# Microsoft PowerPoint - Sound Waves by dfhercbml

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```									                                                                          In this lecture
• Sound waves
Sound Waves                                 • Speed of sound
• Acoustic Pressure
• Acoustic Impedance
• Decibel Scale
• Reflection of sound waves
• Doppler effect

Sound Waves                         (Longitudinal Waves)
Sound
Range                 Frequency
Source,                      Direction of propagation
Vibrating surface                                               Audible Range         15 – 20,000Hz
Propagation                                                                   Child’
Child’s hearing       15 – 40,000Hz
of zones of
alternating
-+-+-+-+-+-                                    Male voice            100 – 1500Hz
compression                                                                   Female voice          150 – 2500Hz
and                                                                           Middle C              262Hz
rarefaction
Pressure

Concert A             440Hz
Bat sounds            50,000 – 200,000Hz
Wavelength, λ                               Medical US            2.5 - 40 MHz
Propagation Speed = number of cycles per second X wavelength                  Max sound freq.       600 MHz
c=fλ

B
Speed of Sound                                        c=                Sound Particle Velocity
ρ
• Speed at which longitudinal displacement of                           • Velocity, v, of the particles in the
particles propagates through medium                                     material as they oscillate to and fro

• Speed governed by mechanical properties of                                                          c

medium
v

• Stiffer materials have a greater Bulk modulus
and therefore a higher speed of sound                                 • Typically several tens of mms-1

1
Acoustic Pressure                                       Acoustic Impedance
• Pressure, p, caused by the pressure changes           • Pressure, p, is applied to a molecule it
induced in the material by the sound energy             will exert pressure the adjacent
molecule, which exerts pressure on its
c
P0            P                           • It is this sequence that causes pressure
to propagate through medium.
• p= P-P0 , (where P0 is normal pressure)
P-
• Typically several tens of kPa

Acoustic Impedance                                      Acoustic Impedance
• Acoustic pressure increases with particle
velocity, v, but also depends upon properties        • Acoustic pressure is analogous to electrical
of the medium                                          resistance:
• Relationship between acoustic pressure and
particle velocity is characterised by the
V =I R
acoustic impedance of the medium
p =v Z
p        • Units: kg m-2 s-1            • Z is a constant for a material (resistance, R)
Z =                  or a rayl                     that inhibits velocity (current, I) for a given
v                                         pressure (voltage, V)

Acoustic Impedance
Acoustic Impedance
• Acoustic impedance is also related to the elasticity
of the medium
• Wave propagation speed depends upon
• Stiffer bonds between molecules increases the             elasticity of medium and density:
pressure exerted by a molecule moving with
velocity v.
• A springy material will have high molecular motion
and absorb sound energy in the bonds                             Z = ρc = B ρ
– less energy will be transferred between molecules

B
Z =
v

2
Acoustic Power                                      Acoustic Intensity
•   Sound energy is measured in Joules (J)          • Acoustic Intensity is measured in W cm-2
•   Sound Power in Js-1 or Watts (W)
• Instantaneous power passing through a
•   Again analogous to electricity
unit area of material

P = pv = v Z           2                 • Typical intensities used for ultrasound
imaging are between 0.01 – 1 mW mm-2

Recap Logarithms                                     Decibel Scale
• Comparative sound intensity is measured using
• If   loga(b) = c                                    decibels
• Logarithmic unit used to describe a ratio
• Then c is the power to which you
have to raise a, in order to get b.                                           ⎛ I2 ⎞
dB = 10 log⎜
⎜
⎟
I1 ⎟
⎝ ⎠
• Put more simply,         ac = b                   •Describe very big ratios using modest numbers

Reflection & Transmission of
Example
Sound Waves
• For an incident ultrasound beam of intensity     • A pulse of sound                                    Z1
of 1 Wcm-2 is reflected with an intensity of    incident on an interface             Incident, II
0.1mW cm-2. Express this power loss in dB.      between media with
different mechanical
Reflected, IR
⎛ I2 ⎞                   properties can undergo
dB = 10 log⎜
⎜
⎟
⎟
⎝ I1 ⎠                   two processes
Z2
⎛ 0.0001 ⎞
= 10 log⎜        ⎟                                                     Transmitted, IT
⎝ 1      ⎠              • Transmission or
= 10 × −4                        Reflection
= −40dB

3
Reflection & Transmission of                   Reflection & Transmission of
Sound Waves                                    Sound Waves
• Amount of reflected                      Z1
• Reflected Intensity                        Z1
and transmitted light    Incident, II                                      Incident, II
depends upon
impedance difference
IR = R × II
Reflected, IR                                     Reflected, IR

I I = IT + I R                         Z2         IR ⎛ Z 1 − Z 2 ⎞
2
Z2
R =     =⎜           ⎟
Transmitted, IT           II ⎜ Z1 + Z2 ⎟
⎝           ⎠        Transmitted, IT

Reflection & Transmission of                   Reflection & Transmission of
Sound Waves                                    Sound Waves
• Transmitted Intensity                   Z1                                                Z1
Incident, II
• If no energy is lost to    Incident, II
medium
IT =T × I I
Reflected, IR                                     Reflected, IR

IT    4Z 1Z 2
T +R =1
Z2                                                Z2
T =      =
I I (Z 1 + Z 2 )2     Transmitted, IT                                   Transmitted, IT

Reflection & Transmission of
Impedance Matching
Sound Waves
• Reflection and                                • To optimise                         Probe
transmission of                                  Transmission of US
sound waves                                      into patient from
forms the basis                                  probe an impedance
of ultrasound                                    matching medium is                   Gel
imaging                                          used

Tissue
Z M = ZT × Z p

4
Example

• If a transducer and tissue have acoustic
impedances of 30x106 & 1.5x106 kgm-2s-1
respectively, what acoustic impedance                             Doppler Effect
should a matching medium have to
minimise reflection?

Stationary Sound Source                                        Moving Sound Source
Speed of sound in air
is constant
340ms-1

c         λ                                           c    λ                 λ   c

f=c/λ                  v               f=c/λ
f=c/λ

Moving Sound Source                                            Example
Definitions                        • A police car travelling at 60mph has a
VS : Velocity of Source                                        siren emitting sound with frequency (fs)
(f
(+ve when source is travelling away from listener)
VL : Velocity of Listener                                      300Hz. What frequency would a
(+ve when listener is travelling away from source)
stationary observer measure if the
police car was travelling away from her?
c          λ

v

⎛ c + vL ⎞
⎜c+v ⎟
fL = fS ⎜        ⎟
⎝      s ⎠

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Super-sonic                                                           Super-sonic

“Bang”

v=c                                                                                   v>c

Reflected Sound                                                       Doppler Ultrasound
source

reflector

listener
⎛   2V ⎞                     VR
f L = f S ⎜1 − R ⎟
⎝    c ⎠
Remember: If reflector is moving
Change in frequency,     away from transceiver, shift is –ve
known as the Doppler     (i.e.. Red or down shift in freq). If
Shift, fDoppler          reflector is moving towards
2V     transceiver, shift is +ve (i.e. blue or
f Doppler = f S ⎛ R ⎞
⎜    ⎟   up-shift in freq).
⎝ c ⎠

Summary                                                               Practice Questions
• Sound waves
1. A sound wave propagates at 300ms-1 through a medium with an acoustic pressure of
• Speed of sound                                                          10 pa. Calculate the acoustic impedance of the medium

• Acoustic Pressure                                                    2. A sound wave propagates at 4080ms-1 through a medium with a density of 1700 kgm-3.
Calculate the acoustic impedance of the medium

• Acoustic Impedance
• Reflection of sound waves
• Decibel Scale
• Doppler effect

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