Sound Wave | Physics Optional Notes for UPSC PDF Download

 Page 1


SOUND WAVE 
LECTURE NOTES 
 
Sound is a type of wave that happens when something vibrates, like a guitar string, vocal 
cords, or a tuning fork. These vibrations create waves that move through stuff like air, 
water, or metal. How fast travels sound depends on what it's moving through. When 
sound travels through air, the air wiggles back and forth, changing how packed together 
it is and how much pressure there is. If the thing making the sound vibrates smoothly, 
the changes in pressure also happen smoothly. When we describe these smooth sound 
waves with math, it's a lot like how we describe waves in a vibrating string. 
PRESSURE (FLUCTUATION) WAVES 
 
When a sound wave travels through something, like air or water, the particles in that 
stuff move back and forth. We can talk about this movement in two ways. One way is by 
looking at how much the particles move from their normal spot. We call this the 
displacement wave. The other way is by checking how much the pressure changes 
because of the squeezing and stretching the wave causes. We call this the pressure wave. 
Let's imagine a sound wave traveling in the x-direction through a medium. At a certain 
time, t, let's say a particle at the spot where nothing's happening, x, moves a distance, y, 
in the x-direction because of the wave. The displacement wave then will be described by 
 ?? =?? sin (???? -???? ) 
?? 2
 =?? (?? 1
+??? 1
??) (??)
?? 1
 =?? (?? 1
??) (??)
??? ?? =
(?? 2
-?? 1
)?? ?????
?
??? ?? = lim
??? ?0
?
?? (?? +??? 1
??)-?? (?? 1
??)
??? ?
??? ?? =
??? ??? ???? =
-?? ??? ??? (??)
?? =?? sin (???? -???? ) (??)
??? =?????? cos (???? -???? ) (??)
???? =??? ?? cos (???? -???? ) (??)
 
? ?P
in
 is amplitude of excess pressure. 
? A sound wave may be considered as either a displacement wave ?? =?? sin (???? -
???? ) or a pressure wave P=P
0
cos ( ot -kx) . 
? The pressure wave is 90
°
 out of phase with respect to displacement wave, i.e, 
displacement will be maximum when pressure is minimum and vice-versa. 
? The amplitude of pressure wave: ?? 0
=?????? =?????? ?? 2
=?? va [???? ?? =v?? /?? ,?? =
?? /?? ] 
Page 2


SOUND WAVE 
LECTURE NOTES 
 
Sound is a type of wave that happens when something vibrates, like a guitar string, vocal 
cords, or a tuning fork. These vibrations create waves that move through stuff like air, 
water, or metal. How fast travels sound depends on what it's moving through. When 
sound travels through air, the air wiggles back and forth, changing how packed together 
it is and how much pressure there is. If the thing making the sound vibrates smoothly, 
the changes in pressure also happen smoothly. When we describe these smooth sound 
waves with math, it's a lot like how we describe waves in a vibrating string. 
PRESSURE (FLUCTUATION) WAVES 
 
When a sound wave travels through something, like air or water, the particles in that 
stuff move back and forth. We can talk about this movement in two ways. One way is by 
looking at how much the particles move from their normal spot. We call this the 
displacement wave. The other way is by checking how much the pressure changes 
because of the squeezing and stretching the wave causes. We call this the pressure wave. 
Let's imagine a sound wave traveling in the x-direction through a medium. At a certain 
time, t, let's say a particle at the spot where nothing's happening, x, moves a distance, y, 
in the x-direction because of the wave. The displacement wave then will be described by 
 ?? =?? sin (???? -???? ) 
?? 2
 =?? (?? 1
+??? 1
??) (??)
?? 1
 =?? (?? 1
??) (??)
??? ?? =
(?? 2
-?? 1
)?? ?????
?
??? ?? = lim
??? ?0
?
?? (?? +??? 1
??)-?? (?? 1
??)
??? ?
??? ?? =
??? ??? ???? =
-?? ??? ??? (??)
?? =?? sin (???? -???? ) (??)
??? =?????? cos (???? -???? ) (??)
???? =??? ?? cos (???? -???? ) (??)
 
? ?P
in
 is amplitude of excess pressure. 
? A sound wave may be considered as either a displacement wave ?? =?? sin (???? -
???? ) or a pressure wave P=P
0
cos ( ot -kx) . 
? The pressure wave is 90
°
 out of phase with respect to displacement wave, i.e, 
displacement will be maximum when pressure is minimum and vice-versa. 
? The amplitude of pressure wave: ?? 0
=?????? =?????? ?? 2
=?? va [???? ?? =v?? /?? ,?? =
?? /?? ] 
? As sound-sensors (e.g. ear or mic) detects pressure changes, description of sound 
as pressure-wave is preferred over displacement wave. 
As the piston oscillates sinusoidal, regions of compression and rarefaction are 
continuously set up. The distance between two successive compressions (or two 
successive rarefactions) equals the wavelength ?? . As these regions travel through the 
tube, any small element of the medium moves with simple harmonic motion parallel to 
the direction of the wave. If ?? (?? ,?? ) is the position or a small element relative to its 
equilibrium position, We can express this harmonic position function as 
?? (?? .??)=?? ina
cos (???? -???? ) 
Where s
max
 is the maximum position of the element relative to equilibrium. This is often 
called the displacement amplitude of the wave. The parameter ?? is the wave number and 
0 is the angular frequency of the piston. Note that the displacement of the element is 
along ?? , in the direction of propagation of the sound wave, which means we are 
describing a longitudinal wave. 
(b) 
 
 
Figure: (a) Displacement amplitude and (b) pressure amplitude versus position for a 
sinusoidal longitudinal wave 
EQUATION OF SOUND WAVES 
Sinusoidal wave equation: 
s=s
0
sin (???? ±kx±Ø) s= Displacement of particle present at (x,0) in x direction 
 
Point (A) 
? Density, pressure (max.) 
? Compression (max.) 
? Maximum compressive stress 
Page 3


SOUND WAVE 
LECTURE NOTES 
 
Sound is a type of wave that happens when something vibrates, like a guitar string, vocal 
cords, or a tuning fork. These vibrations create waves that move through stuff like air, 
water, or metal. How fast travels sound depends on what it's moving through. When 
sound travels through air, the air wiggles back and forth, changing how packed together 
it is and how much pressure there is. If the thing making the sound vibrates smoothly, 
the changes in pressure also happen smoothly. When we describe these smooth sound 
waves with math, it's a lot like how we describe waves in a vibrating string. 
PRESSURE (FLUCTUATION) WAVES 
 
When a sound wave travels through something, like air or water, the particles in that 
stuff move back and forth. We can talk about this movement in two ways. One way is by 
looking at how much the particles move from their normal spot. We call this the 
displacement wave. The other way is by checking how much the pressure changes 
because of the squeezing and stretching the wave causes. We call this the pressure wave. 
Let's imagine a sound wave traveling in the x-direction through a medium. At a certain 
time, t, let's say a particle at the spot where nothing's happening, x, moves a distance, y, 
in the x-direction because of the wave. The displacement wave then will be described by 
 ?? =?? sin (???? -???? ) 
?? 2
 =?? (?? 1
+??? 1
??) (??)
?? 1
 =?? (?? 1
??) (??)
??? ?? =
(?? 2
-?? 1
)?? ?????
?
??? ?? = lim
??? ?0
?
?? (?? +??? 1
??)-?? (?? 1
??)
??? ?
??? ?? =
??? ??? ???? =
-?? ??? ??? (??)
?? =?? sin (???? -???? ) (??)
??? =?????? cos (???? -???? ) (??)
???? =??? ?? cos (???? -???? ) (??)
 
? ?P
in
 is amplitude of excess pressure. 
? A sound wave may be considered as either a displacement wave ?? =?? sin (???? -
???? ) or a pressure wave P=P
0
cos ( ot -kx) . 
? The pressure wave is 90
°
 out of phase with respect to displacement wave, i.e, 
displacement will be maximum when pressure is minimum and vice-versa. 
? The amplitude of pressure wave: ?? 0
=?????? =?????? ?? 2
=?? va [???? ?? =v?? /?? ,?? =
?? /?? ] 
? As sound-sensors (e.g. ear or mic) detects pressure changes, description of sound 
as pressure-wave is preferred over displacement wave. 
As the piston oscillates sinusoidal, regions of compression and rarefaction are 
continuously set up. The distance between two successive compressions (or two 
successive rarefactions) equals the wavelength ?? . As these regions travel through the 
tube, any small element of the medium moves with simple harmonic motion parallel to 
the direction of the wave. If ?? (?? ,?? ) is the position or a small element relative to its 
equilibrium position, We can express this harmonic position function as 
?? (?? .??)=?? ina
cos (???? -???? ) 
Where s
max
 is the maximum position of the element relative to equilibrium. This is often 
called the displacement amplitude of the wave. The parameter ?? is the wave number and 
0 is the angular frequency of the piston. Note that the displacement of the element is 
along ?? , in the direction of propagation of the sound wave, which means we are 
describing a longitudinal wave. 
(b) 
 
 
Figure: (a) Displacement amplitude and (b) pressure amplitude versus position for a 
sinusoidal longitudinal wave 
EQUATION OF SOUND WAVES 
Sinusoidal wave equation: 
s=s
0
sin (???? ±kx±Ø) s= Displacement of particle present at (x,0) in x direction 
 
Point (A) 
? Density, pressure (max.) 
? Compression (max.) 
? Maximum compressive stress 
??? ?? =-
 coefficient of ?? coefficient of ?? 
?
ds
dt
=-v
?? (
?s
?x
) 
? v
p
=-v
?? (
?S
?x
) 
Point (B) 
? Density, pressure (min.) 
? Rare faction (max.) 
? Maximum tensile stress. 
Variation of excess pressure in gas due to propagation of longitudinal wave: 
s is displacement of particle at x. 
s+ds is displacement of particle at x+dx 
Change in volume of element =?v 
Initial volume =Adx 
Final volume =A(ds+dx) 
?y=v
i
-v
i
? Ads 
 
?v
v
=
?s
?x
 
?? =
-??? ??? /?? =-
?? ??? /?? 
?P=P=-B(
?s
?x
) P= Excess pressure ; B= bulk mod. Of medium 
Consider s=s
0
sin (?? t-kx) 
P=-Bs
0
cos (?? t-kx)(-k) 
P=Bks
0
cos (?? t-kx)=Pcos (?? t-kx) 
Page 4


SOUND WAVE 
LECTURE NOTES 
 
Sound is a type of wave that happens when something vibrates, like a guitar string, vocal 
cords, or a tuning fork. These vibrations create waves that move through stuff like air, 
water, or metal. How fast travels sound depends on what it's moving through. When 
sound travels through air, the air wiggles back and forth, changing how packed together 
it is and how much pressure there is. If the thing making the sound vibrates smoothly, 
the changes in pressure also happen smoothly. When we describe these smooth sound 
waves with math, it's a lot like how we describe waves in a vibrating string. 
PRESSURE (FLUCTUATION) WAVES 
 
When a sound wave travels through something, like air or water, the particles in that 
stuff move back and forth. We can talk about this movement in two ways. One way is by 
looking at how much the particles move from their normal spot. We call this the 
displacement wave. The other way is by checking how much the pressure changes 
because of the squeezing and stretching the wave causes. We call this the pressure wave. 
Let's imagine a sound wave traveling in the x-direction through a medium. At a certain 
time, t, let's say a particle at the spot where nothing's happening, x, moves a distance, y, 
in the x-direction because of the wave. The displacement wave then will be described by 
 ?? =?? sin (???? -???? ) 
?? 2
 =?? (?? 1
+??? 1
??) (??)
?? 1
 =?? (?? 1
??) (??)
??? ?? =
(?? 2
-?? 1
)?? ?????
?
??? ?? = lim
??? ?0
?
?? (?? +??? 1
??)-?? (?? 1
??)
??? ?
??? ?? =
??? ??? ???? =
-?? ??? ??? (??)
?? =?? sin (???? -???? ) (??)
??? =?????? cos (???? -???? ) (??)
???? =??? ?? cos (???? -???? ) (??)
 
? ?P
in
 is amplitude of excess pressure. 
? A sound wave may be considered as either a displacement wave ?? =?? sin (???? -
???? ) or a pressure wave P=P
0
cos ( ot -kx) . 
? The pressure wave is 90
°
 out of phase with respect to displacement wave, i.e, 
displacement will be maximum when pressure is minimum and vice-versa. 
? The amplitude of pressure wave: ?? 0
=?????? =?????? ?? 2
=?? va [???? ?? =v?? /?? ,?? =
?? /?? ] 
? As sound-sensors (e.g. ear or mic) detects pressure changes, description of sound 
as pressure-wave is preferred over displacement wave. 
As the piston oscillates sinusoidal, regions of compression and rarefaction are 
continuously set up. The distance between two successive compressions (or two 
successive rarefactions) equals the wavelength ?? . As these regions travel through the 
tube, any small element of the medium moves with simple harmonic motion parallel to 
the direction of the wave. If ?? (?? ,?? ) is the position or a small element relative to its 
equilibrium position, We can express this harmonic position function as 
?? (?? .??)=?? ina
cos (???? -???? ) 
Where s
max
 is the maximum position of the element relative to equilibrium. This is often 
called the displacement amplitude of the wave. The parameter ?? is the wave number and 
0 is the angular frequency of the piston. Note that the displacement of the element is 
along ?? , in the direction of propagation of the sound wave, which means we are 
describing a longitudinal wave. 
(b) 
 
 
Figure: (a) Displacement amplitude and (b) pressure amplitude versus position for a 
sinusoidal longitudinal wave 
EQUATION OF SOUND WAVES 
Sinusoidal wave equation: 
s=s
0
sin (???? ±kx±Ø) s= Displacement of particle present at (x,0) in x direction 
 
Point (A) 
? Density, pressure (max.) 
? Compression (max.) 
? Maximum compressive stress 
??? ?? =-
 coefficient of ?? coefficient of ?? 
?
ds
dt
=-v
?? (
?s
?x
) 
? v
p
=-v
?? (
?S
?x
) 
Point (B) 
? Density, pressure (min.) 
? Rare faction (max.) 
? Maximum tensile stress. 
Variation of excess pressure in gas due to propagation of longitudinal wave: 
s is displacement of particle at x. 
s+ds is displacement of particle at x+dx 
Change in volume of element =?v 
Initial volume =Adx 
Final volume =A(ds+dx) 
?y=v
i
-v
i
? Ads 
 
?v
v
=
?s
?x
 
?? =
-??? ??? /?? =-
?? ??? /?? 
?P=P=-B(
?s
?x
) P= Excess pressure ; B= bulk mod. Of medium 
Consider s=s
0
sin (?? t-kx) 
P=-Bs
0
cos (?? t-kx)(-k) 
P=Bks
0
cos (?? t-kx)=Pcos (?? t-kx) 
P
0
=Bks
0
 
P
0
= Atmospheric pressure 
Consider a thin disk-shaped element of gas whose circular cross section is parallel to the 
piston in figure. This element will undergo changes in position, pressure, and density as 
a sound wave propagates through the gas. From the definition of bulk modulus, the 
pressure variation in the gas is 
?P=-B
?V
V
i
 
The element has a thickness ??? in the horizontal direction and a cross-sectional area A, 
so its volume is V
i
=A?x. The change in volume ?V accompanying the pressure change is 
equal to A?s, where ??? is the difference between the value os ?? at ?? +??? and the value of 
?? at ?? . Hence, we can express ??? as 
?P=-B
?V
V
i
=-B
A?s
A?x
=-B
?s
?x
. 
As ??? approaches zero, the ratio ??? /??? becomes ??? /??? (The partial derivative indicates 
that we are interested in the variation of ?? with position at a fixed time.) Therefore, 
?P=-B
?s
?x
 
If the position function is the simple sinusoidal function given by Equation, we find that 
?P=-B
?
?x
[s
max 
cos (kx--?? t)]=Bs
max 
ksin (kx-?? t) 
?P=?P
max
sin (kx-(ot) 
Thus we can describe sound waves either in terms of excess pressure (equation 1.1) or in 
terms of the longitudinal displacement suffered by the particles of the medium. 
If  ?? =?? 0
sin ?? (?? -?? /?? ) represents a sound wave where 
s= Displacement of medium particle from its mean position at x, then it can be proved 
that 
P=P
0
sin {w(t-x/v)+?? /2} 
Represents that same sound wave where, ?? is excess pressure at position ?? , over and 
above the average atmospheric pressure and the pressure amplitude P
0
 is given by 
P
0
=
B?? s
0
 V
=BKs
0
 
(B= Bulk modulus of the medium, K= angular wave number) 
Page 5


SOUND WAVE 
LECTURE NOTES 
 
Sound is a type of wave that happens when something vibrates, like a guitar string, vocal 
cords, or a tuning fork. These vibrations create waves that move through stuff like air, 
water, or metal. How fast travels sound depends on what it's moving through. When 
sound travels through air, the air wiggles back and forth, changing how packed together 
it is and how much pressure there is. If the thing making the sound vibrates smoothly, 
the changes in pressure also happen smoothly. When we describe these smooth sound 
waves with math, it's a lot like how we describe waves in a vibrating string. 
PRESSURE (FLUCTUATION) WAVES 
 
When a sound wave travels through something, like air or water, the particles in that 
stuff move back and forth. We can talk about this movement in two ways. One way is by 
looking at how much the particles move from their normal spot. We call this the 
displacement wave. The other way is by checking how much the pressure changes 
because of the squeezing and stretching the wave causes. We call this the pressure wave. 
Let's imagine a sound wave traveling in the x-direction through a medium. At a certain 
time, t, let's say a particle at the spot where nothing's happening, x, moves a distance, y, 
in the x-direction because of the wave. The displacement wave then will be described by 
 ?? =?? sin (???? -???? ) 
?? 2
 =?? (?? 1
+??? 1
??) (??)
?? 1
 =?? (?? 1
??) (??)
??? ?? =
(?? 2
-?? 1
)?? ?????
?
??? ?? = lim
??? ?0
?
?? (?? +??? 1
??)-?? (?? 1
??)
??? ?
??? ?? =
??? ??? ???? =
-?? ??? ??? (??)
?? =?? sin (???? -???? ) (??)
??? =?????? cos (???? -???? ) (??)
???? =??? ?? cos (???? -???? ) (??)
 
? ?P
in
 is amplitude of excess pressure. 
? A sound wave may be considered as either a displacement wave ?? =?? sin (???? -
???? ) or a pressure wave P=P
0
cos ( ot -kx) . 
? The pressure wave is 90
°
 out of phase with respect to displacement wave, i.e, 
displacement will be maximum when pressure is minimum and vice-versa. 
? The amplitude of pressure wave: ?? 0
=?????? =?????? ?? 2
=?? va [???? ?? =v?? /?? ,?? =
?? /?? ] 
? As sound-sensors (e.g. ear or mic) detects pressure changes, description of sound 
as pressure-wave is preferred over displacement wave. 
As the piston oscillates sinusoidal, regions of compression and rarefaction are 
continuously set up. The distance between two successive compressions (or two 
successive rarefactions) equals the wavelength ?? . As these regions travel through the 
tube, any small element of the medium moves with simple harmonic motion parallel to 
the direction of the wave. If ?? (?? ,?? ) is the position or a small element relative to its 
equilibrium position, We can express this harmonic position function as 
?? (?? .??)=?? ina
cos (???? -???? ) 
Where s
max
 is the maximum position of the element relative to equilibrium. This is often 
called the displacement amplitude of the wave. The parameter ?? is the wave number and 
0 is the angular frequency of the piston. Note that the displacement of the element is 
along ?? , in the direction of propagation of the sound wave, which means we are 
describing a longitudinal wave. 
(b) 
 
 
Figure: (a) Displacement amplitude and (b) pressure amplitude versus position for a 
sinusoidal longitudinal wave 
EQUATION OF SOUND WAVES 
Sinusoidal wave equation: 
s=s
0
sin (???? ±kx±Ø) s= Displacement of particle present at (x,0) in x direction 
 
Point (A) 
? Density, pressure (max.) 
? Compression (max.) 
? Maximum compressive stress 
??? ?? =-
 coefficient of ?? coefficient of ?? 
?
ds
dt
=-v
?? (
?s
?x
) 
? v
p
=-v
?? (
?S
?x
) 
Point (B) 
? Density, pressure (min.) 
? Rare faction (max.) 
? Maximum tensile stress. 
Variation of excess pressure in gas due to propagation of longitudinal wave: 
s is displacement of particle at x. 
s+ds is displacement of particle at x+dx 
Change in volume of element =?v 
Initial volume =Adx 
Final volume =A(ds+dx) 
?y=v
i
-v
i
? Ads 
 
?v
v
=
?s
?x
 
?? =
-??? ??? /?? =-
?? ??? /?? 
?P=P=-B(
?s
?x
) P= Excess pressure ; B= bulk mod. Of medium 
Consider s=s
0
sin (?? t-kx) 
P=-Bs
0
cos (?? t-kx)(-k) 
P=Bks
0
cos (?? t-kx)=Pcos (?? t-kx) 
P
0
=Bks
0
 
P
0
= Atmospheric pressure 
Consider a thin disk-shaped element of gas whose circular cross section is parallel to the 
piston in figure. This element will undergo changes in position, pressure, and density as 
a sound wave propagates through the gas. From the definition of bulk modulus, the 
pressure variation in the gas is 
?P=-B
?V
V
i
 
The element has a thickness ??? in the horizontal direction and a cross-sectional area A, 
so its volume is V
i
=A?x. The change in volume ?V accompanying the pressure change is 
equal to A?s, where ??? is the difference between the value os ?? at ?? +??? and the value of 
?? at ?? . Hence, we can express ??? as 
?P=-B
?V
V
i
=-B
A?s
A?x
=-B
?s
?x
. 
As ??? approaches zero, the ratio ??? /??? becomes ??? /??? (The partial derivative indicates 
that we are interested in the variation of ?? with position at a fixed time.) Therefore, 
?P=-B
?s
?x
 
If the position function is the simple sinusoidal function given by Equation, we find that 
?P=-B
?
?x
[s
max 
cos (kx--?? t)]=Bs
max 
ksin (kx-?? t) 
?P=?P
max
sin (kx-(ot) 
Thus we can describe sound waves either in terms of excess pressure (equation 1.1) or in 
terms of the longitudinal displacement suffered by the particles of the medium. 
If  ?? =?? 0
sin ?? (?? -?? /?? ) represents a sound wave where 
s= Displacement of medium particle from its mean position at x, then it can be proved 
that 
P=P
0
sin {w(t-x/v)+?? /2} 
Represents that same sound wave where, ?? is excess pressure at position ?? , over and 
above the average atmospheric pressure and the pressure amplitude P
0
 is given by 
P
0
=
B?? s
0
 V
=BKs
0
 
(B= Bulk modulus of the medium, K= angular wave number) 
{Note from equation (3.1) and (3.2) that the displacement of a medium particle and 
excess pressure at any position are out of phase by 
?? 2
. Hence a displacement maxima 
corresponds to a pressure minima and vice-versa.} 
Example. The equation of a sound wave in air is given by 
?? =0.02sin [3000?? -9?? ], where all variables are in S.I. units. 
(a) Find the frequency, wavelength and the speed of sound wave in air. 
(b) If the equilibrium pressure of air is 1.0×10
5
 N/m
2
, what are the maximum and 
minimum pressures at a point as the wave passes through that point? 
Solution: (a) Comparing with the standard form of a travelling wave 
?? =?? 0
sin [?? (?? -?? /?? )] 
we see that ?? 3000 s
-1
. The frequency is 
C=
?? 2?? =
3000
2?? Hz 
Also from the same comparison, (1)/v=9.0 m
-1
. 
or,  ?? =
?? 9.0?? -1
=
3000 s
-1
9.0 m
-1
 
1000
3
 m/s
-1
 
The wavelength is ?? =
v
f
=
1000/3 m/s
3000/2?? Hz
=
2?? 9
 m 
(b) The pressure amplitude is P
0
=0.02 N/m
2
. Hence, the maximum and minimum 
pressures at a point in the wave motion will be (1.01×10
5
±0.02)N/m
2
. 
Example. A sound wave of wavelength 40 cm travels in air. If the difference between the 
maximum and minimum pressure at a given point is 4.0×10
-3
 N/m
2
, find the 
amplitude of vibration of the particles of the medium. The bulk modulus of air is 1.4×
10
5
 N/m
2
. 
Solution: The pressure amplitude is 
p
0
=
4.0×10
-3
 N/m
2
2
=2×10
-3
 N/m
2
 
The displacement amplitude s
0
 is given by 
Or 
p
0
=Bks
0
s
0
 =
p
0
Bk
=
p
0
?? 2?? B
=
2×10
-3
 N/m
2
×(40×10
-2
 m)
2×?? ×14×10
4
 N/m
2
=
200
7?? A
 =.91×10
-3
 m