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’’s hearing 15 – 40,000Hz of zones of Male voice 100 – 1500Hz alternating ------+ + + + + compression Female voice 150 – 2500Hz and Middle C 262Hz rarefaction Concert A 440Hz Pressure 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 = f λ
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 adjacent molecule.
P0 P • It is this sequence that causes pressure to propagate through medium. • p= P-P0 , (where P0 is normal pressure) • 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 V = I R particle velocity is characterised by the 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 = ρ c = 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 2Z • 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 ⎛ I ⎞ have to raise a, in order to get b. ⎜ 2 ⎟ dB = 10log⎜ ⎟ ⎝ I1 ⎠ c • Put more simply, a = 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 • • Z of 1 Wcm-2 is reflected with an intensity of incident on an interface 1 Incident, II 0.1mW cm-2. Express this power loss in dB. between media with different mechanical Reflected, I ⎛ I ⎞ Reflected, IR ⎜ 2 ⎟ properties can undergo dB = 10log⎜ ⎟ ⎝ I1 ⎠ two processes Z ⎛ 0.0001 ⎞ 2 = 10log⎜ ⎟ Transmitted, I ⎝ 1 ⎠ • Transmission or T = 10 × −4 Reflection = −40dB
3 Reflection & Transmission of Reflection & Transmission of Sound Waves Sound Waves
• Amount of reflected • Reflected Intensity • Z • Z and transmitted light 1 1 Incident, II Incident, II depends upon = × impedance difference IR R II Reflected, IR Reflected, IR
II = IT + IR 2 Z2 I ⎛ Z − Z ⎞ Z2 R = R = ⎜ 1 2 ⎟ Transmitted, I ⎜ ⎟ Transmitted, I T II ⎝ Z1 + Z2 ⎠ T
Reflection & Transmission of Reflection & Transmission of Sound Waves Sound Waves
• Transmitted Intensity • Z Z 1 • If no energy is lost to 1 Incident, II Incident, II medium IT =T × II
Reflected, IR Reflected, IR T + R = 1 IT 4Z1Z 2 Z2 Z2 T = = 2 II ()Z1 + Z 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
v f=c/λ 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 (f ) (+ve when source is travelling away from listener) s
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 + v ⎞ ⎜ L ⎟ f L = fS ⎜ ⎟ ⎝ c + vs ⎠
5 Super-sonic Super-sonic
“Bang”
v = c v > c
Reflected Sound Doppler Ultrasound
source
reflector
listener ⎛ 2V ⎞ V f = f ⎜1 − R ⎟ R L S ⎝ 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 transceiver, shift is +ve (i.e. blue or ⎛ 2VR ⎞ f = f ⎜ ⎟ up-shift in freq). Doppler S ⎝ c ⎠
Summary Practice Questions
• Sound waves 1. A sound wave propagates at 300ms-1 through a medium with an acoustic pressure of 10 pa. Calculate the acoustic impedance of the medium • Speed of sound
2. A sound wave propagates at 4080ms-1 through a medium with a density of 1700 kgm-3. • Acoustic Pressure Calculate the acoustic impedance of the medium • Acoustic Impedance • Reflection of sound waves • Decibel Scale • Doppler effect
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