1. Find the Fourier series expansion for the following function [pic] for [pic] [pic]
Hence, deduce that
2. A periodic function of period [pic] is defined over one period by [pic] for [pic]
Determine the Fourier series and illustrate graphically for [pic][pic] Then, deduce the value of [pic]
3. Find a Fourier series expansion of the periodic function
4. A periodic function of period [pic] is defined over one period by
Determine the Fourier series expansion
5. A sinusoidal voltage [pic]is passed through a half-wave rectifier which clips the negative portion of the wave. Expand the resulting periodic function as a Fourier series [pic]
6. For the function defined by the graph OAB, find the half-range Fourier sine series.
7. The damped vibrations of a stretched string are governed by the equation
[pic] equation (1)
where [pic] is the transverse deflection, t is the time, [pic]is the position coordinate along the string, and [pic] and τ are positive constants. A taut elastic string, [pic], is fixed at its end points so that [pic] Show that separation of variable solutions of equation (1) satisfying these boundary conditions are of the form
Show that if the parameters [pic] are such that [pic]the solutions for [pic] are all of the form [pic]
[pic] and [pic] are constants.
Hence find the general solution of equation (1) satisfying the given boundary conditions
[pic] and [pic] , find [pic].
8. In a uniform bar of length [pic] the temperature [pic]...