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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

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

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

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

=

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 =

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Various Intensity Levels

•  Threshold of hearing is 0 dB •  Threshold of pain is 120 dB •  Jet airplanes are about 150 dB •  Multiplying a given intensity by 10 adds 10

dB to the intensity level •  See Fig. 14-14, compare to 14-2

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?

Sound vs. Light

•  Sound waves and light waves are not the same thing!

•  You don’t actually hear radio waves, you hear sound waves that were picked up by the radio

•  Ears are sensitive to vibrations/compressions in the air

•  Eyes are sensitive to light waves (but only a very small portion of them…)

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

Directions in the Doppler Effect

•  v = wave speed, u = source/observer speed

•  Motion towards each other (decreasing distance): higher pitch

•  Motion away from each other (increasing distance): lower pitch

!f =

1±uo v( )1 us v( )

f

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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?

Interference & Superposition

•  Two traveling waves can meet and pass through each other without being destroyed or even altered

•  Waves obey the Superposition Principle –  If two or more traveling waves are moving

through a medium, the resulting wave is found by adding together the displacements of the individual waves point by point

– Actually only true for waves with small amplitudes

Constructive Interference

•  Two waves, a and b, have the same frequency and amplitude –  Are in phase

•  The combined wave, c, has the same frequency and a greater amplitude

Constructive Interference in a String

•  Two pulses are traveling in opposite directions •  The net displacement when they overlap is the

sum of the displacements of the pulses •  Note that the pulses are unchanged after the

interference

Destructive Interference

•  Two waves, a and b, have the same amplitude and frequency

•  They are 180° out of phase

•  When they combine, the waveforms cancel

Destructive Interference in a String

•  Two pulses are traveling in opposite directions •  The net displacement when they overlap is

decreased since the displacements of the pulses subtract

•  Note that the pulses are unchanged after the interference

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Reflection of Waves – Fixed End

•  Whenever a traveling wave reaches a boundary, some or all of the wave is reflected

•  When it is reflected from a fixed end, the wave is inverted

•  The shape remains the same

Reflected Wave – Free End

•  When a traveling wave reaches a boundary, all or part of it is reflected

•  When reflected from a free end, the pulse is not inverted

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

Standing Waves on a String

•  Nodes must occur at the ends of the string because these points are fixed

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

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 much closer should they move towards one speaker (either side)?

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

antinode).

Open-open and closed-closed tubes behave in the same way because they both support symmetrical standing waves.

Tube Open at Both Ends (flute)

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,…

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!