Sound
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A longitudinal
pressure wave.
Requires a
medium.
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„
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D( x, t ) = DM sin (k ′x − ωt )
Pressure changes are the result of a volume change.
ΔP = − B
ΔV
V
ΔP = − B
A ΔD( x, t )
A Δx
Solid, liquid, or gas
We can “sense”
sound.
‰
Sound
We know the displacement is sinusoidal.
Discern properties
Hearing
Other?
Sound Properties
Displacement is sinusoidal.
D( x, t ) = DM sin (k ′x − ωt )
Pressure is sinusoidal but 90o out of phase.
ΔP = −ΔPM cos(k ′x − ωt )
Speed of sound.
v= B
ρ ≈ 343 m / s in air
lim ⇒ ΔP = − B
Δx →0
B = Bulk Modulus
∂D( x, t )
= − Bk ′DM cos(k ′x − ωt )
∂x
ΔP = −ΔPM cos(k ′x − ωt )
ΔPM = Bk ′DM = ρv 2 k ′DM = 2πρvfDM
Sound Intensity
Intensity is a measure of the Power Flux of sound, or in other words,
the Power per Unit Area (W/m2).
Intensity is measured in bel’s (after Alexander Graham Bell) or more
precisely, decibels dB (1/10 of a bel)
β (in dB) = 10 log10
I
Io
Where Io is the threshold of hearing for the average person = 1.0 x 10-12 W/m2
β = 0 ⇒ threshold of hearing
Pitch is a measure of frequency
Loudness is a measure of Intensity
Δβ = 10 ⇒ Intensity has increased by a factor of 10
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Sound Intensity
Standing Sound Waves
A telephone receiver is at your left ear, with the sound of the conversation at a
level of 58 dB. At the same time, your little sister’s screams, at a level of 93
dB, enter your right ear. What is the ratio of the sound intensities that enter
your two ears?
Standing Sound Waves
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Standing Sound Waves
Fourier Decomposition of Waves
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Fourier’s theorem:
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any periodic wave can be
approximated by a sum of
sinusoidal waves with
different frequencies
any pulse can be
approximated by an
integral of sinusoidal waves
At a science museum there is a display called sewer pipe symphony. It consists
of several plastic pipes of various lengths which are open at both ends.
A) If the pipes have lengths of 3.0 m, 2.5 m, 2.0 m, 1.5 m, and 1.0 m, what
frequencies will be heard by a visitor’s ear place near the ends of the pipes?
B) Why does the display work better on a noisy day rather than a quit day?
Example: Triangular Wave
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Left: The first two terms in the
Fourier expansion
Right: The first term and the sum
of the first two terms compared
to the original waveform
General form:
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