The string described in the problem introduction is oscillating in one of its no

Question

The string described in the problem introduction is oscillating in one of its normal modes. Which of the following statements about the wave in the string is correct?

A. The wave is traveling in the +x direction.
B. The wave is traveling in the -x direction.
C. The wave will satisfy the given boundary conditions for any arbitrary wavelength lambda_i.
D. The wavelength lambda_i can have only certain specific values if the boundary conditions are to be satisfied.
E. The wave does not satisfy the boundary condition y_i(0;t)=0.


Part B
Which of the following statements are true?

A. The system can resonate at only certain resonance frequencies f_i and the wavelength lambda_i must be such that y_i(0;t) = y_i(L;t) = 0.
B. A_i must be chosen so that the wave fits exactly on the string.
C. Any one of A_i or lambda_i or f_i can be chosen to make the solution a normal mode.


Part C
Find the three longest wavelengths (call them lambda_1, lambda_2, and lambda_3) that "fit" on the string, that is, those that satisfy the boundary conditions at x=0 and x=L. These longest wavelengths have the lowest frequencies.
Express the three wavelengths in terms of L. List them in decreasing order of length, separated by commas.
A. lambda_1=
B. lambda_2
C. lambda_3 =

Part D
The frequency of each normal mode depends on the spatial part of the wave function, which is characterized by its wavelength lambda_i.

Find the frequency f_i of the ith normal mode.
Express f_i in terms of its particular wavelength lambda_i and the speed of propagation of the wave v.

A. f_i= ?

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Explanation / Answer

Part A

The string described in the problem introduction is oscillating in one of its normal modes. Which of the following statements about the wave in the string is correct?

The string described in the problem introduction is oscillating in one of its normal modes. Which of the following statements about the wave in the string is correct?

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Part B

Which of the following statements are true?

Which of the following statements are true?

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The key factor producing the normal modes is that there are two spatial boundary conditions, yi(0,t)=0 and yi(L,t)=0, that are satisfied only for particular values of ?i.

Part C

Find the three longest wavelengths (call them ?1, ?2, and ?3) that "fit" on the string, that is, those that satisfy the boundary conditions at x=0 and x=L. These longest wavelengths have the lowest frequencies.

Express the three wavelengths in terms of L. List them in decreasing order of length, separated by commas.

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The procedure described here contains the same mathematics that leads toquantization in quantum mechanics.

Part D

The frequency of each normal mode depends on the spatial part of the wave function, which is characterized by its wavelength ?i.

Find the frequency fi of the ith normal mode.

Express fi in terms of its particular wavelength ?i and the speed of propagation of the wave v.

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The frequencies fi are the only frequencies at which the system can oscillate. If the string is excited at one of these resonance frequencies it will respond by oscillating in the pattern given by yi(x,t), that is, with wavelength ?i associated with the fi at which it is excited. In quantum mechanics these frequencies are called the eigenfrequencies, which are equal to the energy of that mode divided by Planck's constant h. In SI units, Planck's constant has the value h=6.63

The wave is traveling in the +xdirection. The wave is traveling in the -xdirection. The wave will satisfy the given boundary conditions for any arbitrary wavelength ?i.

• The wavelength ?i can have only certain specific values if the boundary conditions are to be satisfied.

The wave does not satisfy the boundary condition yi(0;t)=0.

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