| Never enter units in your answers. Be
sure to use the units indicated on the HW submission page.
Enter numbers without commas. 23,546 should be 23546 Always enter a * for multiplication. g(H+h) should be g*(H+h). |
| You are now obtaining input data for CID = |
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Submitted Answers will be graded W 9/24, Th 9/25, F 9/26, and M 9/29 at 12:00 noon.
| Problem 2.1 | |||||||||||||
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What you should learn:
y1(x,t) = A1 sin(θ+ψ1)
is a phasor with magnitude A1 and relative phase ψ1. Phasor 1 has components (A1cosψ1,
A1sinψ1)
Finally, y(x,t) = y1(x,t)+y2(x,t) = A sin(θ+ψ) It's usually easier to form the phasors and add them than it is to plug things into the formulas, but either way works.
This problem and the next two problems involve adding harmonic equations where the waves are traveling in the same direction with the same wavelength. |
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| Problem: Two
harmonic waves are traveling in the same direction along a string. The
functional form of the individual waves is
y1(x,t) = A1 sin(kx-ωt)
Find A the amplitude of the resulting wave. [Special characters used in this problem: omega = ω, phi = φ, psi = ψ, theta = θ] |
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| Constants and fixed variables:
A1 = A2 = 0.005 m k = 12.0 m-1 omega = 360 rad/s |
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Variables that must be changed with each
submission:
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| Hints:
How do I use the wave addition formulas? |
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Results of previous submissions:
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| Answer Range: 0.00900-0.01000 m Submit HW answers. |
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| Problem 2.2 | |||||||||||||
| What you should learn: See Problem 2.1 | |||||||||||||
| Problem: Two
harmonic waves are traveling in the same direction along a string. The
functional form of the individual waves is
y1(x,t) = A1 sin(kx-ωt)
Find psi (ψ) the relative phase of the resulting wave. [Special characters used in this problem: omega = ω, phi = φ, theta = θ] |
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| Constants and fixed variables:
A1 = A2 = 0.005 m k = 12.0 m-1 omega = 360 rad/s |
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Variables that must be changed with each submission:
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Results of previous submissions:
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| Answer Range: 15.0-27.0 degrees Submit HW answers. |
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| Problem 2.3 | |||||||||||||
| What you should learn: See Problem 2.1 | |||||||||||||
| Problem: Two
harmonic waves are traveling in the same direction along a string. The
functional form of the individual waves is
y1(x,t) = A1 sin(kx-ωt)
Find k0 the wavenumber of the resulting wave. [Special characters used in this problem: omega = ω, phi = φ, theta = θ] |
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| Constants and fixed variables:
A1 = A2 = 0.005 m k = 12.0 m-1 omega = 360 rad/s |
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Variables that must be changed with each
submission:
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Results of previous submissions:
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| Answer Range: 10.0 -14.0 m-1 Submit HW answers. |
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| Problem 2.4 | |||||||||||||
| What you should learn:
When two waves of equal amplitude and frequency travel in opposite directions along a string, they produce a standing wave. If the waves are: y1(x,t) = A sin(kx-ωt)
A standing wave has the form:
y((x,t) = 2A cos(ωt) sin(kx) We can interpret this a simple sine function of position with an amplitude that varies in time: 2A cos(ωt). Nodes
are the points on the string that always have zero amplitude. Antinodes
are the points on the string that vibrate with maximum amplitude.
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| Problem: A string oscillates in a standing wave with a frequency f . The wave velocity on the string is v. One node is found at the origin (x = 0). How far from the origin is the second closest antinode along the +x axis? (Call this distance x2) | |||||||||||||
| Constants and fixed variables:
f = 440 Hz |
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Variables that must be changed with each submission:
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| Hints:
Need some help? |
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Results of previous submissions:
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| Answer Range: 1.00 -1.90 m Submit HW answers. |
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| Problem 2.5 | |||||||||||||
| What you should learn:
If a string has both ends fixed, then both ends must be nodes of any standing wave on the string. This restricts oscillations on the strings to certain wavelengths. |
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| Problem: A string
of linear mass density μ has a length L and a tension T. What is the frequency
f of the third lowest oscillation (corresponding to the third
longest wavelength) that will form a regular standing wave (a normal
mode) on the
string? (Call the answer f3.) [Special characters used in this problem: mu = μ] |
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| Constants and fixed variables:
L = 1.24 m mu = 7.00 g/m |
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Variables that must be changed with each
submission :
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Results of previous submissions:
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| Answer Range: 79 - 104 Hz Submit HW answers. |
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| Problem 2.6 | |||||||||||||
| What you should learn:
Interference maxima occur when crests of one wave are on top of the crests of another wave. Conversely, interference minima occur when the crests of one wave are on top of the troughs of another wave. If two identical speakers emit sound produced by the same signal (so both speakers emit crests at the same time), we can tell if a spot will be an interference maximum or minimum by determining the difference in the path length from the observer to each speaker. if the difference in path length is one, two, three, wavelengths, etc., then crests are top of crests and the interference is constructive. If the difference is one-half, one and one-half wavelengths, etc. then crests are on top of troughs and the interference is destructive. |
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| Problem: Two speakers located at the front of a stage emit a sound at frequency f. The speakers are each located at a distance d from the center of the stage, one to the right and the other to the left. Row 16 of the auditorium is located a distance L from the front of the stage. Someone sitting near the center of the row complains that she can't hear the sound well. But everyone between her and the center of the row hears the sound well. How far is she from the center of the row? (Call the answer x.) | |||||||||||||
| Constants and fixed variables:
Velocity of sound in air: v = 340 m/s f = 220 Hz d = 12.4 m |
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Variables that must be changed with each
submission:
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| Hints:
How do I start? |
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Results of previous submissions:
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| Answer Range: 0.700 - 1.40 m Submit HW answers. |
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| Problem 2.7 | |||||||||||||
| What you should learn: See Problem 2.6. | |||||||||||||
| Problem: How far from the center of Row 16 would the next dead spot be located? (Call the answer x.) | |||||||||||||
| Constants and fixed variables:
Velocity of sound in air: v = 340 m/s f = 220 Hz d = 12.4 m |
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Variables that must be changed with each
submission:
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Results of previous submissions:
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| Answer Range: 2.10 - 4.00 m Submit HW answers. |
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| Problem 2.8 | |||||||||||||
| What you should learn: Boundary conditions for standing waves on a string: ends are nodes. | |||||||||||||
| Problem: A harpsichord has a string of length L and a linear mass density of μ . What tension T is necessary to give the frequency f given below? | |||||||||||||
| Constants and fixed variables:
String length L = 70.0 cm Linear density of the string: μ = 1.05 g/m [Special characters used in this problem: mu=μ] |
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Variables that must be changed with each
submission:
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| Hints:
What is the frequency of a string? |
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Results of previous submissions:
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| Answer Range: 510 - 1400 N Submit HW answers. |
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| Problem 2.9 | |||||||||||||
| What you should learn: Boundary conditions
for standing waves in a pipe are:
the fipple or an open end is a pressure node or a displacement antinode a closed end is a displacement node or a pressure antinode (Note, the pipe lengths are little long, so the frequency may be a bit too low to be audible, but don't worry about it...) |
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| Problem: The longest pipe in a given set of open-ended organ pipes has a length L. What is the fundamental frequency of the pipe? | |||||||||||||
| Constants and fixed variables:
Speed of sound v = 330 m/s |
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Variables that must be changed with each
submission:
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| Hints:
How do I determine the wavelength from the pipe length? |
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Results of previous submissions:
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| Answer Range: 16.0 - 24.0 Hz Submit HW answers. |
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| Problem 2.10 | |||||||||||||
| What you should learn: Boundary conditions
for standing waves in a pipe are:
the fipple or an open end is a pressure node or a displacement antinode a closed end is a displacement node or a pressure antinode |
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| Problem: The longest pipe in a given set of closed-ended organ pipes has a length L. What is the fundamental frequency of the pipe? | |||||||||||||
| Constants and fixed variables:
Speed of sound v = 330 m/s |
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Variables that must be changed with each
submission:
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| Hints:
How do I determine the wavelength from the pipe length? |
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Results of previous submissions:
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| Answer Range: 8.00 - 12.0 Hz Submit HW answers. |
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| Problem 2.11 | |||||||||||||
| What you should learn: We can force a string to have a node in a given location by touching it. | |||||||||||||
| Problem: A violin of length L oscillates at the frequency f. When you touch the string a distance L/5 from one end, what frequency do you hear? (Call this frequency f2). | |||||||||||||
Variables that must be changed with each
submission:
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Results of previous submissions:
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| Answer Range: 2500 - 4000 Hz | |||||||||||||
| Problem 2.12 | |||||||||||||
| What you should learn: Harmonics are multiples of the fundamental frequency. Overtones are the actual frequencies produced in addition to the fundamental frequency. Overtones may or may not be harmonic, depending on the system. | |||||||||||||
| Problem: A violin string oscillates at a frequency f . What is the frequency of the second overtone? (Call this frequency f2). | Variables that must be changed with each
submission:
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| Hints:
What are the harmonic characteristics of strings? |
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Results of previous submissions:
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Range of answers:
1400 - 1900 Hz Submit HW answers. |
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| Problem 2.13 | |||||||||||||
| What you should learn: See Problem 2.12. | |||||||||||||
| Problem: An open organ pipe oscillates at a frequency f . What is the frequency of the third harmonic? (Call this frequency f3.) | |||||||||||||
Variables that must be changed with each
submission:
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| Hints:
What are the harmonic characteristics open pipe? |
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Results of previous submissions:
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Range of answers: 1400 - 1900 Hz
Submit HW answers. |
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| Problem 2.14 | |||||||||||||
| What you should learn: See Problem 2.12. | |||||||||||||
| Problem: A closed organ pipe oscillates at a frequency f . What is the frequency of the second overtone? (Call this frequency f2.) | |||||||||||||
Variables that must be changed with each
submission:
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| Hints:
What are the harmonic characteristics of a closed pipe? |
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Results of previous submissions:
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Range of answers: 2500 -3200 Hz
Submit HW answers. |
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| Problem 2.15 | ||||
| What you should learn: A Fourier Transform (FT) gives the frequency spectrum of a sound wave. | ||||
Problem: The
Fourier Transform of a musical instrument is shown to the right. Of the
choices below, the instrument is
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Results of previous submissions:
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| Problem 2.16 | |||||||||||||
| What you should learn: You should learn how to tell what pitch is perceived when a series of harmonic tones is heard. | |||||||||||||
| Problem: Consider the Fourier transform of the previous problem. What pitch would you hear? (Call this frequency f.) | |||||||||||||
| Hints: Check the Power Point slides. | |||||||||||||
Results of previous submissions:
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| Problem 2.17 | |||||||||||||
| What you should learn: When two sound waves of similar frequency are heard simultaneously, they produce a wah-wah sound because crests are on crests at times and crests are on troughs at other times. | |||||||||||||
| Problem: Strings on two violins are tuned to f1 and f2. The interference of the two sound waves produces beats. How many pulses per second do you hear? (Call the answer fbeat.) | |||||||||||||
| Constants and fixed variables:
The frequency of the first string, f1 = 470 Hz |
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Variables that must be changed with each
submission:
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Results of previous submissions:
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| Answer Range: 0.00 - 20.0 Hz Submit HW answers. |
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