Chapter 1: Introduction: Waves and Phasors
Lesson #1
Chapter — Section: Chapter 1
Topics: EM history and how it relates to other fields
Highlights:
• EM in Classical era: 1000 BC to 1900
• Examples of Modern Era Technology timelines
• Concept of “fields” (gravitational, electric, magnetic)
• Static vs. dynamic fields
• The EM Spectrum
Special Illustrations:
• Timelines from CD-ROM
Timeline for Electromagnetics in the Classical Era
ca. 900 Legend has it that while walking 1752 Benjamin Franklin
BC across a field in northern Greece, a (American) invents the
shepherd named Magnus experiences lightning rod and
a pull on the iron nails in his sandals demonstrates that
by the black rock he was standing on. lightning is electricity.
The region was later named Magnesia
and the rock became known as 1785 Charles-Augustin de
magnetite [a form of iron with Coulomb (French) demonstrates that
permanent magnetism]. the electrical force between charges is
proportional to the inverse of the
ca. 600 Greek philosopher Thales square of the distance between them.
BC describes how amber,
after being rubbed 1800 Alessandro Volta
with cat fur, can pick (Italian) develops the
up feathers [static first electric battery.
electricity].
1820 Hans Christian Oersted
ca. 1000 Magnetic compass used as (Danish) demonstrates the
a navigational device. interconnection between
electricity and magnetism
through his discovery that an electric
current in a wire causes a compass
needle to orient itself perpendicular to
the wire.
,2
Lessons #2 and 3
Chapter — Sections: 1-1 to 1-6
Topics: Waves
Highlights:
• Wave properties
• Complex numbers
• Phasors
Special Illustrations:
• CD-ROM Modules 1.1-1.9
• CD-ROM Demos 1.1-1.3
,CHAPTER 1 3
Chapter 1
Section 1-3: Traveling Waves
Problem 1.1 A 2-kHz sound wave traveling in the x-direction in air was observed to
have a differential pressure p x t 10 N/m2 at x 0 and t 50 µs. If the reference
phase of p x t is 36 , find a complete expression for p x t . The velocity of sound
in air is 330 m/s.
Solution: The general form is given by Eq. (1.17),
2t 2x
pxt A cos 0
T
where it is given that 0 36 . From Eq. (1.26), T 1 f 1 2 103 0 5 ms.
From Eq. (1.27),
up 330
0 165 m
f 2 103
Also, since
2 50 10 6
px 0 t 50 µs 10 (N/m2) A cos rad
36
5 10 4 180
A cos 1 26 rad 0 31A
it follows that A 10 0 31 32 36 N/m2. So, with t in (s) and x in (m),
t x
pxt 32 36 cos 2 106 2 103 36 (N/m2)
500 165
3
32 36 cos 4 10 t 12 12x 36 (N/m2)
Problem 1.2 For the pressure wave described in Example 1-1, plot
(a) p x t versus x at t 0,
(b) p x t versus t at x 0.
Be sure to use appropriate scales for x and t so that each of your plots covers at least
two cycles.
Solution: Refer to Fig. P1.2(a) and Fig. P1.2(b).
, 4 CHAPTER 1
p(x,t=0) p(x=0,t)
12. 12.
10. 10.
8. 8.
6. 6.
Amplitude (N/m2)
Amplitude (N/m2)
4. 4.
2. 2.
0. 0.
-2. -2.
-4. -4.
-6. -6.
-8. -8.
-10. -10.
-12. -12.
0.00 k 0.25 k 0.50 k 0.75 k 1.00 k 1.25 k 1.50 k 1.75 k 2.00 k 2.25 k 2.50 0.0 k k 0.2 k k 0.4 k k 0.6 k k 0.8 k k 1.0 k k 1.2 k k 1.4 k k 1.6 k k 1.8 k k 2.0
k 2.75 k 3.00
Time k t k(ms)
Distance kx k(m)
(a) (b)
Figure P1.2: (a) Pressure wave as a function of distance at t
k k k k k k k k k k k 0 and (b) pressure
k k k
wave as a function of time at x
k k 0. k k k k k k
Problem 1.3 A harmonic wave traveling along a string is generated by an oscillator that
k k k k k k k k k k k k k k
completes 180 vibrations per minute. If it is observed that a given crest, or maximum,
k k k k k k k k k k k k k k k
travels 300 cm in 10 s, what is the wavelength?
k k k k k k k k k k
Solution:
180
f 3 Hz k
60
300 cm k
up 0 3 m/s k k
10 s k
up 0 3 0 1 m k
k k 10 cm k
f 3
k
Problem 1.4 Twowaves, y1 t and y2 t , have identical amplitudes and oscillate at
k k k k k k k k k k k k k k
the same frequency, but y2 t leads y1 t by a phase angle of 60 . If
k k k k k k k k k k k k k k k k k
y1 t k 4cos 2 k k 103t
write down the expression appropriate for y2 t and plot both functions over the time
k k k k k k k k k k k k k k
span from 0 to 2 ms.
k k k k k k
Solution:
y2 t k 4cos 2 k k 103t 60
Lesson #1
Chapter — Section: Chapter 1
Topics: EM history and how it relates to other fields
Highlights:
• EM in Classical era: 1000 BC to 1900
• Examples of Modern Era Technology timelines
• Concept of “fields” (gravitational, electric, magnetic)
• Static vs. dynamic fields
• The EM Spectrum
Special Illustrations:
• Timelines from CD-ROM
Timeline for Electromagnetics in the Classical Era
ca. 900 Legend has it that while walking 1752 Benjamin Franklin
BC across a field in northern Greece, a (American) invents the
shepherd named Magnus experiences lightning rod and
a pull on the iron nails in his sandals demonstrates that
by the black rock he was standing on. lightning is electricity.
The region was later named Magnesia
and the rock became known as 1785 Charles-Augustin de
magnetite [a form of iron with Coulomb (French) demonstrates that
permanent magnetism]. the electrical force between charges is
proportional to the inverse of the
ca. 600 Greek philosopher Thales square of the distance between them.
BC describes how amber,
after being rubbed 1800 Alessandro Volta
with cat fur, can pick (Italian) develops the
up feathers [static first electric battery.
electricity].
1820 Hans Christian Oersted
ca. 1000 Magnetic compass used as (Danish) demonstrates the
a navigational device. interconnection between
electricity and magnetism
through his discovery that an electric
current in a wire causes a compass
needle to orient itself perpendicular to
the wire.
,2
Lessons #2 and 3
Chapter — Sections: 1-1 to 1-6
Topics: Waves
Highlights:
• Wave properties
• Complex numbers
• Phasors
Special Illustrations:
• CD-ROM Modules 1.1-1.9
• CD-ROM Demos 1.1-1.3
,CHAPTER 1 3
Chapter 1
Section 1-3: Traveling Waves
Problem 1.1 A 2-kHz sound wave traveling in the x-direction in air was observed to
have a differential pressure p x t 10 N/m2 at x 0 and t 50 µs. If the reference
phase of p x t is 36 , find a complete expression for p x t . The velocity of sound
in air is 330 m/s.
Solution: The general form is given by Eq. (1.17),
2t 2x
pxt A cos 0
T
where it is given that 0 36 . From Eq. (1.26), T 1 f 1 2 103 0 5 ms.
From Eq. (1.27),
up 330
0 165 m
f 2 103
Also, since
2 50 10 6
px 0 t 50 µs 10 (N/m2) A cos rad
36
5 10 4 180
A cos 1 26 rad 0 31A
it follows that A 10 0 31 32 36 N/m2. So, with t in (s) and x in (m),
t x
pxt 32 36 cos 2 106 2 103 36 (N/m2)
500 165
3
32 36 cos 4 10 t 12 12x 36 (N/m2)
Problem 1.2 For the pressure wave described in Example 1-1, plot
(a) p x t versus x at t 0,
(b) p x t versus t at x 0.
Be sure to use appropriate scales for x and t so that each of your plots covers at least
two cycles.
Solution: Refer to Fig. P1.2(a) and Fig. P1.2(b).
, 4 CHAPTER 1
p(x,t=0) p(x=0,t)
12. 12.
10. 10.
8. 8.
6. 6.
Amplitude (N/m2)
Amplitude (N/m2)
4. 4.
2. 2.
0. 0.
-2. -2.
-4. -4.
-6. -6.
-8. -8.
-10. -10.
-12. -12.
0.00 k 0.25 k 0.50 k 0.75 k 1.00 k 1.25 k 1.50 k 1.75 k 2.00 k 2.25 k 2.50 0.0 k k 0.2 k k 0.4 k k 0.6 k k 0.8 k k 1.0 k k 1.2 k k 1.4 k k 1.6 k k 1.8 k k 2.0
k 2.75 k 3.00
Time k t k(ms)
Distance kx k(m)
(a) (b)
Figure P1.2: (a) Pressure wave as a function of distance at t
k k k k k k k k k k k 0 and (b) pressure
k k k
wave as a function of time at x
k k 0. k k k k k k
Problem 1.3 A harmonic wave traveling along a string is generated by an oscillator that
k k k k k k k k k k k k k k
completes 180 vibrations per minute. If it is observed that a given crest, or maximum,
k k k k k k k k k k k k k k k
travels 300 cm in 10 s, what is the wavelength?
k k k k k k k k k k
Solution:
180
f 3 Hz k
60
300 cm k
up 0 3 m/s k k
10 s k
up 0 3 0 1 m k
k k 10 cm k
f 3
k
Problem 1.4 Twowaves, y1 t and y2 t , have identical amplitudes and oscillate at
k k k k k k k k k k k k k k
the same frequency, but y2 t leads y1 t by a phase angle of 60 . If
k k k k k k k k k k k k k k k k k
y1 t k 4cos 2 k k 103t
write down the expression appropriate for y2 t and plot both functions over the time
k k k k k k k k k k k k k k
span from 0 to 2 ms.
k k k k k k
Solution:
y2 t k 4cos 2 k k 103t 60
