Ch. 17: Sound Waves Speed of Soundjaszczak/ph2100lect/ph2100-26.pdf · Ch. 18: Superposition Linear...

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1 Ch. 17: Sound Waves Longitudinal (in fluids) Disturbance is: Density, Pressure Particle Displacement from Equilibrium Medium is vibrating particles Frequency Ranges: Audible: 20 Hz to 20 kHz Infrasonic: < 20 Hz Ultrasonic: > 20 kH Speed of Sound density modulus bulk = = - = ρ ρ dV VdP B B v In air: C T v 273 1 m/s 331 + = Pressure and Displacement ) sin( ) , ( max t kx P t x P ω - Δ = Δ ) cos( ) , ( max t kx S t x S ω - = Pressure: Displacement: From the definition of bulk modulus we can show: ΔP max = BkS max = ρωvS max Energy and Intensity E λ = K λ + U λ = ½ρA(ωS max ) 2 λ Power P = E λ = ½ρA( ωS max ) 2 v Area Intensity = Power/Area I = ½ρ( ωS max ) 2 v v P I ρ 2 2 max Δ =

Transcript of Ch. 17: Sound Waves Speed of Soundjaszczak/ph2100lect/ph2100-26.pdf · Ch. 18: Superposition Linear...

Page 1: Ch. 17: Sound Waves Speed of Soundjaszczak/ph2100lect/ph2100-26.pdf · Ch. 18: Superposition Linear Wave Equation: Addition of two or more independent solutions (waves) is also a

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Ch. 17: Sound WavesLongitudinal (in fluids)Disturbance is:

Density, PressureParticle Displacement from Equilibrium

Medium is vibrating particlesFrequency Ranges:

Audible: 20 Hz to 20 kHzInfrasonic: < 20 HzUltrasonic: > 20 kH

Speed of Sound

density

modulusbulk

=

=−≡

=

ρ

ρ

dVVdP

B

Bv

In air:

CTv

�2731m/s331 +=

Pressure and Displacement

)sin(),( max tkxPtxP ω−∆=∆

)cos(),( max tkxStxS ω−=

Pressure:

Displacement:

From the definition of bulk modulus we can show:

∆Pmax = BkSmax = ρωvSmax

Energy and Intensity

Eλ = Kλ + Uλ = ½ρA(ωSmax)2λ

Power P = Eλ/Τ = ½ρA(ωSmax)2v

Area

Intensity = Power/AreaI = ½ρ(ωSmax)2v

vP

Iρ2

2max∆=

Page 2: Ch. 17: Sound Waves Speed of Soundjaszczak/ph2100lect/ph2100-26.pdf · Ch. 18: Superposition Linear Wave Equation: Addition of two or more independent solutions (waves) is also a

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Hearing

At f = 103 Hz:Faintest sound:

∆Pmax ≅ 3×10−5 N/m2

I = 10−12 W/m2

Loudest tolerable sound:∆Pmax ≅ 3×101 N/m2

I = 1 W/m2

NOTE: 12 orders of magnitude!

Decibel Sound Level

���

���≡

0log10 I

Iβ DecibelsdB

I0 = human hearing threshold: 10-12 W/m2

β = 1 dB

Pain threshold: I = 1 W/m2

β = 10 log (1/10-12) = 120 dB

Audio

Example:

What is the difference in intensity between the two sound levels, one dB apart?

Point sound source

Sound radiates equally in all directions unless otherwise specified.

Sound spreads out over SPHERICAL area….

ASPHERE= 4ππππR2

Audio

Page 3: Ch. 17: Sound Waves Speed of Soundjaszczak/ph2100lect/ph2100-26.pdf · Ch. 18: Superposition Linear Wave Equation: Addition of two or more independent solutions (waves) is also a

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Spherical Waves

Power supplied by the source is distributed uniformly over a spherical surface:

24/

rP

API ave

π== I ∝ 1/r2

Since I ∝ S2max Smax ∝ 1/r

Spherical Wave Function: )sin(),( 0 tkrrS

tx ωψ −=Q1

Doppler Effect

Shift in observed frequency due to the motion of the wave source and observer. Q2

���

����

� ±=s

o

vvvv

ff�

'•Use upper signs:

motion is toward each otherfrequency increases

•Use lower signs:motion is away from each otherfrequency decreases

Superposition

Two or more traveling waves in the same medium lead to a resultant wave function that is the algebraic sum of the wave functions of the individual waves.

Linear waves (linear media; wave eqn.)Small amplitudes

�A telling property of waves is interference:�Two waves at the same place at the same

time can add together constructively or destructively.

Constructive: individual displacements in the same direction

Destructive: individual displacements in opposite directions

Page 4: Ch. 17: Sound Waves Speed of Soundjaszczak/ph2100lect/ph2100-26.pdf · Ch. 18: Superposition Linear Wave Equation: Addition of two or more independent solutions (waves) is also a

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Interferenceconstructive destructive

Ch. 18: Superposition

Linear Wave Equation: Addition of two or more independent solutions (waves) is also a solution (wave).

)sin(

)sin(

2

1

φωω

+−=−=

tkxAy

tkxAy

Sum of two waves:

)sin()2/cos(2 221φωφ +−=+ tkxAyy

Resultant wave is:(1) Algebraic sum of all displacements(2) Sinusoidal with the same k and ω(3) Amplitude is 2Acos(φ/2)

Page 5: Ch. 17: Sound Waves Speed of Soundjaszczak/ph2100lect/ph2100-26.pdf · Ch. 18: Superposition Linear Wave Equation: Addition of two or more independent solutions (waves) is also a

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Amplitude: 2Acos(φ/2)

Constructive interference:

φ = 0, 2π, 4π, 6π, etc

Destructive interference:

Maximum amplitude

φ = π, 3π, 5π, etcMinimum amplitude

Sound Interference

Trombone: Path lengths: r1, r2

Phase difference:φ = k|r1-r2| = k∆r

Path difference controlsthe phase difference!

∆r = nλ: constructive

∆r = (2n+1)λ/2: destructive