Chapter 10, Waves in solids

Example 1, page 406

In [2]:
from __future__ import division
# Young's modulus of steel
#given data :
p=7.8*10**3 # in kg/m**3
v=5200 # m/s
Y=p*v**2 
print "Young modulus of steel, Y = %0.1e N/m^2 " %Y

Young modulus of steel, Y = 2.1e+11 N/m^2

Example 2, page 406

In [4]:
from math import sqrt
# Velocity and wavelength
#given data :
Y=8*10**10 # in N/m**2
p=5000 # in kg/m**3
v=sqrt(Y/p) 
print "(1) The velocity, v = %0.f m/s " %v
f=400 # in vibration/sec
lamda=v/f 
print "(2) The wavelength = %0.f m " %lamda

(1) The velocity, v = 4000 m/s 
(2) The wavelength = 10 m

Example 3, page 406

In [7]:
# Velocity and wavelength
#given data :
Y=7*10**10 # in N/m**2
p=2.8*10**3 # in kg/m**3
v=sqrt(Y/p) 
print "(1) The velocity, v = %0.e m/s " %v
f=500 # in vibration/sec
lamda=v/f 
print "(2) The wavelength = %0.f m " %lamda

(1) The velocity, v = 5e+03 m/s 
(2) The wavelength = 10 m

Example 4, page 410

In [9]:
# Young's modulus
#given data :
l=3 # in m
n=600 # in Hz
p=8.3*10**3 # in kg/m**3
Y=p*n**2*(2*l)**2 
print "Youngs modulus, Y = %0.3e N/m^2 " %Y

Youngs modulus, Y = 1.076e+11 N/m^2

Example 5, page 411

In [11]:
# Frequency
#given data :
Y=2*10**11 # in N/m**2
p=8*10**3 # in kg/m**3
l=0.25 # in m
n=sqrt(Y/p)/(2*l) 
print "The frequency, n = %0.e vibrations/s " %n

The frequency, n = 1e+04 vibrations/s

Example 6, page 411

In [13]:
# Area of cross section
#given data :
n1BYn2=20 
T=20*9.8 # in N
Y=19.6*10**10 # in N/m**2
alfa=n1BYn2**2*T/Y 
print "Area of cross section, alfa = %0.e m^2  " %alfa

Area of cross section, alfa = 4e-07 m^2

Example 7, page 412

In [14]:
# Velocity and Young modulus
#given data :
n=2600 # in Hz
l=1 # in m
p=7.8*10**3 # kg/m**3
v=2*n*l 
print "The velocity, v = %0.f m/s " % v
Y=v**2*p 
print "Youngs modulus, Y = %0.2e N/m^2 " %Y

The velocity, v = 5200 m/s 
Youngs modulus, Y = 2.11e+11 N/m^2

Example 8, page 412

In [15]:
# Frequencies
#given data :
Y=7.1*10**10 # in N/m**2
p=2700 #in kg/m**3
l=1.5 # in m
r1=1 
r2=3 
r3=5 
n1=(r1/(4*l))*sqrt(Y/p) 
n2=(r2/(4*l))*sqrt(Y/p) 
n3=(r3/(4*l))*sqrt(Y/p) 
print "Frequency of first harmonic, n1 = %0.2f Hz " %n1
print "Frequency of second harmonic, n2 = %0.2f Hz " %n2
print "Frequency of third harmonic, n3 = %0.2f Hz " %n3

Frequency of first harmonic, n1 = 854.67 Hz 
Frequency of second harmonic, n2 = 2564.00 Hz 
Frequency of third harmonic, n3 = 4273.33 Hz

Example 9, page 428

In [17]:
from math import pi
# Frequency
#given data :
l=1.2 # in m
v=5150 # in m/s
d=0.006 # in m
k=d/sqrt(12) 
v1=pi*v*k*3.011**2/(8*l**2) 
print "The frequency, v1 = %0.2f Hz " %v1

The frequency, v1 = 22.05 Hz

Example 10, page 429

In [18]:
from math import pi
# Frequencies
#given data :
l=2 # in m
v=3560 # in m/s
r=0.004 # in m
k=r/2 
v1=pi*v*k*3.011**2/(8*l**2) 
print "The frequency, v1 = %0.2f Hz " %v1
v2=pi*v*k*5**2/(8*l**2) 
print "The frequency of first overtone, v2 = %0.2f Hz" %v2
v3=pi*v*k*7**2/(8*l**2) 
print "The frequency of second overtone, v3 = %0.2f Hz" %v3

The frequency, v1 = 6.34 Hz 
The frequency of first overtone, v2 = 17.48 Hz
The frequency of second overtone, v3 = 34.25 Hz

Example 11, page 429

In [20]:
# Frequency
#given data :
Y=7.1*10**10 # in N/m**2
p=2.7*10**3 # in kg/m**3
r=0.005 # in m
vu=sqrt(Y/p) 
k=r/2 
v=vu/(2*pi*k) 
print "The frequency, v = %0.2e Hz " %v

The frequency, v = 3.26e+05 Hz