Surface Waves
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
[pic],
where [pic]is the magnitude of the acceleration due to gravity and [pic]is the wavenumber.
This relationship applies only when the following three conditions hold:
1. The water is several times deeper than the wavelength.
2. The wavelength is large enough that the surface tension of the waves can be neglected.
3. 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 [pic]of water waves in terms of the wavelength [pic].
Express the speed in terms of [pic], [pic], and [pic].
|[pic|sqrt(g*lambda/(2*pi)) | | |
|]= | | | |
B. Find the speed [pic]of a wave of wavelength [pic].
Give your answer in meters per second to a precision of two significant figures.
|[pic]= |3.5 (+/- 2%) |m/s | |
| | | | |
C. Find the period [pic]for a wave of wavelength [pic].
Express the period in terms of [pic], [pic], and [pic].
|[pic]|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 [pic]and speed [pic]of these waves.
Express your answers numerically as an ordered pair separated by a comma. Give an accuracy of two significant figures.
|[pic], |25 (+/- 3%) |m, m/s | |
|[pic]= |, | | |
| |6.25 (+/- 1.5%) | | |
| | | | |
E. [pic]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 [pic]and speed [pic]of these waves.
Express your answers numerically as an ordered pair separated by a comma. Give an accuracy of two significant figures.
|[pic], |350 (+/- 2%) |m, m/s | |
|[pic]= |, | | |
| |23 (+/- 4%) | | |
| | | | |
[pic] [ Print ]
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
[pic].
All the questions in this problem refer to this formula and to the wave it describes.
A. Which of the following are independent variables?
|[pic][pic]only | | |
|[pic][pic]only | | |
|[pic][pic]only | | |
|[pic][pic]only | | |
|[pic][pic]only | | |
|[pic][pic]and [pic] | | |
|[pic][pic]and [pic] | | |
|[pic][pic]and [pic]and | | |
|[pic] | | |
B. Which of the following are parameters that determine the characteristics of the wave?
|[pic][pic]only | | |
|[pic][pic]only | | |
|[pic][pic]only | | |
|[pic][pic]only | | |
|[pic][pic]only | | |
|[pic][pic]and [pic] | | |
|[pic][pic]and [pic] | | |
|[pic][pic]and [pic]and | | |
|[pic] | | |
C. What is the phase [pic]of the wave?
Express the phase in terms of one or more given variables ([pic], [pic], [pic], [pic], and [pic]) and any needed constants like [pic].
|[pic]= |k*x-omega*t | | |
| | | | |
D. What is the wavelength [pic]of the wave?
Express the wavelength in terms of one or more given variables ([pic], [pic], [pic], [pic], and [pic]) and any needed constants like [pic].
|[pic|2*pi/k | | |
|]= | | | |
E. What is the period [pic]of this wave?
Express the period in terms of one or more given variables ([pic], [pic], [pic], [pic], and [pic]) and any needed constants like [pic].
|[pic]|2*pi/omega | | |
|= | | | |
F. What is the speed of propagation [pic]of this wave?
Express the speed of propagation in terms of one or more given variables ([pic], [pic], [pic], [pic], and [pic]) and any needed constants like [pic].
|[pic|omega/k | | |
|]= | | | |
[pic] [ Print ]
Ant on a Tightrope
A large ant is standing on the middle of a circus tightrope that is stretched with tension [pic]. The rope has mass per unit length [pic]. 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 [pic]and amplitude [pic]. Assume that the magnitude of the acceleration due to gravity is [pic].
A. What is the minimum wave amplitude [pic]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 [pic], [pic], [pic], and [pic].
|[pic]= |mu*(lambda/(2*pi))^2*g/T_s | | |
| | | | |
[pic] [ Print ]
Breaking Storm Waves
Large waves on the deep ocean propagate at the speed
[pic],
where [pic]is the magnitude of the acceleration due to gravity and [pic]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 [pic]for the wave to stay in one piece. Because gravity is the only significant vertical force on the water, the acceleration cannot exceed [pic], 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 [pic]is propagating in the +x direction is
[pic].
A. Find the angular frequency [pic]of water waves.
Express the angular frequency in terms of the wavenumber [pic]and constants like [pic]and [pic].
|[pic|sqrt(g*k) | | |
|]= | | | |
B. [pic]Find the vertical acceleration [pic]of this wave at position [pic]and time [pic].
Express the vertical acceleration in terms of [pic], [pic], [pic], [pic], and the independent variables [pic]and [pic].
|[pic]= |-A*omega^2*cos(k*x-omega*t) | | |
| | | | |
C. [pic]Now find the critical ratio of [pic]to [pic]such that this wave will have a maximum acceleration at the top of [pic]. Then the water at the top would have to accelerate faster than [pic]to stay connected with the wave, which is clearly not possible!
Express [pic]as a function of [pic]and constants like [pic]and [pic].
|[pic]|1/(2*pi) | | |
|= | | | |
[pic] [ Print ]
Why the Highest Piano Notes Have Short Strings
The steel used for piano wire has a breaking (tensile) strength [pic]of about [pic]and a density [pic]of [pic].
A. What is the speed [pic]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.
|[pic|620 (+/- 0.1%) |m/s | |
|]= | | | |
B. Imagine that the wire described in the problem introduction is used for the highest C on a piano ([pic]). 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.
|[pic|7.75 (+/- 0.1%) |cm | |
|]= | | | |
[pic] [ 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
[pic].
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?
|[pic]This wave is traveling in the +x direction. | | |
|[pic]This wave is traveling in the -x direction. | | |
|[pic]This wave is oscillating but not traveling. | | |
|[pic]This wave is traveling but not oscillating. | | |
B. What is the mathematical expression [pic]for a wave of the same amplitude that is traveling in the opposite direction? At time [pic]this new wave should have the same displacement as [pic], the wave described in the problem introduction.
|[pic]= |[pic][pic] | | |
| |[pic][pic] | | |
| |[pic][pic] | | |
| |[pic][pic] | | |
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 [pic]. These waves have been chosen so that their sum can be written as follows:
[pic].
This form is significant because [pic]--called the envelope--depends only on position, and [pic]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 [pic].
A. Find [pic]and [pic]. Remember that [pic]should be a trigonometric function of unit amplitude.
Express your answers in terms of [pic], [pic], [pic], [pic], and [pic]. Separate the two functions with a comma.
|[pic], [pic]= |2*A*sin(k*x) | | |
| |, | | |
| |cos(omega*t) | | |
| | | | |
B. Which one of the following statements about the superposition wave [pic]is correct?
|[pic]This wave is traveling in the +x direction. | | |
|[pic]This wave is traveling in the -x direction. | | |
|[pic]This wave is oscillating but not traveling. | | |
|[pic]This wave is traveling but not oscillating. | | |
C. [pic]At the position [pic], what is the displacement of the string (assuming that the standing wave [pic]is present)?
Express your answer in terms of parameters given in the introduction.
|[pic]= |0 | | |
| | | | |
D. At certain times, the string will be perfectly straight. Find the first time [pic]when this is true.
Express [pic]in terms of [pic], [pic], and necessary constants.
|[pic]|pi/(2*omega) | | |
|= | | | |
E. [pic]From the previous part we know that the string is perfectly straight at time [pic]. 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. Although the string is straight at time [pic], parts of the string have nonzero velocity. Therefore, there is energy stored in the string.
d. The total mechanical energy in the string oscillates but is constant if averaged over a complete cycle.
|[pic]a [pic]b [pic]c [pic]d | | |
[pic] [ Print ]
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 [pic]represents the transverse displacement of a string that extends along the x direction. Here is a common mathematical form for such a wave:
[pic]
where [pic]is the maximum transverse displacement of the string (the amplitude of the wave), which is assumed to be nonzero, [pic]is the wavenumber, [pic]is the angular frequency of the wave, and [pic]is time.
A. Which one of the following statements about wave [pic]is correct?
|[pic]This wave is traveling toward +x. | | |
|[pic]This wave is traveling toward -x. | | |
|[pic]This wave is oscillating but not traveling. | | |
|[pic]This wave is traveling but not oscillating. | | |
B. At time [pic], what is the displacement of the string [pic]?
Express your answer in terms of [pic], [pic], and other previously introduced quantities.
|[pic]= |0 | | |
| | | | |
C. What is the displacement of the string as a function of [pic]at time [pic], where [pic]is the period of oscillation of the string?
Express the displacement in terms of the given variables.
|[pic]= |A*cos(k*x) | | |
| | | | |
D. At which three points [pic], [pic], and [pic]closest to [pic]but with [pic]will the displacement of the string [pic]be zero for all times? These are the first three nodal points.
Express the first three nonzero nodal points as multiples of the wavelength [pic], using constants like [pic]. List them in increasing order, separated by commas.
|[pic], [pic], |1/4 |[pi| |
|[pic]= |, |c] | |
| |3/4 | | |
| |, | | |
| |5/4 | | |
| | | | |
[pic] [ Print ]
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 [pic]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 [pic]is
[pic],
where [pic]is a reference intensity. For sound waves, [pic]is taken to be [pic]. Note that [pic]refers to the logarithm to the base 10.
A. What is the sound intensity level [pic], in decibels, of a sound wave whose intensity is 10 times the reference intensity (i.e., [pic])?
Express the sound intensity numerically to the nearest integer.
|[pic|10 |dB | |
|]= | | | |
B. What is the sound intensity level [pic], in decibels, of a sound wave whose intensity is 100 times the reference intensity (i.e. [pic])?
Express the sound intensity numerically to the nearest integer.
|[pic|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 [pic]to be the factor of increase of the physical intensity.
A. Calculate the change in decibels ([pic], [pic], and [pic]) corresponding to [pic], [pic], and [pic].
Give your answers, separated by commas, to the nearest integer--this will give an accuracy of 20%, which is good enough for sound.
|[pic], [pic], [pic]= |3 |dB | |
| |, | | |
| |6 | | |
| |, | | |
| |9 | | |
| | | | |
[pic]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:
[pic].
In this formula, [pic]is a reference intensity, which, for sound waves, is taken to be [pic]. 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 [pic]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 [pic]are in units of watts per meter squared; intensities denoted by [pic]are in units of decibels.
A. What is the peak intensity [pic]in decibels at a distance of 1 m from the firecracker?
Express [pic]in decibels to the nearest integer.
|[pic]= |140 |dB | |
| | | | |
B. [pic]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 [pic], in meters, to one decimal place.
|[pic]= |0.1 (+/- 9%) |m | |
| | | | |
C. [pic]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.
|[pic]yes [pic]no | | |
[pic] [ Print ]
Doppler Shift
Learning Goal: To understand the terms in the Doppler shift formula.
The Doppler shift formula gives the frequency [pic]at which a listener L hears the sound emitted by a source S at frequency [pic]:
[pic],
where [pic]is the speed of sound in the medium, [pic]is the velocity of the listener, and [pic]is the velocity of source.
A. The velocity of the source is positive if the source is ______________.
|[pic]traveling in the +x direction | | |
|[pic]traveling toward the listener | | |
|[pic]traveling away from the listener | | |
B. [pic]The velocity of the source is measured with respect to the ________.
|[pic]medium (such as air or water) | | |
|[pic]listener | | |
C. [pic]The velocity of the listener is positive if the listener is _____________.
|[pic]traveling in the +x direction | | |
|[pic]traveling toward the source | | |
|[pic]traveling away from the source | | |
D. [pic]The velocity of the listener is measured with respect to the ________.
|[pic]source | | |
|[pic]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 [pic]direction. If the listener is stationary, what value does [pic]approach as the source's speed approaches the speed of sound moving to the right?
|[pic]0 | | |
|[pic][pic] | | |
|[pic][pic] | | |
|[pic]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 [pic]direction. If the source is stationary, what value does [pic]approach as the listener's speed (moving in the [pic]direction) approaches the speed of sound?
|[pic]0 | | |
|[pic][pic] | | |
|[pic][pic] | | |
|[pic]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 [pic]when the sound waves reach the listener?
|[pic]0 | | |
|[pic][pic] | | |
|[pic][pic] | | |
|[pic]It approaches infinity. | | |
[pic]
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