematical Description

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

electromagnetic, sound, on string, in water…)

),( txy

T

txA

2cos

tkxA 2cos

tv

xfA 2cos

tv

xAcos

wave function

222

),(1),(

t

txy

vx

txy

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)

F

v

22

2

1AFPav

21

22

2

1

r

r

I

I

Reflection

Boundary conditions – whetherthe 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 x=0, λ/2, λ, 3λ/2,…

• Where do antinodes occur?

Constructiveinterference

destructiveinterference

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

2

nL

12nf

L

vnfn

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?

TF

Lf

2

11

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 =

BkAp max

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?

B

v

Y

v

M

RTv

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.

22

21 ABI

0log)10( I

IdB

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?

a aa n n

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?

L

nvfn 2

Stopped Resonators

• Stopped pipe – one open end, one closed

• Antinode at the open end and node at closed end

•

• Ex – Oboe, clarinet

L

nvfn 4

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 λ (λ, 2λ, 3λ, …)

• Destructive/ out of phase: waves differ by an integer multiple of λ/2 (λ/2, 3λ/2, 5λ/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

• babeat fff

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

SS

LL f

vv

vvf

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?