Show that f(v)dv = 1 ,where f(v) is the maxwell boltzmann distribution equation
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A) A. 380 kHz
To clerly see the image of the fetus, the wavelength of the ultrasound must be 1/4 of the size of the fetus, therefore
The frequency of a wave is given by
where
v is the speed of the wave
is the wavelength
For the ultrasound wave in this problem, we have
v = 1500 m/s is the wave speed
is the wavelength
So, the frequency is
B) B. f(c+v)/c−v
The formula for the Doppler effect is:
where
f' is the apparent frequency
v is the speed of the wave
is the velocity of the receiver (positive if the receiver is moving towards the source, negative if it is moving away from the source)
is the speed of the source (positive if the source is moving away from the receiver, negative if it is moving towards the receiver)
f is the original frequency
In this problem, we have two situations:
- at first, the ultrasound waves reach the blood cells (the receiver) which are moving towards the source with speed
(positive)
- then, the reflected waves is "emitted" by the blood cells (the source) which are moving towards the source with speed
also
v = c = speed of sound in the blood
So the formula becomes
C. A. The gel has a density similar to that of skin, so very little of the incident ultrasonic wave is lost by reflection
The reflection coefficient is
where Z1 and Z2 are the acoustic impedances of the two mediums, and R represents the fraction of the wave that is reflected back. The acoustic impedance Z is directly proportional to the density of the medium, .
In order for the ultrasound to pass through the skin, Z1 and Z2 must be as close as possible: therefore, a gel with density similar to that of skin is applied, in order to make the two acoustic impedances Z1 and Z2 as close as possible, so that R becomes close to zero.