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17

r chapters; th

Figure 3.1).

Figure 3.2

Copyright 2008 by Taylor and Francis Group, LLC

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18 Industrial Process Sensors

section 3.4.6).

Compression

Waves

Figure 3.1 Te ringing o a mechanical bell creates sound waves.

0 ms

0.73 ms

1.47 ms

2.21 ms

2.94 ms

P(x)

P(x)

P(x)

P(x)

P(x)

0 1 2 Meters

Figure 3.2 Te propagation o pressure waves through air, shown at ve dierent instants in

time. Te dots represent molecules o air, which move orward and backward along the direction

o propagation. Te local compression o air molecules causes variations in the pressure P(x),

which is shown as a unction o position x.


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Sound and Wave Phenomena 19

see chapter 4)

section 3.4.3)

Figure 3.4.

fT

= 1

(a)

( b)

Figure 3.3 wo types o wave: (a) longitudi-

nal; (b) transverse.

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T

(a)

(b)

Time Axis

Position Axis

A

Peak-to-Peak Amplitude

Figure 3.4 A continuous wave at a single requency: (a) the time dependence o a wave mea-

sured at a xed point; the amplitude A and the peak-to-peak amplitudes are shown; the period

T is the time interval between zero crossings at the axis (or equivalently, the interval between

any two successive points o the wave that have the same phase); (b) Te position dependence o

a wave at a single instant in time; the wavelength is the distance between zero crossings at the

axis (or equivalently, the distance between any two successive points o the wave that have the

same phase).

Continuous Wave (CW)

Pulse

Tone Burst

Asymmetric Pulse

Figure 3.5 Four varieties o wave shape, as discussed in the text.


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in chapter 8.

section 3.6)

2

2 2

2

2

1

x c t=

( , ) sin( )x t A kx t = - +

( , ) sin( )x t A kx t = + +

ck

=

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= =2 2fT

k =2

c f=

( , ) ( )x t ei kx t = - +

e ii = +cos sin

( ) ( ) ( )t A e e d i i t=-

12

A e t e di i t( ) ( )( )

= --

12


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It should be evident that the wave shapes shown in Figure 3.5 can be transormed into

Fourier components using equation 3.11; since each o the components is a solution o

the wave equation, it ollows that these other wave shapes are also solutions o the wave

equation.

ck

g =

sed inchapter 8).

z c=

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cn

nc2

1

21=

22

11=

n

n

sin sin 21

21=

n

n

11

2

n1

n2

Nodes

Antinodes

3Beat

(a) (b)

(c) (d)

(e) (f )

Figure 3.6 Generic wave phenomena: (a) reection and reraction; (b) the superposition o two

colliding waves; (c) the intererence o two waves (the dotted and dashed lines) causes cancellation

(solid line) at this moment in time; (d) a standing wave; (e) the beat signal caused by combining

two signals o dierent requencies (as in heterodyning); () the resonance o a cavity.


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cn

n=

arcsin 2

1

= +1 2

sin( ) sin( ) sin( ) [ sin(kx t kx t kx t kx - + - + = - + - -- =t)] 0

sin( ) sin( ) cos sinkx t kx t kx - + - + =

+

22

tt+

2


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sin( ) sin( ) sin si

1 21 22

2t t t+ =

-

nn

1 22

+

t

A B

Dead Zones

Figure 3.7 wo speakers that emit the same tone create an intererence pattern that includes

dead zones in which the sound is reduced.


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fc c

cT v T

c

c v T

c

c vd

d s s s

= =-

=-

=-

( )

1

fs

AB C

Wavefronts

Figure 3.8 A Doppler shif caused by motion o the sourceA moving away rom observer B and

toward observer C.


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fc v

c vfd

d

ss=

+-

fc v

c vfd

d

ss

=-+

f fd s= -+

11

f fc v

cfd s s= - =

-

( )1

f fd s= -12


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Nodal Line

Beam

DiffractedBeam

Aperture

IncidentWave

HuygensWaves

Figure 3.9 Diraction o a wave by an aperture. Wave ronts are represented by solid lines,

and troughs are represented by dashed lines. Many o the Huygens waves have been omitted or

clarity.


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Wavefronts

Huygens Waves

Direction ofPropagation

Figure 3.10 Reconstruction o waveronts by Huygens waves.

Angle

Intensity

Figure 3.11 A representative polar plot. Te

solid line represents the intensity o the wave as

a unction o angle.


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PD D= ( )

=

dP

d0


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D D D = +x x xx

0 0( ) ( )

(

D D DDx x xx

= + +

= DD DD D D D + + +x x xx

xx

) ( ) ( )

0 0

= + + +

0 0 0D D( )

x x

D D= - + -( )0 0

x x

x

(x, t)

(x + x, t)

x

Figure 3.12 Te volume element o air used

in the derivation o the wave equation or sound.

Te dimension along the x axis is greatly exag-

gerated or clarity.


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D D DDPP

xx

P

xx= - = -

( ) ( )

0

2

2( ) ( )D D

D

x t

P

x x= -

0

2

2 0t

P

x x x x = - = - = - -

=D D

0

2

2x

2

2 2

2

2

1

x c t=

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