Surface Waves
The waves on the ocean are surface waves: They occur at the interface of water and air, extending
down into the water and up into the air at the expense of becoming exponentially reduced in amplitude.
They are neither transverse nor longitudinal. The water both at and below the surface travels in vertical
circles, with exponentially smaller radius as a function of depth.
Both empirical measurements and calculations beyond the scope of introductory physics give the
propagation speed of water waves as
,
where
is the magnitude of the acceleration due to gravity and is the wavenumber.
This relationship applies only when the following three conditions hold:
1.
2.
3.
The water is several times deeper than the wavelength.
The wavelength is large enough that the surface tension of the waves can be neglected.
The ratio of wave height to wavelength is small.
The restoring force (analogous to the tension in a string) that restores the water surface to flatness is
due to gravity, which explains why these waves are often called "gravity waves."
A. Find the speed of water waves in terms of the wavelength
Express the speed in terms of ,
, and
.
.
= sqrt(g*lambda/(2*pi))
B. Find the speed of a wave of wavelength
.
Give your answer in meters per second to a precision of two significant figures.
= 3.5 (+/- 2%) m/s
C. Find the period
for a wave of wavelength
Express the period in terms of
,
.
, and .
= sqrt(2*pi*lambda/g)
D. On the East Coast of the United States, the National Weather Service frequently reports waves
with a period of 4.0 s. Find the wavelength
and speed of these waves.
Express your answers numerically as an ordered pair separated by a comma. Give an
accuracy of two significant figures.
, = 25 (+/- 3%) , 6.25 (+/- 1.5%) m, m/s
E.
On the West Coast of the United States, the National Weather Service frequently reports
waves (really swells) with a period of 15 s. Find the wavelength
and speed of these waves.
Express your answers numerically as an ordered pair separated by a comma. Give an
accuracy of two significant figures.
, = 350 (+/- 2%) , 23 (+/- 4%) m, m/s
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Standard Expression for a Traveling Wave
Learning Goal: To understand the standard formula for a sinusoidal traveling wave.
One formula for a wave with a y displacement (e.g., of a string) traveling in the x direction is
.
All the questions in this problem refer to this formula and to the wave it describes.
A. Which of the following are independent variables?
only
only
only
only
only
and
and
and and
B. Which of the following are parameters that determine the characteristics of the wave?
only
only
only
only
only
and
and
and and
C. What is the phase
of the wave?
Express the phase in terms of one or more given variables (
needed constants like .
, ,
, , and
) and any
= k*x-omega*t
D. What is the wavelength
of the wave?
Express the wavelength in terms of one or more given variables (
any needed constants like .
, ,
, , and
) and
= 2*pi/k
E. What is the period
of this wave?
Express the period in terms of one or more given variables (
needed constants like .
, ,
, , and
) and any
= 2*pi/omega
F.
What is the speed of propagation of this wave?
Express the speed of propagation in terms of one or more given variables (
and ) and any needed constants like .
, ,
, ,
= omega/k
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Ant on a Tightrope
A large ant is standing on the middle of a circus tightrope that is stretched with tension . The rope
has mass per unit length . Wanting to shake the ant off the rope, a tightrope walker moves her foot up
and down near the end of the tightrope, generating a sinusoidal transverse wave of wavelength
amplitude
and
. Assume that the magnitude of the acceleration due to gravity is .
A. What is the minimum wave amplitude
such that the ant will become momentarily
weightless at some point as the wave passes underneath it? Assume that the mass of the ant is
too small to have any effect on the wave propagation.
Express the minimum wave amplitude in terms of
= mu*(lambda/(2*pi))^2*g/T_s
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,
,
, and .
Breaking Storm Waves
Large waves on the deep ocean propagate at the speed
,
where is the magnitude of the acceleration due to gravity and is the wavenumber.
Seafaring mariners report that in great storms when the average peak-to-peak wave height becomes
about 1/7 of the wavelength, the tops of the largest ocean waves can become separated from the rest of
the wave. They claim that the wind and the wave's forward velocity cause huge "hunks" of water to
tumble down the face of the wave. Some are reportedly large enough to damage or capsize small
vessels.
The reason these "rogue waves" appear is that the amplitude of the water waves becomes so large that
the acceleration of the water in the top of the wave would have to be greater than for the wave to stay
in one piece. Because gravity is the only significant vertical force on the water, the acceleration cannot
exceed , so instead the water at the top of the wave breaks off and is blown down the side of the
wave.
In this problem, you will compute the ratio of amplitude to wavelength of a rogue wave.
The analytic expression for the vertical displacement of the water surface when an ocean wave of
amplitude
is propagating in the +x direction is
.
A. Find the angular frequency
of water waves.
Express the angular frequency in terms of the wavenumber and constants like and
= sqrt(g*k)
B.
Find the vertical acceleration
of this wave at position
Express the vertical acceleration in terms of
, , ,
and time .
, and the independent variables
and
.
= -A*omega^2*cos(k*x-omega*t)
C.
Now find the critical ratio of
to
such that this wave will have a maximum acceleration
at the top of
. Then the water at the top would have to accelerate faster than to stay
connected with the wave, which is clearly not possible!
Express
as a function of and constants like and
.
.
=
1/(2*pi)
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Why the Highest Piano Notes Have Short Strings
The steel used for piano wire has a breaking (tensile) strength
of
of about
and a density
.
A. What is the speed of a wave traveling down such a wire if the wire is stretched to its
breaking point?
Express the speed of the wave numerically, in meters per second, to the nearest integer.
= 620 (+/- 0.1%) m/s
B. Imagine that the wire described in the problem introduction is used for the highest C on a
piano (
). If the wire is in tune when stretched to its breaking point, what must
the vibrating length of the wire be?
Express the length numerically, in centimeters, using three significant figures.
= 7.75 (+/- 0.1%) cm
[ Print ]
Creating a Standing Wave
Learning Goal: To see how two traveling waves of the same frequency create a standing wave.
Consider a traveling wave described by the formula
.
This function might represent the lateral displacement of a string, the local electric field, the position of
the surface of the water, or many other physical manifestations of waves.
A. Which one of the following statements about this wave is correct?
This wave is traveling in the +x direction.
This wave is traveling in the -x direction.
This wave is oscillating but not traveling.
This wave is traveling but not oscillating.
B. What is the mathematical expression
traveling in the opposite direction? At time
displacement as
=
for a wave of the same amplitude that is
this new wave should have the same
, the wave described in the problem introduction.
The principle of superposition states that if two functions each separately satisfy the wave equation
then the sum (or difference) also satisfies the wave equation. This principle follows from the fact that
every term in the wave equation is linear in the amplitude of the wave.
Consider the sum of these two waves
sum can be written as follows:
. These waves have been chosen so that their
.
This form is significant because
--called the envelope--depends only on position, and
depends only on time. Traditionally, the time function is taken to be a trigonometric function with unit
amplitude; that is, the overall amplitude of the wave is written as part of
A. Find
and
amplitude.
. Remember that
Express your answers in terms of
comma.
,
.
should be a trigonometric function of unit
, ,
,
, and . Separate the two functions with a
= 2*A*sin(k*x) , cos(omega*t)
B. Which one of the following statements about the superposition wave
is correct?
This wave is traveling in the +x direction.
This wave is traveling in the -x direction.
This wave is oscillating but not traveling.
This wave is traveling but not oscillating.
C.
At the position
wave
, what is the displacement of the string (assuming that the standing
is present)?
Express your answer in terms of parameters given in the introduction.
= 0
D. At certain times, the string will be perfectly straight. Find the first time
Express
in terms of
= pi/(2*omega)
, , and necessary constants.
when this is true.
E.
From the previous part we know that the string is perfectly straight at time
. What
does this imply about the energy stored in the string?
a. There is no energy stored in the string--the string will remain straight for all
subsequent time.
b. Energy will flow into the string, causing the standing wave to form at a later time.
c.
d.
Although the string is straight at time
, parts of the string have nonzero
velocity. Therefore, there is energy stored in the string.
The total mechanical energy in the string oscillates but is constant if averaged over a
complete cycle.
a
b
c
d
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Nodes of a Standing Wave (Cosine)
Learning Goal: To understand the concept of nodes of a standing wave.
The nodes of a standing wave are points where the displacement of the wave is zero at all times. Nodes
are important for matching boundary conditions, for example that the point at which a string is tied to a
support has zero displacement at all times (i.e., the point of attachment does not move).
Consider a standing wave, where represents the transverse displacement of a string that extends along
the x direction. Here is a common mathematical form for such a wave:
where
is the maximum transverse displacement of the string (the amplitude of the wave), which is
assumed to be nonzero, is the wavenumber,
is the angular frequency of the wave, and is time.
A. Which one of the following statements about wave
is correct?
This wave is traveling toward +x.
This wave is traveling toward -x.
This wave is oscillating but not traveling.
This wave is traveling but not oscillating.
B. At time
, what is the displacement of the string
Express your answer in terms of
?
, , and other previously introduced quantities.
= 0
C. What is the displacement of the string as a function of
oscillation of the string?
at time
Express the displacement in terms of the given variables.
, where
is the period of
=
A*cos(k*x)
D. At which three points
the string
,
, and
closest to
but with
will the displacement of
be zero for all times? These are the first three nodal points.
Express the first three nonzero nodal points as multiples of the wavelength
constants like . List them in increasing order, separated by commas.
,
,
, using
= 1/4 , 3/4 , 5/4
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The Decibel Scale
Learning Goal: To understand the decibel scale.
The decibel scale is a logarithmic scale for measuring the sound intensity level. Because the decibel
scale is logarithmic, it changes by an additive constant when the intensity as measured in
changes by a multiplicative factor. The number of decibels increases by 10 for a factor of 10 increase in
intensity. The general formula for the sound intensity level, in decibels, corresponding to intensity is
,
where is a reference intensity. For sound waves,
to the logarithm to the base 10.
A. What is the sound intensity level
is taken to be
. Note that
refers
, in decibels, of a sound wave whose intensity is 10 times
the reference intensity (i.e.,
)?
Express the sound intensity numerically to the nearest integer.
= 10 dB
B. What is the sound intensity level
, in decibels, of a sound wave whose intensity is 100 times
the reference intensity (i.e.
)?
Express the sound intensity numerically to the nearest integer.
= 20 dB
One often needs to compute the change in decibels corresponding to a change in the physical intensity
measured in units of power per unit area. Take
A. Calculate the change in decibels (
.
to be the factor of increase of the physical intensity.
,
, and
) corresponding to
,
, and
Give your answers, separated by commas, to the nearest integer--this will give an
accuracy of 20%, which is good enough for sound.
,
= 3 , 6 , 9 dB
,
Ear Damage from a Small Firecracker
Learning Goal: To understand how to convert between different sound intensity scales and how the
decibel intensity of a sound changes with distance.
The decibel scale is logarithmic in intensity:
.
In this formula, is a reference intensity, which, for sound waves, is taken to be
. This
constant must be used to convert a particular physical intensity into a sound intensity level measured in
decibels.
Once we know the sound intensity level (in decibels) at a certain reference distance from a sound
source, the
decrease of intensity with distance can be accounted for by subtracting the decibel
value appropriate to the ratio of the new distance to the reference distance.
In this problem you will use the decibel scale to analyze a small firecracker that emits 1200 W of peak
power. To avoid confusion, intensities denoted by are in units of watts per meter squared; intensities
denoted by
are in units of decibels.
A. What is the peak intensity
Express
in decibels at a distance of 1 m from the firecracker?
in decibels to the nearest integer.
= 140 dB
B.
It takes a sound intensity of about 160 dB to rupture the human eardrum. How close must
the firecracker described in the introduction be to the ear to rupture the eardrum?
Express the distance
, in meters, to one decimal place.
= 0.1 (+/- 9%) m
C.
Will this firecracker produce temporary loss of hearing in someone who sets if off and
stands 3 m away from the explosion? Momentary sounds above 120 dB produce such loss.
yes
no
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Doppler Shift
Learning Goal: To understand the terms in the Doppler shift formula.
The Doppler shift formula gives the frequency
source S at frequency
at which a listener L hears the sound emitted by a
:
,
where is the speed of sound in the medium,
source.
is the velocity of the listener, and
is the velocity of
A. The velocity of the source is positive if the source is ______________.
traveling in the +x direction
traveling toward the listener
traveling away from the listener
B.
The velocity of the source is measured with respect to the ________.
medium (such as air or water)
listener
C.
The velocity of the listener is positive if the listener is _____________.
traveling in the +x direction
traveling toward the source
traveling away from the source
D.
The velocity of the listener is measured with respect to the ________.
source
medium
E. Imagine that the source is to the right of the listener, so that the positive reference direction
(from the listener to the source) is in the
does
direction. If the listener is stationary, what value
approach as the source's speed approaches the speed of sound moving to the right?
0
It approaches infinity.
F.
Now, imagine that the source is to the left of the listener, so that the positive reference
direction is in the
direction. If the source is stationary, what value does
the listener's speed (moving in the
approach as
direction) approaches the speed of sound?
0
It approaches infinity.
G. In this last case, imagine that the listener is stationary and the source is moving toward the
listener at the speed of sound. (Note that it is irrelevant whether the source is moving to the
right or to the left.) What is
0
It approaches infinity.
when the sound waves reach the listener?