To understand the concept of normal modes of oscillation and to derive some properties of normal modes of waves on a string. 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. In this problem, you will investigate the shape of the normal modes and then their frequency. 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)=Aisin(2πxλi)sin(2πfit).

a) Find the three longest wavelengths (call them λ1, λ2, and λ3) 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.

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

Find the frequency fi of the ith normal mode.

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

c) Find the three lowest normal mode frequencies f1, f2, and f3.

Express the frequencies in terms of L, v, and any constants. List them in increasing order, separated by commas.