Answer :
These are the conditions for the a, b, c and d satisfy these conditions.
What is wave ?
A wave is an energetic disturbance in a medium without a net movement of particles. It could manifest as elastic deformation, pressure changes, shifts in electric or magnetic field strength, changes in electric potential, or changes in temperature.
What is oscillating ?
Oscillation is the process of any quantity or measure repeatedly varying about its equilibrium value in time. Another definition of oscillation is the periodic variation of a substance between two values or around its core value.
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?
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.
Part - B
The system can resonate at only certain resonance frequencies fi and the wavelength ?i must be such that yi(0;t)=yi(L;t)=0.
Ai must be chosen so that the wave fits exactly on the string.
Any one of Ai or ?i or fi can be chosen to make the solution a normal mode.
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.
?1, ?2, ?3 = 2L,L,2L/3
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.
fi = v/?i
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
Therefore, these are the conditions for the a, b, c and d satisfy these conditions.
Learn more about waves from the given link.
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