Workshop 5 (Chapter 15)
Properties of Ocean Waves
A fisherman notices that his boat is moving up and down periodically without any horizontal motion, owing to waves
on the surface of the water. It takes a time of 2.60 s for the boat to travel from its highest point to its lowest, a total
distance of 0.650 m . The fisherman sees that the wave crests are spaced a horizontal distance of 6.00 m apart.
Part A
How fast are the waves traveling?
Express the speed v in meters per second using three significant figures.
Hint 1. How to approach the problem
Calculate the period of the ocean waves, using the fisherman's observations. Then, use the period and
wavelength to calculate the speed of the waves.
Hint 2. Calculate the period of the waves
Calculate the period T of the ocean waves.
Express your answer in seconds using three significant figures.
Hint 1. Definition of period
The period of a wave is the time it takes for one full wavelength to pass a particular point. This is
also the time it takes to go from one crest to the next, or from one trough to the next.
ANSWER:
T
= 5.20 s Hint 3. Equation for the speed of a wave
The speed of a wave is given by v = f λ , where f is the frequency of the waves and λ = 6.00 m is the
wavelength. The frequency is simply the reciprocal of the period, or f = 1/T .
ANSWER:
v
= 1.15 m/s Part B
A
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What is the amplitude A of each wave?
Express your answer in meters using three significant figures.
Hint 1. Definition of amplitude
The amplitude of a wave is the vertical distance from the top of the crest to the neutral position, halfway
between the crest and trough. Equivalently, the amplitude is the vertical distance from the bottom of the
trough to the neutral position.
ANSWER:
A
= 0.325 m The fisherman does not simply move up and down as the waves pass by. In fact, the motion of the
fisherman will be roughly circular with both upward and forward components (with respect to the direction of
the wave) as the wave rises and downward and backward components as the wave falls. The water that
comprises the ocean wave itself moves in this same way. Thus, an ocean wave is not a purely transverse
wave; it also has a longitudinal component.
Exercise 15.10
A water wave traveling in a straight line on a lake is described by the equation y(x, t)
=
(2.75 cm) cos(0.410 rad/cm x + 6.20 rad/s t)
where y is the displacement perpendicular to the undisturbed surface of the lake.
Part A
How much time does it take for one complete wave pattern to go past a fisherman in a boat at anchor?
Express your answer with the appropriate units.
ANSWER:
t
= 1.01 s
Part B
What horizontal distance does the wave crest travel in that time?
Express your answer with the appropriate units.
ANSWER:
s
= 0.153 m
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Workshop 5 (Chapter 15)
Part C
What is the wave number?
ANSWER:
k
= 0.410 rad/cm Part D
What is the number of waves per second that pass the fisherman?
ANSWER:
f
= 0.987 waves per second Part E
How fast does a wave crest travel past the fisherman?
Express your answer with the appropriate units.
ANSWER:
v
= 0.151 m
s
Part F
What is the maximum speed of his cork floater as the wave causes it to bob up and down?
Express your answer with the appropriate units.
ANSWER:
v max
= 0.171 m
s
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Workshop 5 (Chapter 15)
Exercise 15.15
One end of a horizontal rope is attached to a prong of an electrically driven tuning fork that vibrates at a frequency
115 Hz . The other end passes over a pulley and supports a mass of 1.50 kg . The linear mass density of the rope
is 0.0510 kg/m .
Part A
What is the speed of a transverse wave on the rope?
Express your answer with the appropriate units.
ANSWER:
v
= 17.0 m
s
Part B
What is the wavelength?
Express your answer with the appropriate units.
ANSWER:
λ
= 0.148 m
Part C
What is the speed of a transverse wave on the rope if the mass were increased to 3.00 kg ?
Express your answer with the appropriate units.
ANSWER:
v
= 24.0 m
s
Part D
What is the wavelength of a transverse wave on the rope if the mass were increased to 3.00 kg ?
Express your answer with the appropriate units.
ANSWER:
λ
= 0.209 m
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Workshop 5 (Chapter 15)
Exercise 15.17
The upper end of a 3.30­m­long steel wire is fastened to the ceiling, and a 53.0­kg object is suspended from the
lower end of the wire. You observe that it takes a transverse pulse 0.0492 s to travel from the bottom to the top of
the wire.
Part A
What is the mass of the wire?
Express your answer with the appropriate units.
ANSWER:
m
= 0.381 kg
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