Question

ro A wave gauge that measures the free surface elevation at a given point was used to take data in a wave tank. Waves were generated by a vertical board in sinusoidal motion; the generated waves are therefore periodic. However, due to a so-called nonlinear effect (occurs when the wave amplitude becomes large), higher harmonic waves were generated so the waves become steeper at the crest and flatter at the trough. Use the program you wrote in Problem 1 to compute the amplitude spectrum of the wave data and discuss what frequencies (including the fundamental and high harmonic) are involved in the waves. You will need to plot the time domain data (the wave data) and the frequency domain data (the amplitude spectrum) out. You should be able to estimate the fundamental frequency (period) by plotting out the waves and check if your calculation is correct. Note that the data file has two columns: the first column is time (unit: second) and the second column is water elevation (unit: m) Problem 3 The record of the Kobe earthquake is measured using an accelerometer. Use the program you wrote in Problem 1 to compute the amplitude spectrum of the Kobe earthquake data and discuss what frequencies are dominant. You will need to plot the time domain data and the frequency domain data (the amplitude spectrum) out. Note that the data file has two columns: the first column is time and the second column is acceleration.

Program from problem 1: (Using MATLAB)

% Sampling frequency and sampling period
fs = 10000;
ts = 1/fs;
% Number of samples, assume 1000 samples
l = 1000;
t = 0:1:l-1;
t = t.*ts; % Convert the sample index into time for generation and plotting of signal
% Frequency and amplitude of the sensor
f1 = 110;
a1 = 1.0;
% Frequency and amplitude of the power grid noise
f2 = 60;
a2 = 0.7;
% Generating the sinusoidal waves
x1 = a1*sin(2*pi*f1*t); % Sensor wave
x2 = a2*sin(2*pi*f2*t); % power grid wave
% Generating the zero-mean random noise
noise = randn(1,l);
% Adding all the waves and random noise
signal = x1 + x2 + noise;
% Plotting the signal
figure,plot(t,signal)
xlabel('Time in seconds')
ylabel('Amplitude')
title('Noisy signal with 110 Hz sensor signal and 60 Hz power grid noise')
% Performing FFT on the signal
xk = fft(signal, l);
% Calculation of magnitude of the spectrum coefficients
xkabs = abs(xk);
% Plotting the single sided magnitude spectrum
f = 0:1:500;
figure,plot(f*(fs/l),xkabs(1:501))
xlabel('Frequency in Hz')
ylabel('Magnitude')
title('FFT of Noisy signal')

Answer 1

% Sampling frequency and sampling period
fs = 10000;
ts = 1/fs;
% Number of samples, assume 1000 samples
l = 1000;
t = 0:1:l-1;
t = t.*ts; % Convert the sample index into time for generation and plotting of signal
% Frequency and amplitude of the sensor
f1 = 110;
a1 = 1.0;
% Frequency and amplitude of the power grid noise
f2 = 60;
a2 = 0.7;
% Generating the sinusoidal waves
x1 = a1*sin(2*pi*f1*t); % Sensor wave
x2 = a2*sin(2*pi*f2*t); % power grid wave
% Generating the zero-mean random noise
noise = randn(1,l);
% Adding all the waves and random noise
signal = x1 + x2 + noise;
% Plotting the signal
figure,plot(t,signal)
xlabel('Time in seconds')
ylabel('Amplitude')
title('Noisy signal with 110 Hz sensor signal and 60 Hz power grid noise')
% Performing FFT on the signal
xk = fft(signal, l);
% Calculation of magnitude of the spectrum coefficients
xkabs = abs(xk);
% Plotting the single sided magnitude spectrum
f = 0:1:500;
figure,plot(f*(fs/l),xkabs(1:501))
xlabel('Frequency in Hz')
ylabel('Magnitude')
title('FFT of Noisy signal')