1
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∆=
2
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
3
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
4
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)
5
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