14
THEORY OF MEASURING ULTRASOUND POWER WITH THE RADIATION FORCE METHOD
Sound is a form of energy that sets the particles in the isonated medium into vibrational motion. The particles then
possess a kinetic energy. If dP
m
is the rate of the flow of this energy about an area dA, then the mean acoustic energy
is:
Eq. 1 I = dP
m
/dA I = Acoustic intensity at a point in that area, Watts/cm
2
When a plane sound wave propagates through a uniform non-absorbent medium, the intensity must be the same for
all points in the wave. Let E represent the energy density, i.e., the energy per unit volume. When the sound energy
passes through a unit cross-sectional area with a speed c, the intensity is:
Eq. 2 I = cE E = Energy density per unit volume, ergs/cm
3
c = Ultrasound wave velocity, cm/sec
The radiation pressure effect can be explained by analogy to the application of an alternating electric voltage to a non-
linear load. With the non-linear load it appears that both AC and DC components are present. In ultrasonics the non-
linear element is the density of the fluid and hence acoustic impedance (load) varies in the same periodical manner as
the density. Therefore in ultrasound the two components of pressure, one alternating and the other direct are present.
The average AC pressure per cycle is zero, but the DC pressure of radiation is:
Eq. 3 P = I/C P = Pressure of Radiation, ergs/cm
3
Therefore, from the above two equations, the pressure of radiation (P) is equal to the energy density (E).
Eq. 4 P = E
It is this DC pressure of radiation that can be measured. At low frequencies, below 100KHz, a standard high fre-
quency hydrophone can be used. For higher frequencies, generally used in medical applications, 1-15 MHz, hydro-
phones are not available. At these frequencies the force can be measured using a precision balance and a radiation
force target that is perfectly absorptive. The conversion from force to power can be accomplished using the equation:
Eq. 5 p = Wgc W = measured force, grams
g = acceleration, dynes
c = velocity of ultrasound, cm/sec
p = power, ergs/sec
By combining all constants together and converting from ergs/sec to watts, we obtain a simplified equation that is used
to calculate the ultrasonic power once the force is measured:
P = w(14.65) P = Ultrasonic power in watts
w = Ultrasonic force in grams
To determine the ultrasonic watt density (watts/cm
2
or watts/in
2
) of a given transducer the P is divided by the cross
sectional area of the transducer.
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