14.1
WAVE MOTION
E
XERCISES
Section 14.1 Waves and Their Properties
16.
Wave crests (adjacent wavefronts) take a time of one period to pass a fixed point, traveling at the wave speed (or
phase velocity) for a distance of one wavelength. Thus
T
v
=
=
=
λ
/
( .
/ )
.
.
18
5 3
3 40
m/
m s
s
17.
I
NTERPRET
This problem is about wave propagation. Given the speed and frequency of the ripples, we are asked
to compute the period and the wavelength.
D
EVELOP
Equation 14.1 relates the speed of the wave to its period, frequency, and wavelength:
v
T
f
=
=
λ
λ
This is the equation we shall use to solve the problem.
E
VALUATE
Equation 14.1 gives
(a)
T
f
=
=
=
1
1
5 2
0 192
.
.
Hz
s,
and
(b)
λ
=
=
=
v
f
34
5 2
6 54
cm s
Hz
cm.
/
.
.
A
SSESS
The unit of frequency is Hz, with
1
1
1
Hz
s
=
−
.
If the frequency is kept fixed, then increasing the
wavelength will increase the speed of propagation.
18.
From Equation 14.1,
λ
=
=
×
×
=
v f
/
(
/ )/(
.
)
.
.
3
10
88 7
10
3 38
8
6
m s
Hz
m
19.
I
NTERPRET
This problem is about wave propagation. Given the speed and frequency of various electromagnetic
waves, we are asked to compute their wavelength.
D
EVELOP
Equation 14.1 relates the speed of the wave to its period, frequency, and wavelength:
v
T
f
v
f
=
=
→
=
λ
λ
λ
This is the equation we shall use to solve the problem.
E
VALUATE
Since the speed of propagation of electromagnetic waves in vacuum is simply equal to the speed of
light,
v
c
=
=
×
3 0
10
8
.
m/s,
Equation 14.1 gives
(a)
λ
=
=
=
×
c
f
3 10
10
8
6
300
m s
Hz
m
/
;
(b)
λ
=
=
=
×
×
c
f
3 10
190 10
8
6
1 58
m s
Hz
m
/
.
;
(c)
λ
=
=
=
=
×
c
f
3 10
10
8
10
0 03
3
m s
Hz
m
cm
/
.
;
(d)
λ
µ
=
=
=
×
=
×
×
−
c
f
3 10
4 10
6
8
13
7 5
10
7 5
m s
Hz
m
m
/
.
.
;
(e)
λ
=
=
=
×
=
×
×
−
c
f
3 10
6 10
7
8
14
5 0
10
500
m s
Hz
m
nm
/
.
;
(f )
λ
=
=
=
×
=
×
×
−
c
f
3 10
1 0 10
10
8
18
3 0
10
3
m s
Hz
o
m
A
/
.
.
(See Appendix C on units.)
A
SSESS
If the speed of propagation is kept fixed, then a higher frequency means a shorter wavelength.
20.
The wave speed can be calculated from the distance and the travel time, which, together with the frequency and
Equation 14.1, gives a wavelength of
λ
=
=
=
×
×
=
v f
d t f
/
( / )/
(
. )
.
.
1200
5
60
3 1
1 29
km/
km
14
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14.2
Chapter 14
Section 14.2 Wave Math
21.
I
NTERPRET
This problem is about the ultrasound wave. Given its frequency, and wavelength, we want to find its
angular frequency, wave number, and wave speed.
D
EVELOP
The relationships between the speed of the wave, its wave number, frequency, and wavelength are
given by Equations 13.6, 14.1, and 14.2:
f
v
T
f
k
=
=
=
=
ω
π
λ
λ
π
λ
2
2
,
E
VALUATE
(a)
Equation 13.6 gives
ω
π
π
=
=
=
×
−
2
2
4 8
3 02
10
7
1
f
( .
)
.
.
MHz
s
(b)
Equation 14.2 gives
k
=
=
=
×
−
2
2
0 31
4
1
2 03
10
π
λ
π
.
.
.
mm
m
(c)
Using Equation 14.1, the speed of the ultrasound wave is
v
f
=
=
×
×
=
×
−
λ
( .
)( .
)
.
4 8
10
0 31
10
1 49
10
6
3
3
Hz
m
m/s
A
SSESS
The speed of the wave can also be computed as
v
k
=
=
×
×
=
×
−
−
ω
3 02
10
2 03
10
1 49
10
7
1
4
1
3
.
.
.
s
m
m/s
Thus, we see that the pairs
f
,
λ
and
ω
,
k
are equivalent ways to describe the same wave.
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 Fall '10
 Pheong
 Frequency, Wavelength

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