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Agenda

•  Today: HW #12 Quiz, power and energy in waves and decibel scale

•  Thursday: Doppler effect, more superposition & interference, closed vs. open tubes

•  Next week: Last HW quiz, post-test, review for final exam!

Chapter 14, Problem 24

A 200 g ball is tied to a string. It is pulled to an angle of 8.00º and released to swing as a pendulum. A student with a stopwatch finds that 10 oscillations take 12.0 s. How long is the string?

f = 12π

gL

Intensity of Sound Waves

•  The average intensity of a wave is the rate at which the energy flows through a unit area, A, oriented perpendicular to the direction of travel of the wave

•  The rate of energy transfer is the power •  Units are W/m2

I = 1AΔEΔt

=PA

Various Intensities of Sound •  Threshold of hearing

–  Faintest sound most humans can hear –  About 1 x 10-12 W/m2

•  Threshold of pain –  Loudest sound most humans can tolerate –  About 1 W/m2

•  The ear is a very sensitive detector of sound waves –  It can detect pressure fluctuations as small as about 3

parts in 1010

Intensity Level of Sound Waves

•  The sensation of loudness is logarithmic in the human hear

•  β is the intensity level or the decibel level of the sound

•  Io is the threshold of hearing, 1.0 x 10-12 W/m2

10 logo

II

β =

Various Intensity Levels

•  Threshold of hearing is 0 dB •  Threshold of pain is 120 dB •  Jet airplanes are about 150 dB •  Table 15.3 lists intensity levels of various

sounds – Multiplying a given intensity by 10 adds 10 dB

to the intensity level

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Spherical Waves •  A spherical wave

propagates radially outward from the oscillating sphere

•  The energy propagates equally in all directions

•  The intensity is

I =Pav

A=Pav

4πr2

Plane Wave

•  Far away from the source, the wave fronts are nearly parallel planes

•  The rays are nearly parallel lines

•  A small segment of the wave front is approximately a plane wave

Intensity of a Point Source

•  Since the intensity varies as 1/r2, this is an inverse square relationship

•  The average power is the same through any spherical surface centered on the source

•  To compare intensities at two locations, the inverse square relationship can be used

21 2

22 1

I rI r

=

Decibel Scale Example

Two people hear a songbird singing. One person, only 1.00 m from the bird, hears the sound with an intensity of 2.80 x 10-6 W/m2. •  What is the decibel level of the sound? •  What intensity is heard by a second

person standing 4.25 m from the bird (assuming no reflected sound).

•  What is the power output of the bird’s song?

The Doppler Effect •  Definition: “The change in wavelength of

radiation due to relative radial motion between the source and the observer.”

•  Radial: Doppler effect only works for line-of-sight motion

•  Relative: Effect will happen whether the source, observer, or both are moving.

Doppler Effect

Demo: http://astro.unl.edu/classaction/animations/light/dopplershift.html

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Directions in the Doppler Effect

•  Motion towards each other (decreasing distance): - direction

•  Motion away from each other (increasing distance): + direction

wave

source

source vv

=Δ

λλ

Example: A dog whistle produces sound waves with a frequency of

30 kHz. If the human range of hearing begins around 20 kHz, how

fast would someone have to run with a dog whistle for a human

being to be able to hear it? Should they run towards the listener or

away from them?

Agenda

•  Today: More superposition & interference, beats

•  Tuesday: HW #13 quiz, post-test •  Next Thursday: Review for final exam

Standing Waves on a String

•  The lowest frequency of vibration (b) is called the fundamental frequency

•  ƒ1, ƒ2, ƒ3 form a harmonic series

ƒn= mƒ

1=m2L

Fµ

Standing Sound Waves

There are three categories of standing sound waves, depending on their structure: •  Open-open (both ends are nodes) •  Closed-closed (both ends are antinodes) •  Open-closed (one end is a node, other is

an antinode).

Tube Open at Both Ends (flute)

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Harmonics in Open-open or Closed-closed tube

•  In a pipe open at both ends, the natural frequency of vibration forms a series whose harmonics are equal to integral multiples of the fundamental frequency

•  This also works for closed-closed tubes!

ƒm= m v2L

= mƒ1m = 1, 2, 3,…

Tube Closed at One End (clarinet)

Resonance in an Air Column Closed at One End

•  The closed end must be a node •  The open end is an antinode

•  There are no even multiples of the fundamental harmonic

fm= m v

4L= mƒ

1m = 1, 3, 5,…

The human ear canal is similar to a pipe that is closed at one end (by the eardrum, or tympanic membrane). A typical ear canal has a length of about 2.4 cm. a.  What are the fundamental frequency and

wavelength of the ear canal? b.  Find the frequency and wavelength of the

third harmonic.

Suppose a person had a shorter-than-average ear canal (less than 2.4

cm). How would their fundamental frequency be changed?

A. Their fundamental frequency would be higher than average.

B. Their fundamental frequency would be lower than average.

C. We cannot tell unless we know the circumference of their ear canal as well.

Quality of Sound –Tuning Fork

•  Tuning fork produces only the fundamental frequency

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Quality of Sound – Flute

•  The same note played on a flute sounds different

•  Notice that the second harmonic is very strong

Quality of Sound –Clarinet

•  The fifth harmonic is very strong

•  The first and fourth harmonics are very similar, with the third being close to them

Timbre

•  In music, the characteristic sound of any instrument is referred to as the quality of sound, or the timbre, of the sound

•  The quality depends on the mixture of harmonics in the sound

Interference of Sound Waves •  Constructive interference occurs when

the path difference between two waves’ motion is zero or some integer multiple of wavelengths – path difference: Δd = nλ

•  Destructive interference occurs when the path difference between two waves’ motion is an odd half wavelength – path difference: Δd = (n + ½)λ

Sound Wave Interference Example

Two speakers on opposite ends of a basketball court (28m wide) emit sound waves in phase with a frequency of 171.5 Hz. 1.  If someone stood exactly halfway between

the two speakers, would they hear constructive or destructive interference?

2.  If the person wanted to hear destructive interference, how close could they stand to one speaker (either side)?

Beats •  Beats are variations in loudness due to interference •  Two waves have slightly different frequencies and

the time between constructive and destructive interference alternates

•  The beat frequency equals the difference in frequency between the two sources:

ƒb= ƒ

1− ƒ

2

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Beats Example

The second-chair saxophone player is trying to get in tune, listening to the first-chair player. The first-chair player is playing a note at 440 Hz, while the second-chair player is a little off at 442 Hz. What is the beat frequency?

Doppler Effect and Beats: Speeding!

On a separate sheet of paper, answer the following:

Next week on Thursday we will be doing review for the final exam. Pick 1 or 2 topics from the entire semester that you would most like to review. Keep in mind that the review will likely take the format of a quick reminder of basic principles, an overview of problem-solving strategies specific to that topic, and/or practice problems in small groups. Please pick the TOP 1-2 you want to review!