Sound and ultrasound
Mechanical wave (requires medium, contrast with
EMW)
Longitudinal and transverse wave types can be realized
in solids, however, in liquids and gases only
longitudinal waves can occur (because of the absence
of dilatational forces).
For the description of sound-waves, we use the
pressure-differences (∆p) as a periodic function of
position (x) and time (t).
c is the velocity of sound (not to be confused with the
velocity of light), which depends on the properties of
the transmitting medium (e.g. approx.1500 m/s in water,
330 m/s in air).
Fourier analysis is a decomposition of a waveform
(sound) into the series of harmonics of determined
frequencies, amplitudes and phase.
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Characterization of sound (subjective, objective)
Pitch is a physiological sensation of the highness or
lowness of the note that is a logarithmic function of
frequency
n octave = log 2
f2
f1
Tone is determined by the amplitudes of the
fundamental frequency and harmonics in the Fourier
analysis of the waveform.
Loudness is psychophysical quantity, its value
represents how loud we feel a sound of given intensity
J ~ (∆pmax)2
Audible sound and ultrasound (US) differ merely in
their frequencies. It is usual to regard the bounds of the
range of (human) hearing approximately as 20 Hz and
20 000 Hz.
US-s are the sounds of frequency exceeding 20kHz.
The frequency of ultrasounds used in the medical
diagnostics and therapy is 2-10 MHz.
The velocity of sound is independent of frequency.
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Propagation of sound in media
Velocity of sound
c=
1
ρκ
where ρ is the density of the medium and κ is the
compressibility.
The compressibility (κ) of a medium is defined by:
κ=−
1 ∆V
V ∆p
This quantity gives the relative decrease of the volume
caused by a unit increase of the pressure
Acoustic impedance (acoustic hardness) is a resistance
type of quantity, characterizing the sound conducting
medium:
Z = cρ ,
where c is the velocity of sound in the medium, ρ is the
density of the medium. Z can also be calculated in the
following way:
ρ
Z=
κ
With the aid of the acoustic impedance the average
intensity
J=
1
2
∆pmax
2Z
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Loss of energy during sound propagation
As US energy is lost due to friction and heat generation
the intensity of radiation decreases: absorption. The
general law describing the attenuation of radiation
observed in media holds with a good accuracy:
J = J 0e − µx
In the US diagnostic frequency-range, the absorption
coefficient is approximately proportional to the
frequency: µ ∼ f.
In practical applications, the so called damping (α) is
often used for the characterisation of the decrease of
intensity. This quantity can be given in terms of units of
decibel (dB) by the following formula:
α = 10 lg
J0
J
At the boundary of two domains of different acoustic
impedances, new phenomena may arise such as
reflection or refraction
US diagnostics is based on the reflection of US waves.
Coefficient of reflection (R) is a ratio of the reflected
and incident sound intensity. It can be calculated from
known acoustic impedances of the media as follows: .
2
J R  Z1 − Z 2 

= 
J 0  Z1 + Z 2 
Need connecting medium to avoid “full” reflection
R=
4
Relationship between stimulus and sensation
A stimulus, can be described by more than one physical quantity:
Φ(φ1, φ2, φ3, …)
The components of the psychophysical experience:
Ψ(ψ1, ψ2, ψ3, …)
The relationship between the above quantities can be expressed
mathematically:
Ψ(ψ1, ψ2, ψ3, …) = f [Φ(φ1, φ2, φ3, …)],
where φ1, φ2, φ3, ... and ψ1, ψ2, ψ3, … are the corresponding
parameters of stimulus and sensation, respectively, and f is the
function that gives the connection between them.
Psychophysical laws express a special case of the above general
relation.
Weber-Fechner law
∆ψ = const.
∆φ
φ
The logarithmic relationship called the ‘psychophysical law of
Weber and Fechner’ is derived from the solution of the above
equation:
ψ = const. log
φ
φ0
where φ0 is the absolute threshold-stimulus.
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Stevens-law
∆ψ
ψ
= const.
∆φ
φ
If we solve the equation, we can derive the Stevens-law, which is a
power function:
n
φ 

 φ0  .
ψ = const.
The n exponent is a constant specific to the type of sensation and
φ0 is the absolute threshold-stimulus, as basis for comparison.
If n < 1, the function is called compressive, if n > 1, it is called
expansive (see also the phon and sone scale in the manual:
AUDIOMETRY).
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Audiometry
People with hearing loss have a higher auditory threshold than
healthy people.
auditory threshold audiogram the decrease of hearing is given in
decibels (dB) at different measured frequencies
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