# Chapter 4: Acoustics and Ultrasonics¶

## Example 4.11.1,Page number 4-17¶

In :
import math

#Given data
d=8900.0                                  #density
Y=20.8*10**10                           #Young's modulus
n=40*10**3                              #frequency of wave
k=1.0                                   #consider 1st harmonic

l=(k/(2*n))*math.sqrt(Y/d)              #arranging formula of natural frequency

print"length =",round(l,4),"meter"

length = 0.0604 meter


## Example 4.12.1,Page number 4-20¶

In :
import math

#Given data

d=2.65*10**3                            #density
Y=8*10**10                               #Young's modulus
n=1*10**6                                #frequency of wave
k=1.0                                   #consider 1st harmonic

t=(k/(2*n))*sqrt(Y/d)                   #arranging formula of natural frequency

print"thickness =",round(t,4),"meter"

thickness = 0.0027 meter


## Example 4.15.1,Page number 4-25¶

In :
import math

#Given Data
l=20                                    #length of room
b=15                                    #bredth of room
h=10                                    #height of room
V=l*b*h                                 #volume of room
a=0.106                                 #absorption coefficient

S=2*(l*b+b*h+h*l)                       #surface area of hall

T=(0.161*V)/(a*S)                       #Reverberation time,using Sabine's formula

print"Reverberation time =",round(T,4),"sec"

Reverberation time = 3.5051 sec


## Example 4.15.2,Page number 4-26¶

In :
import math

#given data

m=1j                                  #original sound intensity
n=1000*1j                              #increased intensity value

l=10*log10(n/m)                         #change in intensity level

print"change in intensity level =",l,"dB"

change in intensity level = (30+0j) dB


## Example 4.15.3,Page number 4-26¶

In :
import math

#given data

S1=220                                  #wall area
a1=0.03                                 #absorption coefficient for the wall
S2=120                                  #floor area
a2=0.8                                  #absorption coefficient for the floor
S3=120                                  #ceiling area
a3=0.06                                 #absorption coefficient for the ceiling
V=600                                   #volume of room

S=S1+S2+S3                              #total surface area

a=(a1*S1+a2*S2+a3*S3)/S                 #average sound absorption coefficient

print"1) average sound absorption coefficient =",round(a,4)

T=(0.161*V)/(a*S)                       #Reverberation time,using Sabine's formula

print"2) Reverberation time =",round(T,4),"sec"

1) average sound absorption coefficient = 0.2387
2) Reverberation time = 0.8798 sec


## Example 4.15.4,Page number 4-27¶

In :
import math

#Given data

V=5500                                  #volume
T=2.3                                   #Reverberation time
S=750                                   #sound absorption coefficient
a=(0.161*V)/(S*T)                       #using Sabine's formula

print"average absorption coefficient =",round(a,4)

average absorption coefficient = 0.5133


## Example 4.15.5,Page number 4-27¶

In :
import math

#given data

l=20                                    #length of room
b=12                                    #bredth of room
h=12                                    #height of room
V=l*b*h                                 #volume of room
S=2*(l*b+b*h+h*l)                       #surface area of hall
T1=2.5                                  #Reverberation time

a=(0.161*V)/(T1*S)                      #using Sabine's formula

print"1) average absorption coefficient =",round(a,4)

a1=0.5                                  #absorption coefficient
T2=2                                    #Reverberation time

S1=(0.161*V/(a1-a))*(1.0/T2-1.0/T1)

print"2) carpet area required =",round(S1,4),"m^2"

1) average absorption coefficient = 0.1486
2) carpet area required = 131.958 m^2


## Example 4.15.6,Page number 4-28¶

In :
import math

#given data

Ac=10*12                                #area of carpet covering entire floor
ac=0.06                                 #absorption coefficient of carpet

aS1=Ac*ac                               #absorption due to carpet

Af=10*12                                #area of false celling
af=0.03                                 #absorption coefficient of celling

aS2=Af*af                               #absorption due to celling

As=100*1                                #area of cushioned sets
a_cush=1                                    #absorption coefficient of cushion sets

aS3=As*a_cush                               #absorption due to cusion sets

Aw=346*1                                #area of walls covered with absorbent
aw=0.2                                  #absorption coefficient of walls

aS4=Aw*aw                               #absorption due to walls

Ad=346*1                                #area of wooden door
ad=0.2                                  #absorption coefficient of wooden door

aS5=Ad*ad                               #absorption due to wooden door

aS=aS1+aS2+aS3+aS4                      #total absorption

ap=0.46                                 #absorption coefficient of audience/person
l=12                                    #assuming length of wall
b=10                                    #assuming breadth of wall
h=8                                     #assuming height of wall

V=l*b*h                                 #volume of hall

#case 1 :(no one inside/emptey  hall)

T1=(0.161*V)/aS                         #reverberation time

print" 1)reverberation time of empty hall =",round(T1,4),"sec"

#case 2 :(50 person inside hall)

T2=(0.161*V)/(aS+50*0.46)               #reverberation time

print" 2)reverberation time of hall with 50 person =",round(T2,4),"sec"

#case 2 :(100 person inside hall/full capacity of hall)

T3=(0.161*V)/(aS+100*0.46)              #reverberation time

print" 3)reverberation time of hall with 100 person =",round(T3,4),"sec"

 1)reverberation time of empty hall = 0.8587 sec
2)reverberation time of hall with 50 person = 0.7614 sec
3)reverberation time of hall with 100 person = 0.6839 sec


## Example 4.15.7,Page number 4-30¶

In :
import math

#given data

l=20                                    #length of room
b=15                                    #bredth of room
h=5                                     #height of room

V=l*b*h                                 #volume of room
S=2*(l*b+b*h+h*l)                       #surface area of hall

T=3.5                                   #Reverberation time

a=(0.161*V)/(T*S)                       #using Sabine's formula

print"1) average absorption coefficient =",round(a,4)

avg=a*S                                 #average total absorption

print"2) average total absorption =",round(avg,4),"m^2.s"

1) average absorption coefficient = 0.0726
2) average total absorption = 69.0 m^2.s


## Example 4.15.8,Page number 4-30¶

In :
import math

#given data

l=20                                    #length of room
b=15                                    #bredth of room
h=10                                    #height of room

V=l*b*h                                 #volume of room

a=0.1                                   #absorption coefficient

S=2*(l*b+b*h+h*l)                       #surface area of hall

T1=(0.161*V)/(a*S)                      #Reverberation time,using Sabine's formula

print"1) Reverberation time =",round(T1,4),"sec"

a2=0.66                                 #absorption coefficient of curtain cloth

S2=100                                  #surface area of a curtain cloth

T2=(0.161*V)/(a*S+a2*S2*2)              #Reverberation time,using Sabine's formula

T=T1-T2                                 #change in Reverberation time

print"2) change in Reverberation time =",round(T,4),"sec"

1) Reverberation time = 3.7154 sec
2) change in Reverberation time = 1.8719 sec


## Example 4.15.9,Page number 4-30¶

In :
import math

#given data

S1=220                                  #wall area
a1=0.03                                 #absorption coefficient for the wall
S2=120                                  #floor area
a2=0.8                                  #absorption coefficient for the floor
S3=120                                  #ceiling area
a3=0.06                                 #absorption coefficient for the ceiling
V=600                                   #volume of room

S=S1+S2+S3                              #total surface area
a=(a1*S1+a2*S2+a3*S3)/S                 #average sound absorption coefficient

print"1) average sound absorption coefficient =",round(a,4)

T=(0.161*V)/(a*S)                       #Reverberation time,using Sabine's formula

print"2) Reverberation time =",round(T,4),"sec"

1) average sound absorption coefficient = 0.2387
2) Reverberation time = 0.8798 sec


## Example 4.15.10,Page number 4-31¶

In :
import math

#given data

f=0.07*10**6                             #frequency
t=0.65                                  #time
v=1700                                  #velocity of sound

d=v*t/2                                 #depth of seabed

print"1) depth of seabed =",round(d,4),"meter"

lamda=v/f                                   #wavelength

print"2) wavelength =",round(lamda,4),"meter"

1) depth of seabed = 552.5 meter
2) wavelength = 0.0243 meter


## Example 4.15.11,Page number 4-31¶

In :
import math

#given data

t=1*10**-3                               #thicknesss of crystal
d=2.65*10**3                             #density
Y=8*10**10                               #Young's modulus
k=1                                     #consider 1st harmonic

n=(k/(2*t))*sqrt(Y/d)                   #formula of natural frequency

print" natural frequency =","{0:.3e}".format(n),"Hz"

 natural frequency = 2.747e+06 Hz


## Example 4.15.12,Page number 4-32¶

In :
import math

#given data
d=2650                                  #density
Y=8*10**10                               #Young's modulus
k=1.0                                     #consider 1st harmonic

#case 1

n1=3.8*10**6                             #frequency of wave

t1=(k/(2*n1))*sqrt(Y/d)                 #arranging formula of natural frequency

print"1) thickness =","{0:.3e}".format(t1),"meter"

#case 2

n2=300*10**3                             #frequency of wave

t2=(k/(2*n2))*sqrt(Y/d)                 #arranging formula of natural frequency

print"2) thickness =","{0:.3e}".format(t2),"meter"

1) thickness = 7.230e-04 meter
2) thickness = 9.157e-03 meter


## Example 4.15.13,Page number 4-32¶

In :
import math

#given data

d=2650                                  #density
Y=8*10**10                               #Young's modulus
n=2*10**6                                #frequency of wave
k=1.0                                     #consider 1st harmonic

t=(k/(2*n))*sqrt(Y/d)                   #arranging formula of natural frequency

print"thickness =","{0:.3e}".format(t),"meter"

thickness = 1.374e-03 meter


## Example 4.15.14,Page number 4-33¶

In :
import math

#given data

f=50*10**3                               #frequency
v1=348                                  #velocity of ultrasound in atmosphere
v2=1392                                 #velocity of ultrasound in sea water
t=2.0                                     #time difference

#distance is constant hence v1*t1=v2*t2

m=v2/v1                                 #assuming constant as m

#(t1-t2=d) and (t1=m*t2) therefore

t2=t/(m-1)

d=v2*t2                                 #distance between two ship

print"distance between two ships =",round(d,4),"meter"

distance between two ships = 928.0 meter


## Example 4.15.15,Page number 4-34¶

In :
import math

#given data

#for case1
t1=2*10**-3                              #thicknesss of plate
d=2.65*10**3                             #density
Y=8*10**10                               #Young's modulus
k=1.0                                     #consider 1st harmonic

n1=(k/(2*t1))*sqrt(Y/d)                 #formula of natural frequency

print"1)natural frequency =","{0:.3e}".format(n1),"Hz"

#for case2

n2=3*10**6                               #frequency

t2=(k/(2*n2))*sqrt(Y/d)                 #arranging formula of natural frequency

t=t1-t2                                 #change in thickness

print"2)change in thickness =","{0:.3e}".format(t),"meter"

1)natural frequency = 1.374e+06 Hz
2)change in thickness = 1.084e-03 meter


## Example 4.15.16,Page number 4-34¶

In :
import math

#given data

S=10                                    #salinity
t=2                                     #time
T=20                                    #temperature

v=1510+1.14*S+4.21*T-0.037*T**2          #velocity of ultrasound in sea

d=v*t/2                                 #depth of sea bed

print"depth of sea bed =",round(d,4),"meter"

depth of sea bed = 1590.8 meter


## Example 4.15.17,Page number 4-35¶

In :
import math

#given data

S=29                                    #salinity
t=2                                     #time
l=0.01                                  #wavelength
T=30                                    #temperature

v=1510+1.14*S+4.21*T-0.037*T**2          #velocity of ultrasound in sea

d=v*t/2                                 #depth of sea bed

print"1)depth of sea bed =",round(d,4),"meter"

f=v/l                                   #frequency

print"2) frequency =","{0:.3e}".format(f),"Hz"

1)depth of sea bed = 1636.06 meter
2) frequency = 1.636e+05 Hz


## Example 4.15.18,Page number 4-35¶

In :
import math

#given data

v1=5.9*10**3                             #velocity of UW in mild steel
v2=4.3*10**3                             #velocity of UW in brass
t2=15*10**-3                             #thickness of brass plate

t1=v2*t2/v1                             #since ve;ocity is inversly proportional to thickness

print"real thickness =","{0:.3e}".format(t1),"meter"

real thickness = 1.093e-02 meter


## Example 4.15.19,Page number 4-36¶

In :
import math

#given data

t1=4*10**-3                              #thickness of 1st crystal
n1=400*10**3                             #frequency of 1st crystal
n2=500*10**3                             #frequency of 2nd crystal

t2=n1*t1/n2                             #since frquency is inversly proportional to thickness

print"thickness of 2nd crystal =","{0:.3e}".format(t2),"meter"

thickness of 2nd crystal = 3.200e-03 meter


## Example 4.15.20,Page number 4-36¶

In :
import math

#given data

t2=30*10**-6                             #pulse arrival time of defective steel bar
t1=80*10**-6                             #pulse arrival time of non defective steel bar
d=40*10**-2                              #bar thickness

x=(t2/t1)*d

print"distance at which defect has occurred =",round(x,4),"meter"

distance at which defect has occurred = 0.15 meter


## Example 4.15.21,Page number 4-37¶

In :
import math

#given data

d=18*10**-3                              #thickness
v=5.9*10**3                              #velocity

t=(2*d)/v                               #echo time

print"echo time =","{0:.3e}".format(t),"sec"

echo time = 6.102e-06 sec


## Example 4.15.22,Page number 4-37¶

In :
import math

#given data

t=1*10**-3                               #thickness of quartz crystal

#given t=l/2

l=t*2                                   #wavelength
Y=7.9*10**10                             #young's module of crystal
p=2650                                  #density of crystal

v=sqrt(Y/p)                             #velocity of vibration

n=v/l                                   #frequency of vibration

print"frquency of vibration =","{0:.3e}".format(n),"Hz"

frquency of vibration = 2.730e+06 Hz


## Example 4.15.23,Page number 4-38¶

In :
import math

#given data

d=7.23*10**3                             #density
Y=11.6*10**10                            #Young's modulus
n=20*10**3                               #frequency of wave
k=1.0                                    #consider 1st harmonic

l=(k/(2*n))*sqrt(Y/d)                   #arranging formula of natural frequency

print"length =","{0:.3e}".format(l),"meter"

length = 1.001e-01 meter


## Example 4.15.24,Page number 4-38¶

In :
import math

#given data

#for case1
t1=2*10**-3                              #thicknesss of plate
d=2.65*10**3                             #density
Y=8*10**10                               #Young's modulus
k=1.0                                    #consider 1st harmonic

n1=(k/(2*t1))*sqrt(Y/d)                 #formula of natural frequency

print"1)natural frequency =","{0:.3e}".format(n1),"Hz"

#for case2

n2=3*10**6                               #frequency

t2=(k/(2*n2))*sqrt(Y/d)                 #arranging formula of natural frequency

t=t1-t2                                 #change in thickness

print"2)change in thickness =","{0:.3e}".format(t),"meter"

1)natural frequency = 1.374e+06 Hz
2)change in thickness = 1.084e-03 meter


## Example 4.15.25,Page number 4-39¶

In :
import math

#given data

l=20                                    #length of room
b=15                                    #bredth of room
h=10                                    #height of room

V=l*b*h                                 #volume of room
S=2*(l*b+b*h+h*l)                       #surface area of hall

T=3                                     #Reverberation time

a=(0.161*V)/(T*S)                       #using Sabine's formula

print"1) average absorption coefficient =",round(a,4)

m=a*S                                   #total absorption

print"2) total absorption of surface =",round(m,4),"m**2/sec"

1) average absorption coefficient = 0.1238
2) total absorption of surface = 161.0 m**2/sec


## Example 4.15.26,Page number 4-39¶

In :
import math

#given data

#for case1
t1=1.8*10**-3                            #thicknesss of plate
d=2.65*10**3                             #density
Y=8*10**10                               #Young's modulus
k=1.0                                    #consider 1st harmonic

n1=(k/(2*t1))*sqrt(Y/d)                 #formula of natural frequency

print"1)natural frequency =","{0:.3e}".format(n1),"Hz"

#for case2

n2=2*10**6                               #frequency

t2=(k/(2*n2))*sqrt(Y/d)                 #arranging formula of natural frequency

t=t1-t2                                 #change in thickness

print"2)change in thickness =","{0:.3e}".format(t),"meter"

1)natural frequency = 1.526e+06 Hz
2)change in thickness = 4.264e-04 meter


## Example 4.15.27,Page number 4-39¶

In :
import math

#given data

n=0.4999*10**6                           #frequency
t=5.5*10**-3                             #thicknesss of plate
d=2.65*10**3                             #density
k=1.0                                     #consider 1st harmonic

Y=4*(t**2)*(n**2)*d/k                     #arranging formula of natural frequency

print"Youngs modulus =","{0:.3e}".format(Y),"N/m**2"

Youngs modulus = 8.013e+10 N/m**2

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