6.3
6.3
The Universal Wave Equation
If you hold a piece of rope in your hand, you can create a crest along the rope by
moving your hand through one-half of a cycle (Figure 1). When you move your
hand in the opposite direction, a trough is produced that also travels along the
rope behind the crest. If the motion is continued, a series of crests and troughs
moves along the rope at a uniform speed. One cycle of the source produces one
crest and one trough. The frequency of the wave is defined as the number of
crests and troughs, or complete cycles, that pass a given point in the medium per
unit of time (usually 1 s). The frequency of the wave is exactly the same as that
of the source. It is the source alone that determines the frequency of the wave.
Once the wave is produced, its frequency never changes, even if its speed and
wavelength do change. This behaviour is characteristic of all waves.
A
B
C
D
E
A
B
C
D
E
C
D
E
D
E
rest position
end of first quarter-cycle
A
B
B
end of first half-cycle
C
end of three-quarters of a cycle
A
C
D
E
amplitude
end of first cycle
amplitude
B
A
distance travelled by wave during
one cycle or period of the source
Figure 1
Creating a crest and a trough along a string
Vibrations and Waves 209
When a wave is generated in a spring or a rope, the wave travels one wavelength (l) along the rope in the time required for one complete vibration of the
source. Recall that this time is defined as the period (T) of the source. Since
distance
, we can say
speed (v) = time
l
v T
universal wave equation: v f l
or
vfl
This equation is known as the universal wave equation. It applies to all waves,
visible and invisible.
Universal Wave Equation
v = fl
Sample Problem 1
The wavelength of a water wave in a ripple tank is 0.080 m. If the frequency of
the wave is 2.5 Hz, what is its speed?
Solution
l 0.080 m
f 2.5 Hz
v?
vfl
(2.5Hz)(0.080 m)
v 0.20 m/s
The speed of the wave is 0.20 m/s.
Sample Problem 2
The distance between successive crests in a series of water waves is 4.0 m, and the
crests travel 9.0 m in 4.5 s. What is the frequency of the waves?
Solution
d 9.0 m
t 4.5 s
l 4.0 m
f?
v fl
v
f l
2.0 m/s
4.0 m
f 0.50 Hz
The frequency of the waves is 0.50 Hz.
210 Chapter 6
d
v t
9.0 m
4.5 s
v 2.0 m/s
6.3
Sample Problem 3
The period of a sound wave from a piano is 1.18 × 10–3 s. If the speed of the wave
in the air is 3.4 × 102 m/s, what is its wavelength?
Solution
T 1.18 × 10–3 s
v 3.4 × 102 m/s
l?
l
v T
l vT
(3.4 × 102 m/s) (1.18 × 10–3 s)
l 0.40 m
The wavelength is 0.40 m.
Practice
Understanding the Concepts
1. Calculate the speed (in metres per second) of the waves for each of
the following:
(a) f = 18 Hz, l = 2.7 m
(b) f = 2.1 × 104 Hz, l = 2.0 × 105 cm
(c) T = 4.5 × 10–4 s , l = 9.0 × 104 m
(d) T = 2.0 ms, l = 3.4 km
Answers
1. (a) 49 m/s
(b) 4.2 × 107 m/s
(c) 2.0 × 108 m/s
(d) 1.7 × 106 m/s
2. Write an equation for each of the following:
(a) f in terms of v and l
(b) T in terms of v and l
(c) l in terms of v and f
(d) l in terms of v and T
SUMMARY
The Universal Wave Equation
• One vibration of the source produces one complete wavelength.
• The frequency and the period of a wave are the same as those of the
source, and they are not affected by changes in the speed of the wave.
• The universal wave equation, v = f l, applies to all waves.
Section 6.3 Questions
Understanding Concepts
1. A vibrator in a ripple tank with a frequency of 20.0 Hz produces
waves with a wavelength of 3.0 cm. What is the speed of the
waves?
2. A wave in a skipping rope travels at a speed of 2.5 m/s. If the
wavelength is 1.3 m, what is the period of the wave?
3. Waves travel along a wire at a speed of 10.0 m/s. Find the frequency and the period of the source if the wavelength is 0.10 m.
4. A crest of a water wave requires 5.2 s to travel between two
points on a fishing pier located 19 m apart. It is noted in a series
of waves that 20 crests pass the first point in 17 s. What is the
wavelength of the waves?
(continued)
Vibrations and Waves 211
5. The period of a sound wave emitted by a vibrating guitar string is
3.0 × 10–3 s. If the speed of the sound wave is 343 m/s, what is its
wavelength?
6. Bats emit ultrasonic sound to help them locate obstacles. The
waves have a frequency of 5.5 × 104 Hz. If they travel at 350 m/s,
what is their wavelength?
7. What is the speed of a sound wave with a wavelength of 3.4 m
and a frequency of 1.0 × 102 Hz?
8. An FM station broadcasts radio signals with a frequency of
102 MHz. These radio waves travel at a speed of 3.00 × 108 m/s.
What is their wavelength?
6.4
Figure 1
A pulse from a heavy spring (left) to a light
spring (right)
fixed-end reflection: reflection from a
rigid obstacle when a pulse is inverted
free-end reflection: reflection where the
new medium is free to move and there is no
inversion
Transmission and Reflection
Water waves and waves in long springs travel at a uniform speed as long as the
medium they are in does not change. But if two long springs that differ in stiffness are joined together, the speed of a wave changes abruptly at the junction
between the two springs (Figure 1). The speed change corresponds to a wavelength change. This wavelength change is predicted by the universal wave equation. Since the frequency of a wave remains constant, the wavelength is directly
proportional to the speed; that is, v.
When a wave travels from a light rope into a heavy rope having the same tension, the wave slows down and the wavelength decreases. On the other hand, if
the wave travels from a heavy rope to a light rope, both the speed and the wavelength increase. These properties are true of all waves. A change in medium
results in changes both in the speed of the wave and in its wavelength. In a
1
medium where the speed is constant, the relationship v = f l predicts that f.
In other words, if the frequency of a wave increases, its wavelength decreases, a
fact easily demonstrated when waves of different frequencies are generated in a
rope or spring.
As you saw in Investigation 6.2.1, one-dimensional waves, such as those in a
spring or rope, behave in a special way when they are reflected. In the case of
reflection from a rigid obstacle, usually referred to as fixed-end reflection, the
pulse is inverted. A crest is reflected as a trough and a trough is reflected as a crest
(Figures 2 and 3). If the reflection occurs from a free end, where the medium is
free to move, there is no inversion—crests are reflected as crests and troughs as
troughs (Figure 2). In both fixed-end and free-end reflection there is no change
in the frequency or wavelength. Nor is there any change in the speed of the pulse,
since the medium is the same.
fixed
end
Figure 2
Fixed-end and free-end reflection
212 Chapter 6
free
end