A normal mode of a closed system is an oscillation of the system in which all parts oscillate at a single frequency. In general there are an infinite number of such modes, each one with a distinctive frequency fi and associated pattern of oscillation. Consider an example of a system with normal modes: a string of length L held fixed at both ends, located at x=0 and x=L. Assume that waves on this string propagate with speed v. The string extends in the x direction, and the waves are transverse with displacement along the y direction. The normal modes of this system are products of trigonometric functions. (For linear systems, the time dependance of a normal mode is always sinusoidal, but the spatial dependence need not be.) Specifically, for this system a normal mode is described by

yi(x, t)=Ai sin(2π*x/λi)sin(2πfi*t)

1. 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 λ_i.
d. The wavelength λ_i can have only certain specific values if the boundary conditions are to be satisfied.
e. The wave does not satisfy the boundary condition yi(0; t)=0

2. Find the three longest wavelengths (call them λ_1, λ_2, and λ_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.
3. The frequency of each normal mode depends on the spatial part of the wave function, which is characterized by its wavelength λ_i. Find the frequency f_i of the ith normal mode.