L-5
Sound
1.16.1 Introduction and Classification:
Sound waves are mechanical, compression waves which are in general longitudinal in
nature-meaning that the particles vibrate parallel to the direction of the wave’s velocity. Sound
waves are divided into three categories that cover different frequency ranges.
(1) Audible waves
They within the range of sensitivity of the human ear. The range of human hearing
stretches between 20-20000 Hertz. They can be generated in a variety of ways, such as by
musical instruments human voices, or loud speakers
(2) Infrasonic waves
These waves have frequencies below the audible range, that is less than 20 Hertz.
Elephants can use infrasonic waves to communicate with each other, even when separated by
many kilometers.
(3) Ultrasonic waves
They have frequencies above the audible range, that is greater than 20000 Hertz. Some
animals can emit these sounds. Bats, for example, emit and hear ultrasound waves, which they
use for locating prey and for navigating.
1.16.2 Speed of sound waves
The speed of sound waves in a medium depends on the compressibility and density of
the medium.
The speed of all mechanical waves follows an expression of the general form
v
elastic property
inertial property
The speed of sound also depends on the temperature of the medium. For sound traveling
through air, the relationship between wave speed and medium temperature is
v  (331m / s) 1 
Tc
273C
where 331m/s is the speed of sound in air at 0°C and the T c is the air temperature in degree
Celsius.
1.16.3 Decibel (dB) scale
The range of sound powers and sound pressures is very wide. In order to cover this wide
range while maintaining accuracy, the logarithmic decibel (dB) scale was selected. The intensity
Properties of Matter and Sound
1.2
of the faintest sound that the normal person can hear is about 0.0000000000001 watts/m2, while
the intensity of the sound produced by a Saturn rocket at liftoff is greater than 100,000,000 watts/
m2. This is a range of 100,000,000,000,000,000,000. Given this extremely large range in values,
there needed to be a better way to express or represent these numbers. By using logarithms of
these numbers, as compared to a reference value, we can form a new measurement scale in which
an increase of 1.0 represents a tenfold increase in the ratio (also called a 1.0 bel increase). The
application of logarithms is evolved to the use of 10 subdivisions of a log value.Decibels is
abbreviated to the term dB. The lower case “d” represents deci, or 1/10th of a bel. The capital
“B” stands for bel, named after Alexander Graham Bell, inventor of the telephone.
Decibel isa dimensionless unit related to the logarithm of the ratio of a measured quantity to a
reference quantity.
Sound power level is the acoustical power radiated by a source with respect to the
standard reference of 10-12 watts.
Lw = 10 Log (W/Wre)
The international reference for power is 10-12 watts. Now because we are converting a
sound power into a Level, or dB, the formula is as shown above. The term Lw is used to represent
the Sound Power Level. The “w” subscript identifies the fact this equation deals with power in
units of watts.
1.16.4 Doppler Effect
The Doppler effect is a phenomenon observed whenever the source of waves is moving
with respect to an observer. The Doppler effect can be defined as the effect produced by a
moving source of waves in which there is an apparent upward shift in frequency for the observer
and the source are approaching and an apparent downward shift in frequency when the observer
and the source are receding.
Although the Doppler effect is most typically experienced with sound waves, it is a
phenomenon that is common to all waves. For example, the relative motion of source and
observer produces a frequency shift in light waves. The Doppler effect is used in police radar
systems to measure the speeds of motor vehicles. Likewise, astronomers use the effect to
determine the speeds of stars, galaxies, and other celestial objects relative to the earth.
1.17
Shockwaves
Definition
Shockwave is a wave formed of a zone of extremely high pressure within a fluid,
especially the atmosphere, that propagates through the fluid at a speed in excess of the speed of
sound.
Types
Properties of Matter and Sound
1.3
Shockwaves in supersonic flow may be classified as normal or oblique according to
whether the orientation of the surface of the abrupt change is perpendicular or at an angle to the
direction of flow
Description
Now consider what happens when the speed vs of a source exceeds the wave speed v.
This situation is depicted graphically in Fig.1.18

Fig.1.18 Shock waves
The circles represent spherical wave fronts emitted by the source at various times during
its motion. At t = 0, the source is at S0 and at a later time t, the source is at Sn. At the time t, the
wave front centered at S0 reaches a radius of vt. In this same time interval, the source travels a
distance vst to Sn. At the instant the source is at Sn, waves are just beginning to be generated at
this location, and hence the wave front has zero radius at this point. The tangent line drawn from
Sn to the wave front centered on S0 is tangent to all other wave fronts generated at intermediate
times. Thus, we see that the envelope of these wave fronts is a cone whose apex half-angle  (the
“Mach angle”) is given by
sin  
vt v

vs t vs
and the conical wave front produced when vs > v (supersonic speeds) is known as a shock wave.
An interesting analogy to shock waves is the V-shaped wave fronts producted by a boat (the bow
wave) when the boat’s speed exceeds the speed of the surface-water waves
Sonic Boom
Properties of Matter and Sound
1.4
Jet airplanes traveling at supersonic speeds produce shock waves, which are responsible
for the loud “sonic boom” one hears. The shock wave carries a great deal of energy concentrated
on the surface of the cone, with correspondingly great pressure variations. Such shock waves are
unpleasant to hear and can cause damage to buildings when aircraft fly supersonically at low
altitudes. In fact, an airplane flying at supersonic speeds produces a double boom because two
shock waves are formed, one from the nose of the plane and one from the tail. People near the
path of the space shuttle as it glides toward its landing point often report hearing what sounds
like two very closely spaced cracks of thunder.
Applications
Shock waves have applications outside of aviation. They are used to break up kidney
stones and gallstones without invasive surgery, using a technique with the impressive name
extracorporeal shock-wave lithotripsy. A shock wave produced outside the body is focused by a
reflector or acoustic lens so that as much of it as possible converges on the stone. When the
resulting stresses in the stone exceed its tensile strength, it breaks into small pieces and can be
eliminated.
1.18
Mach Number
Mach number is a dimensionless measure of relative speed. It is defined as the speed of
an object relative to a fluid medium, divided by the speed of sound in that medium.
M
v
vs
where M is the Mach number, v is the speed of the object relative to the medium and vs is the
speed of sound in the medium.
Mach number is named after Austrian physicist and philosopher Ernst Mach. It can be
shown that the mach number is also the ratio of inertial forces (also referred to aerodynamic
forces).
The square of the Mach number is Cauchy number.
M2 = C, Cauchy number.
High speed flights can be classified in five categories
i.
Sonic
:
ii.
Subsonic
iii.
Transonic :
0.8 < M < 1.2
iv.
Supersonic :
1.2 < M<5
v.
Hypersonic :
M>5
:
M=1
M<1
Properties of Matter and Sound
1.5
For supersonic and hypersonic flows, small disturbances are transmitted downstream
within a cone as shown in Fig.1.19
Properties of Matter and Sound
1.6

Fig.1.19 Shock wave
v

vs
Fig 1.20 Mach angle
The wave front is a cone with angle α called the Mach angle as given in Fig. 1.20.
sin  
v
vs
Mach number M 
:. sin  
vs
v
1
M
The speed of sound depends primarily on the fluid temperature around it and is given as
v
RT
Properties of Matter and Sound
1.7
where T is the temperature (Kelvin), R is the gas constant of fluid and γ is the adiabatic index of
the gas (that is the ratio of specific heats of a gas at constant pressure and volume).
For most calculations, standard air conditions are assumed and a value of γ = 1.4 and
R = 287 J/(kg K) are used.
M 
vs
RT
The Mach number is commonly used both with objects traveling at high speed in a fluid,
and with high speed fluid flows inside channels such as nozzles, diffusers or wind tunnels. At a
temperature of 15 degree Celsius and at sea level, Mach 1 is 340 3m/s(1,225 km/h) in the Earth’s
atmosphere. The speed represented by Mach 1 is not a constant, it is temperature dependent.
Hence in the stratosphere it remains about the same regardless of height, though the air pressure
changes with height.
Since the speed of sound increases as the temperature increases, the actual speed of an
object traveling at Mach 1 will depend on the fluid temperature around it. Mach number is useful
because the fluid behaves in a similar way at the same Mach number. So, an aircraft traveling at
a Mach 1 at sea level will experience shock waves in much the same manner as when it is
traveling at Mach 1 at 11,000 m when it is traveling at Mach 1 at 11,000 m even though it is
traveling at 295 m/s( 1.062 km/h, 86% of its speed at sea level).
Critical Mach number
A critical mach number is the speed of an aircraft (below Mach 1)when the air flowing
over some area of the airfoil has reached the speed of sound. For instance, if the air flowing over
a wing reaches Mach 1 when the wing is only moving at Mach 0.8, then the wing’s critical Mach
number is 0.8.
Mach Tuck
For a subsonic aircraft traveling significantly below Mach 1.0, Mach tuck is an
aerodynamic effect, whereby the nose of an aircraft tends to pitch downwards as the air flow
around the wing reaches supersonic speeds.
Mach meter
A Mach meter is an aircraft instrument that shows the ratio of the speed of sound to the
true airspeed, a dimensionless quantity called Mach number. That is, Mach meter is an aircraft
instrument that indicates speed in Mach numbers.
Worked Example 1.12 An aircraft is flying at speed 370m/s at an attitude where the speed of
sound is 320m/s. Calculate the Mach number
Mach number =
Aircraft speed

370m / s
Speed of sound 320m / s
1.156
Properties of Matter and Sound
1.8
Worked Example 1.13: The Concorde is flying at Mach 1.25 at an altitude where the speed of
sound is 325 m/s. Calculate the speed of the Concorde
Concorde speed = Mach number × speed of sound
= 1.25 × 325m/s = 406.25 m/s
Worked Example 1.14: The Concorde is flying at Mach 1.75 at an altitude of 8000 m, where
the speed of sound is 320 m/s. How long after the plane passes directly
overhead will you hear the sonic boom?
The shockwave forms a cone trailing backward from the airplane, so the
problem is really asking for how much time elapses from when the
Concorde flies overhead to when the shockwave reaches you.
vS = Mach 1.75
vS
8000m
Shock wave

L
a
vst
The angle α of the shock cone is
 = arcsin
1
 34.8 o The speed of the plane is the speed of sound
1.75
multiplied by the Mach number
vs = (1.75) (320 m/s) = 560 m/s
From figure, we have, tan α =
t
8000m

(560m / s) tan 34.8
8000 m
vs t
o  20.5s

Properties of Matter and Sound
1.9
Worked Example 1.15: A plane is flying at supersonic speed at an altitude where the speed of
sound is 320m/s. The shock angle makes an angle of 33.5° with the
direction of the plane. What is the plane’s speed and its Mach number
Mach number 
1
1

1.81
sin  sin 33.5
Aircraft speed = Mach number × speed of sound
= (1.81) ( 320 m/s) = 579.2 m/s
Worked Example 1.16: A sonic boom is heard 20.5s after the Concorde passes overhead.
Assuming the Mach 1.75 and speed of sound is 320 m/s, calculate the
distance traveled by the flight at this time.
Distance traveled = speed of flight × time
= (560 m/s) (20.5s)
= 11500 m
Worked Example 1.17: Determine the velocity of a bullet fired in the air if the Mach angle is
observed to be 30°. Given that the temperature of the air is 22°C
Take γ = 1.4 and R = 287.43 J/kg.K
T = 273.15 +22 = 295.15 K
RT  (1.4) (287.4) (295.15 )
Sonic velocity =
=344.6 m/s
For the Mach cone, Sin α =
1
 0.5  M  2.0
M
 Bullet velocity = (2.0) (344.6 m/s) = 689.2 m/s
Worked Example 1.18: A Mach cone of Mach angle π/6 radian is observed for a fighter
aircraft at an altitude where the temperature is 280K Calculate the
aircraft velocity.
Sonic velocity =
=
RT
(11.4) (287) (280) = 336 m/s
1
 sin   sin ( / 6)  0.5
M
Properties of Matter and Sound
1.10
M=2
Aircraft velocity = (2) (336 m/s) = 772 m/s
Worked Example 1.19: An observer on the ground hears the sonic boom of a plane 15km
above when the plane has gone 20km ahead of him. Estimate the speed
of flight of the plane.
T
20km
Plane

15 km
Observer
The observer hears the sound after the plane has passed over his head
because he must be in the zone of silence before he hears it.
As soon as he falls within the conical zone of awareness, he hears it.In that
position
sin α =
1
M
tan α =
15 km
 0.75
20 km
α = 36.87°
sin 36.87° = 0.6 = 1/M
M = 1.67
The plane must be flying at a supersonic speed corresponding to a
local mach number of 1.67.
Exercise Problem1.5: An aircraft is flying horizontally at Mach 1.8 over a flat desert. A sonic
boom is heard on the ground 8.1s after the aircraft has passed directly
overhead. Assume the speed of sound in the air is 350 m/s. At what
altitude is the aircraft flying?
Hint : Altitude = vst sinα = 283lm
Properties of Matter and Sound
1.11
Exercise Problem 1.6: The speed of the first Indian satellite Aryabhatta at an altitude of
8000m was 1000 m/s. Take R = 287.3J/kg.K, γ = 1.4. Calculate the
Mach number for local conditions at T = 240K and the angle of the
Mach cone.
Hint: Sonic velocity =
M
RT
satellite velocity
 3.22
sonic velocity
sin α = 1/M
α = 18.1°
Exercise Problem1.7: A supersonic fighter plane moves with a Mach number of 1.5 in
atmosphere at an altitude of 500m above the ground level. What is the
time that lapses, by which the acoustic disturbance reaches an observer
on the ground after it is directly overhead? Take T = 20°C, γ = 1.4 and
R = 287 J/kg.K
Hint: Sonic velocity =
Time elapsed = 1.09 s
RT