Final answer:
To determine the Fourier Series for the given periodic function, one would need to find the Fourier coefficients by integrating over one period of the function. The student's question relates to the analysis of wave equations and their representation via sinusoidal functions, typically expressed mathematically with amplitude, wavelength, period, and phase shift parameters.
Step-by-step explanation:
The student is asking for the continuous Fourier Series representation for a given periodic function. This involves determining the coefficients for the sine and cosine terms that will replicate the given waveform over each period. To solve for these coefficients, generally denoted as an, bn, and a0, we integrate over one period of the function. For wave equations, this would translate into finding the amplitude, wavelength, period, and sometimes the phase shift, which describes the sinusoidal wave completely.
To find the Fourier coefficients for a function that oscillates between +1 and -1, we need to express our function in terms of mathematical pieces that fit into the Fourier series framework. The sine function does this naturally, and the problem seems to describe a square wave, which can be expanded into a series of sine functions with varying coefficients. We would extend this analysis over the period specified by the user.
For analyzing wave equations, like the one given as y2(x, t) = A sin(2kx + 2ωt), we use the wave characteristics to deduce the continuous Fourier series. The period (T) can be given by T = 1/f where f is the frequency, and the wave speed (v) is related to the wavelength (λ) and frequency by v = λf. To confirm the solution for a superposition of two waves, we utilize trigonometric identities to show that the sum of two waves with a phase shift can be represented by a single resultant wave with its own amplitude and phase.