# Chapter 8: Ultrasonics¶

## Example 8.1, Page 429¶

In [1]:
#Variable declaration
d = 8e-004;  # Thickness of the piece of piezoelectric crystal, m
v = 5760;  # Velocity of ultrasonic waves in the piece of piezoelectric crystal, m/s

#Calculations
n = v/(2*d);    # The frequency of the fundamental mode of ultrasonic wave, Hz

#Result
print "The frequency of the fundamental mode of ultrasonic wave = %3.1f MHz"%(n/1e+006)


The frequency of the fundamental mode of ultrasonic wave = 3.6 MHz


## Example 8.2, Page 430¶

In [2]:
from math import *

#Variable declaration
d = 2e-003;  # Thickness of the piece of quarts crystal, m
rho = 2650;  # Density of the crystal, kg/meter-cube
Y = 7.9e+010;  # Value of Youngs Modulus, N/metre-square

#Calculations
n = 1/(2*d)*sqrt(Y/rho);    #The frequency of the fundamental mode of vibration, Hz

#Result
print "The frequency of the fundamental mode of vibration in quatrz crystal = %5.3f Hz"%(n/1e+006)

The frequency of the fundamental mode of vibration in quatrz crystal = 1.365 Hz


## Example 8.3, Page 430¶

In [3]:
#Variable declaration
v = 5e+003;  # Velocity of ultrasonic beam in steel plate, m/s
n = 25e+003;  # Difference between two neighbouring harmonic frequencies (Nm - Nm_minus1), Hz

#Calculations
d = v/(2*n);    # The thickness of steel plate, m

#Result
print "The thickness of steel plate = %3.1f m"%d

The thickness of steel plate = 0.1 m


## Example 8.4, Page 430¶

In [4]:
from math import *

#Variable declaration
n = 1e+006;  # Frequency of Ultrasonic waves, Hz
C = 2.5e-014;  # Capcitance of capacitor, F

#Calculations
# Frequency of elecric oscillations is given by n = 1/(2*%pi)*sqrt(1/(L*C)), solving for L
L = 1/(4*pi**2*n**2*C);    # The inductance of an inductor to produce ultrasonic waves, henry

#Result
print "The inductance of an inductor to produce ultrasonic waves = %d henry"%L


The inductance of an inductor to produce ultrasonic waves = 1 henry


## Example 8.5, Page 431¶

In [6]:
#Variable declaration
d = 50e-002;  # Thickness of the metallic rod, m
t1 = 30e-006;  # Arrival time for first pulse, s
t2 = 80e-006;      # Arrival time for second pulse, s

#Calculations&Results
v = 2*d/t2;  # Velocity of ultrasonic waves, m/s
print "The velocity of pulse inside the rod = %4.2e m/s"%v
x = t1*v/2;
print "The position of pulse inside the rod = %6.4f m"%x

The velocity of pulse inside the rod = 1.25e+04 m/s
The position of pulse inside the rod = 0.1875 m


## Example 8.6, Page 431¶

In [8]:
from math import *

#Variable declaration
I = 2.5e+004;  # Sound intensity, W/meter-square
v = 1480;  # Sound velocity, m/s
rho_w = 1000;  # Density of water, kg/meter-cube
rho_c = 2650;  # Density of crystal of transducer, kg/meter-cube
d = 0.001;  # Thickness of the quartz, m
f = 20e+003;  # Frequency of sound in water, Hz

#Calculations&Results
# As sound intensity, I = p^2/(2*rho1*v), solving for p
p = sqrt(2*rho_w*v*I);    # Pressure in the medium, N/metre-square
a = p/(d*rho_c);    # Maximum acceleration of the quartz ultrasonic transducer, metre/second-square
print "The maximum acceleration produced in quartz transducer = %4.2e metre/second-square"%a
y = a/(2*pi*f)**2;    # Maximum displacement of the quartz transducer, m
print "The maximum displacement of quartz transducer = %3.1f micrometer"%(y/1e-006)

The maximum acceleration produced in quartz transducer = 1.03e+05 metre/second-square
The maximum displacement of quartz transducer = 6.5 micrometer


## Example 8.7, Page 432¶

In [9]:
#Variable declaration
L = 0.2;  # Length of a magnetostrictive hydrophone, m
lamda = 2*L;  # Wavelength of ultrasonic wave, m
v = 4900;  # Velocity of ultrasonic beam in water, m/s

#Calculations
f = v/lamda;  # Fundamental frequency of ultrasonic, KHz

#Result
print "The fundamental frequency of a magnetostrictive hydrophone = %4.2f KHz"%(f/1e+03)

The fundamental frequency of a magnetostrictive hydrophone = 12.25 KHz


## Example 8.8, Page 432¶

In [10]:
#Variable declaration
v = 3700;  # Velocity of ultrasonic beam in copper, m/s
t = 1e-006;  # Delay time for ultrasonic beam, s

#Calculations
L = v*t;    # # Length of a copper wire required for a delay, m

#Result
print "The length of a copper wire required for a delay = %6.4f m"%L

The length of a copper wire required for a delay = 0.0037 m