The water surface elevation x(t) was measured at a certain point during a representative time interval...

  

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The water surface elevation x(t) was measured at a certain point during a representative time interval (0,To), where To = 2 sec. The results of the measurements were approximated by the formula x(t) = ho- At(Tot), where 0 < t < To, ho = 50 cm and A = 0.75 m sec-². (1) Using the formulas for the Fourier series and its coefficients, see equations (9) and (10) in the lecture notes on Series, calculate the coefficients of the Fourier series of the function x(t). (ii) Write down the obtained Fourier series in the form, 00 x(t) = An cos(w₂t + En) n=1 and give formulas for An, n and En. Give values of A₁, A2, A3, A4, and A5. (iii) Calculate the wave spectrum S(@) at w = wn by the formula, A Aw = Wn+1 - Wn 24w' and plot the spectrum S(w) by points in the interval 0 ≤ ≤ 5n sec-¹. S(wn) = The water surface elevation x(t) was measured at a certain point during a representative time interval (0,To), where To = 2 sec. The results of the measurements were approximated by the formula x(t) = ho- At(Tot), where 0 < t < To, ho = 50 cm and A = 0.75 m sec-². (1) Using the formulas for the Fourier series and its coefficients, see equations (9) and (10) in the lecture notes on Series, calculate the coefficients of the Fourier series of the function x(t). (ii) Write down the obtained Fourier series in the form, 00 x(t) = An cos(w₂t + En) n=1 and give formulas for An, n and En. Give values of A₁, A2, A3, A4, and A5. (iii) Calculate the wave spectrum S(@) at w = wn by the formula, A Aw = Wn+1 - Wn 24w' and plot the spectrum S(w) by points in the interval 0 ≤ ≤ 5n sec-¹. S(wn) =