Dowry

ASSIGNMENT 1

1. Find the Fourier series expansion for the following function [pic] for [pic] [pic]

Hence, deduce that [pic]

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

[pic] [pic] [pic]

4. A periodic function of period [pic] is defined over one period by

[pic]. 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

[pic]

where

[pic]

Show that if the parameters [pic] are such that [pic]the solutions for [pic] are all of the form [pic]

where

[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]