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How to Fit a Wave Equation to Data

Posted 11 years ago
 I have a dataset which I will input below with my attempt. I am attempting to fit a wave equation to it but I am running into issues with getting the model to fit the data...... which is recorded from a torsional oscillator,,,,, any suggestions? The data I am interested in is stored in the third column of the data set which is why I use data[[All,3]].data = Import["C:\\Users\\Physics2\\Desktop\\PRINT_00.CSV"]; (*The data is given bellow*)L1 = ListPlot[{data[[All, 3]]}]s1 = FindFit[data[[All, 2]],   A*Cos[(2*Pi)/T*t -phi], {A, T, phi}, t]p1 = Plot[A*Cos[(2*Pi)/T*t - phi] /. s1, {t, 0, 10}]Show[L1, p1]below is a hard copy of the data set I am working with:  {{-3.02, 0.08, 0.96}, {-3.01, 0.08, 0.88}, {-3.01, 0.08,     0.88}, {-3.01, 0.08, 0.88}, {-3.01, 0.08, 0.88}, {-3., 0.12,     0.8}, {-3., 0.08, 0.88}, {-3., 0.08, 0.88}, {-3., 0.08,     0.8}, {-2.99, 0.12, 0.8}, {-2.99, 0.12, 0.8}, {-2.99, 0.12,     0.72}, {-2.99, 0.12, 0.72}, {-2.98, 0.12, 0.72}, {-2.98, 0.12,     0.72}, {-2.98, 0.12, 0.72}, {-2.98, 0.16, 0.72}, {-2.97, 0.12,     0.64}, {-2.97, 0.12, 0.72}, {-2.97, 0.12, 0.64}, {-2.97, 0.12,     0.64}, {-2.96, 0.12, 0.64}, {-2.96, 0.12, 0.56}, {-2.96, 0.16,     0.56}, {-2.96, 0.16, 0.56}, {-2.95, 0.12, 0.56}, {-2.95, 0.12,    0.56}, {-2.95, 0.12, 0.56}, {-2.95, 0.16, 0.56}, {-2.94, 0.16,    0.48}, {-2.94, 0.12, 0.48}, {-2.94, 0.16, 0.48}, {-2.94, 0.12,    0.4}, {-2.93, 0.16, 0.32}, {-2.93, 0.12, 0.32}, {-2.93, 0.12,    0.32}, {-2.93, 0.12, 0.32}, {-2.92, 0.16, 0.4}, {-2.92, 0.12,    0.32}, {-2.92, 0.12, 0.4}, {-2.92, 0.16, 0.24}, {-2.91, 0.16,    0.32}, {-2.91, 0.16, 0.24}, {-2.91, 0.16, 0.24}, {-2.91, 0.16,    0.24}, {-2.9, 0.16, 0.24}, {-2.9, 0.12, 0.24}, {-2.9, 0.16,    0.24}, {-2.9, 0.16, 0.16}, {-2.89, 0.16, 0.08}, {-2.89, 0.12,    0.08}, {-2.89, 0.16, 0.08}, {-2.89, 0.16, 0.08}, {-2.88, 0.12,    0.08}, {-2.88, 0.16, 0.08}, {-2.88, 0.16, 0.}, {-2.88, 0.16,    0.08}, {-2.88, 0.16, 0.}, {-2.87, 0.12, -0.08}, {-2.87,    0.16, -0.08}, {-2.87, 0.16, -0.08}, {-2.87, 0.16, -0.08}, {-2.86,    0.16, -0.08}, {-2.86, 0.16, -0.08}, {-2.86, 0.12, -0.08}, {-2.86,    0.12, -0.08}, {-2.85, 0.16, -0.08}, {-2.85, 0.16, -0.16}, {-2.85,    0.16, -0.32}, {-2.85, 0.12, -0.32}, {-2.84, 0.16, -0.32}, {-2.84,    0.16, -0.24}, {-2.84, 0.12, -0.24}, {-2.84, 0.16, -0.32}, {-2.83,    0.16, -0.32}, {-2.83, 0.12, -0.32}, {-2.83, 0.12, -0.24}, {-2.83,    0.12, -0.4}, {-2.82, 0.12, -0.4}, {-2.82, 0.16, -0.4}, {-2.82,    0.16, -0.4}, {-2.81, 0.12, -0.4}, {-2.81, 0.16, -0.4}, {-2.81,    0.12, -0.4}, {-2.81, 0.12, -0.4}, {-2.8, 0.12, -0.48}, {-2.8,    0.12, -0.48}, {-2.8, 0.12, -0.56}, {-2.8, 0.12, -0.56}, {-2.79,    0.12, -0.56}, {-2.79, 0.12, -0.56}, {-2.79, 0.12, -0.56}, {-2.79,    0.08, -0.64}, {-2.78, 0.12, -0.56}, {-2.78, 0.12, -0.64}, {-2.78,    0.12, -0.64}, {-2.78, 0.08, -0.64}, {-2.77, 0.12, -0.64}, {-2.77,    0.08, -0.72}, {-2.77, 0.08, -0.72}, {-2.77, 0.08, -0.72}, {-2.76,    0.08, -0.72}, {-2.76, 0.08, -0.72}, {-2.76, 0.08, -0.72}, {-2.76,    0.08, -0.72}, {-2.75, 0.08, -0.72}, {-2.75, 0.08, -0.72}, {-2.75,    0.08, -0.8}, {-2.75, 0.08, -0.8}, {-2.74, 0.08, -0.88}, {-2.74,    0.08, -0.88}, {-2.74, 0.08, -0.96}, {-2.74, 0.08, -0.88}, {-2.73,    0.08, -0.88}, {-2.73, 0.08, -0.88}, {-2.73, 0.08, -0.96}, {-2.73,    0.08, -0.96}, {-2.72, 0.08, -0.96}, {-2.72, 0.08, -0.96}, {-2.72,    0.08, -0.96}, {-2.72, 0.04, -1.04}, {-2.71, 0.04, -1.04}, {-2.71,    0.08, -0.96}, {-2.71, 0.04, -1.04}, {-2.71, 0.08, -1.04}, {-2.7,    0.04, -1.04}, {-2.7, 0.04, -1.04}, {-2.7, 0.04, -1.04}, {-2.7,    0.04, -1.04}, {-2.69, 0.08, -1.04}, {-2.69, 0.04, -1.04}, {-2.69,    0.08, -1.12}, {-2.69, 0.04, -1.12}, {-2.68, 0.04, -1.2}, {-2.68,    0.04, -1.04}, {-2.68, 0.04, -1.2}, {-2.68, 0.04, -1.2}, {-2.67,    0.04, -1.2}, {-2.67, 0.04, -1.2}, {-2.67, 0.04, -1.2}, {-2.67,    0.04, -1.2}, {-2.66, 0.04, -1.2}, {-2.66, 0.04, -1.2}, {-2.66,    0., -1.2}, {-2.66, 0.04, -1.2}, {-2.65, 0.04, -1.2}, {-2.65,    0., -1.28}, {-2.65, 0., -1.2}, {-2.65, 0.04, -1.2}, {-2.64,    0., -1.2}, {-2.64, 0., -1.2}, {-2.64, 0.04, -1.2}, {-2.64,    0., -1.2}, {-2.63, 0., -1.2}, {-2.63, 0., -1.28}, {-2.63,    0., -1.36}, {-2.63, 0., -1.28}, {-2.63, 0., -1.28}, {-2.62,    0., -1.36}, {-2.62, 0., -1.36}, {-2.62, 0., -1.28}, {-2.62,    0., -1.36}, {-2.61, 0., -1.28}, {-2.61,    0., -1.28}, {-2.61, -0.04, -1.36}, {-2.61, 0., -1.36}, {-2.6,    0., -1.36}, {-2.6, -0.04, -1.36}, {-2.6, 0., -1.36}, {-2.6,    0., -1.36}, {-2.59,    0., -1.36}, {-2.59, -0.04, -1.36}, {-2.59, -0.04, -1.36}, {-2.59,    0., -1.36}, {-2.58, -0.04, -1.36}, {-2.58, -0.04, -1.36}, {-2.58, \ -0.04, -1.36}, {-2.58, -0.04, -1.36}, {-2.57, -0.04, -1.36}, {-2.57, \ -0.04, -1.28}, {-2.57, -0.04, -1.28}, {-2.57, -0.04, -1.28}, {-2.56, \ -0.04, -1.28}, {-2.56, -0.04, -1.36}, {-2.56, -0.04, -1.36}, {-2.55, \ -0.04, -1.36}, {-2.55, -0.08, -1.28}, {-2.55, -0.08, -1.28}, {-2.55, \ -0.04, -1.28}, {-2.54, -0.04, -1.28}, {-2.54, -0.08, -1.28}, {-2.54, \ -0.08, -1.28}, {-2.54, -0.08, -1.28}, {-2.53, -0.08, -1.28}, {-2.53, \ -0.08, -1.2}, {-2.53, -0.08, -1.28}, {-2.53, -0.08, -1.28}, {-2.52, \ -0.08, -1.28}, {-2.52, -0.08, -1.28}, {-2.52, -0.08, -1.2}, {-2.52, \ -0.08, -1.2}, {-2.51, -0.08, -1.2}, {-2.51, -0.08, -1.2}, {-2.51, \ -0.08, -1.2}, {-2.51, -0.08, -1.2}, {-2.5, -0.08, -1.2}, {-2.5, \ -0.08, -1.2}, {-2.5, -0.08, -1.2}, {-2.5, -0.08, -1.2}, {-2.49, \ -0.08, -1.2}, {-2.49, -0.08, -1.12}, {-2.49, -0.08, -1.2}, {-2.49, \ -0.08, -1.2}, {-2.48, -0.08, -1.12}, {-2.48, -0.08, -1.2}, {-2.48, \ -0.08, -1.12}, {-2.48, -0.08, -1.12}, {-2.47, -0.12, -1.12}, {-2.47, \ -0.12, -1.12}, {-2.47, -0.12, -1.04}, {-2.47, -0.12, -1.04}, {-2.46, \ -0.12, -1.04}, {-2.46, -0.12, -1.04}, {-2.46, -0.12, -1.04}, {-2.46, \ -0.12, -1.04}, {-2.45, -0.12, -0.96}, {-2.45, -0.12, -1.04}, {-2.45, \ -0.12, -0.96}, {-2.45, -0.12, -0.96}, {-2.44, -0.12, -0.96}, {-2.44, \ -0.12, -0.88}, {-2.44, -0.12, -0.88}, {-2.44, -0.12, -0.88}, {-2.43, \ -0.16, -0.88}, {-2.43, -0.16, -0.88}, {-2.43, -0.12, -0.96}, {-2.43, \ -0.12, -0.96}, {-2.42, -0.16, -0.88}, {-2.42, -0.12, -0.8}, {-2.42, \ -0.12, -0.8}, {-2.42, -0.16, -0.8}, {-2.41, -0.12, -0.88}, {-2.41, \ -0.16, -0.72}, {-2.41, -0.16, -0.72}, {-2.41, -0.16, -0.72}, {-2.4, \ -0.16, -0.72}, {-2.4, -0.16, -0.72}, {-2.4, -0.16, -0.72}, {-2.4, \ -0.16, -0.72}, {-2.39, -0.16, -0.72}, {-2.39, -0.16, -0.64}, {-2.39, \ -0.16, -0.64}, {-2.39, -0.16, -0.56}, {-2.38, -0.16, -0.64}, {-2.38, \ -0.16, -0.64}, {-2.38, -0.16, -0.56}, {-2.38, -0.16, -0.56}, {-2.38, \ -0.16, -0.56}, {-2.37, -0.16, -0.56}, {-2.37, -0.16, -0.48}, {-2.37, \ -0.16, -0.56}, {-2.37, -0.16, -0.48}, {-2.36, -0.16, -0.56}, {-2.36, \ -0.16, -0.4}, {-2.36, -0.16, -0.48}, {-2.36, -0.16, -0.4}, {-2.35, \ -0.16, -0.4}, {-2.35, -0.16, -0.4}, {-2.35, -0.16, -0.4}, {-2.35, \ -0.16, -0.4}, {-2.34, -0.16, -0.32}, {-2.34, -0.16, -0.32}, {-2.34, \ -0.2, -0.32}, {-2.34, -0.16, -0.4}, {-2.33, -0.16, -0.32}, {-2.33, \ -0.16, -0.24}, {-2.33, -0.16, -0.24}, {-2.33, -0.16, -0.32}, {-2.32, \ -0.16, -0.24}, {-2.32, -0.16, -0.24}, {-2.32, -0.16, -0.24}, {-2.32, \ -0.16, -0.32}, {-2.31, -0.16, -0.08}, {-2.31, -0.2, -0.08}, {-2.31, \ -0.16, -0.08}, {-2.31, -0.16, -0.08}, {-2.3, -0.16, -0.08}, {-2.3, \ -0.16, -0.08}, {-2.3, -0.16, -0.08}, {-2.29, -0.2, -0.08}, {-2.29, \ -0.16, 0.}, {-2.29, -0.2, 0.}, {-2.29, -0.16, 0.}, {-2.28, -0.16,   0.08}, {-2.28, -0.16, 0.08}, {-2.28, -0.16, 0.08}, {-2.28, -0.16,   0.08}, {-2.27, -0.16, 0.08}, {-2.27, -0.16, 0.16}, {-2.27, -0.16,   0.16}, {-2.27, -0.16, 0.16}, {-2.26, -0.16, 0.16}, {-2.26, -0.16,   0.16}, {-2.26, -0.16, 0.24}, {-2.26, -0.16, 0.24}, {-2.25, -0.16,   0.24}, {-2.25, -0.16, 0.24}, {-2.25, -0.16, 0.24}, {-2.25, -0.16,   0.32}, {-2.24, -0.16, 0.32}, {-2.24, -0.16, 0.4}, {-2.24, -0.16,   0.4}, {-2.24, -0.16, 0.32}, {-2.23, -0.16, 0.4}, {-2.23, -0.16,   0.32}, {-2.23, -0.16, 0.4}, {-2.23, -0.16, 0.48}, {-2.22, -0.16,   0.56}, {-2.22, -0.16, 0.48}, {-2.22, -0.12, 0.56}, {-2.22, -0.16,   0.56}, {-2.21, -0.16, 0.56}, {-2.21, -0.16, 0.56}, {-2.21, -0.16,   0.56}, {-2.21, -0.16, 0.56}, {-2.2, -0.12, 0.64}, {-2.2, -0.16,   0.56}, {-2.2, -0.12, 0.64}, {-2.2, -0.12, 0.56}, {-2.19, -0.12,   0.64}, {-2.19, -0.12, 0.64}, {-2.19, -0.12, 0.72}, {-2.19, -0.12,   0.72}, {-2.18, -0.12, 0.72}, {-2.18, -0.12, 0.72}, {-2.18, -0.12,   0.72}, {-2.18, -0.12, 0.8}, {-2.17, -0.12, 0.72}, {-2.17, -0.12,   0.72}, {-2.17, -0.12, 0.8}, {-2.17, -0.12, 0.8}, {-2.16, -0.12,   0.88}, {-2.16, -0.08, 0.88}, {-2.16, -0.12, 0.88}, {-2.16, -0.12,   0.88}, {-2.15, -0.12, 0.88}, {-2.15, -0.12, 0.88}, {-2.15, -0.08,   0.88}, {-2.15, -0.08, 0.88}, {-2.14, -0.08, 0.88}, {-2.14, -0.08,   0.88}, {-2.14, -0.08, 0.88}, {-2.14, -0.08, 0.88}, {-2.13, -0.08,   0.96}, {-2.13, -0.08, 1.04}, {-2.13, -0.08, 0.96}, {-2.13, -0.08,   1.04}, {-2.13, -0.08, 1.04}, {-2.12, -0.08, 1.04}, {-2.12, -0.08,   1.04}, {-2.12, -0.08, 1.04}, {-2.12, -0.08, 0.96}, {-2.11, -0.08,   1.12}, {-2.11, -0.08, 1.04}, {-2.11, -0.08, 1.04}, {-2.11, -0.08,   1.04}, {-2.1, -0.08, 1.04}, {-2.1, -0.08, 1.12}, {-2.1, -0.08,   1.12}, {-2.1, -0.08, 1.12}, {-2.09, -0.08, 1.2}, {-2.09, -0.08,   1.2}, {-2.09, -0.08, 1.2}, {-2.09, -0.08, 1.2}, {-2.08, -0.08,   1.2}, {-2.08, -0.08, 1.2}, {-2.08, -0.08, 1.2}, {-2.08, -0.08,   1.2}, {-2.07, -0.04, 1.2}, {-2.07, -0.08, 1.2}, {-2.07, -0.04,   1.2}, {-2.07, -0.04, 1.2}, {-2.06, -0.04, 1.2}, {-2.06, 0.,   1.2}, {-2.06, -0.04, 1.2}, {-2.06, -0.04, 1.2}, {-2.05, -0.04,   1.2}, {-2.05, -0.04, 1.2}, {-2.05, -0.04, 1.2}, {-2.04, -0.04,   1.2}, {-2.04, -0.04, 1.2}, {-2.04, 0., 1.2}, {-2.04, -0.04,   1.28}, {-2.03, 0., 1.28}, {-2.03, 0., 1.28}, {-2.03, 0.,   1.28}, {-2.03, -0.04, 1.28}, {-2.02, 0., 1.2}, {-2.02, 0.,   1.2}, {-2.02, 0., 1.28}, {-2.02, 0., 1.2}, {-2.01, 0.,   1.28}, {-2.01, 0., 1.28}, {-2.01, 0., 1.28}, {-2.01, 0.,   1.28}, {-2., 0., 1.28}, {-2., 0., 1.28}, {-2., 0., 1.2}, {-2., 0.,   1.28}, {-2., 0., 1.28}, {-1.99, 0., 1.2}, {-1.99, 0., 1.2}, {-1.99,   0., 1.28}, {-1.99, 0., 1.2}, {-1.98, 0., 1.28}, {-1.98, 0.,   1.28}, {-1.98, 0., 1.2}, {-1.97, 0., 1.2}, {-1.97, 0.04,   1.2}, {-1.97, 0., 1.2}, {-1.97, 0., 1.2}, {-1.96, 0., 1.2}, {-1.96,   0.04, 1.2}, {-1.96, 0.04, 1.2}, {-1.96, 0.04, 1.2}, {-1.95, 0.04,   1.2}, {-1.95, 0.04, 1.2}, {-1.95, 0.04, 1.2}, {-1.95, 0.04,   1.2}, {-1.94, 0.04, 1.2}, {-1.94, 0.04, 1.2}, {-1.94, 0.04,   1.2}, {-1.94, 0.04, 1.2}, {-1.93, 0.04, 1.2}, {-1.93, 0.04,   1.2}, {-1.93, 0.04, 1.2}, {-1.93, 0.04, 1.2}, {-1.92, 0.04,   1.2}, {-1.92, 0.08, 1.2}, {-1.92, 0.08, 1.12}, {-1.92, 0.08,   1.12}, {-1.91, 0.08, 1.12}, {-1.91, 0.08, 1.12}, {-1.91, 0.08,   1.04}, {-1.91, 0.08, 1.12}, {-1.9, 0.08, 0.96}, {-1.9, 0.04,   1.04}, {-1.9, 0.08, 1.04}, {-1.9, 0.08, 0.96}, {-1.89, 0.08,   1.04}, {-1.89, 0.08, 1.04}, {-1.89, 0.08, 1.04}, {-1.89, 0.08,   1.04}, {-1.88, 0.08, 1.04}, {-1.88, 0.08, 0.96}, {-1.88, 0.08,   0.96}, {-1.88, 0.08, 1.04}, {-1.88, 0.08, 0.96}, {-1.87, 0.08,   1.04}, {-1.87, 0.08, 0.88}, {-1.87, 0.08, 0.96}, {-1.87, 0.08,   0.88}, {-1.86, 0.08, 0.88}, {-1.86, 0.08, 0.88}, {-1.86, 0.08,   0.88}, {-1.85, 0.08, 0.88}, {-1.85, 0.08, 0.88}, {-1.85, 0.08,   0.88}, {-1.85, 0.08, 0.88}, {-1.84, 0.08, 0.88}, {-1.84, 0.08,   0.8}, {-1.84, 0.08, 0.8}, {-1.84, 0.08, 0.8}, {-1.83, 0.12,   0.8}, {-1.83, 0.12, 0.72}, {-1.83, 0.12, 0.72}, {-1.83, 0.12,   0.72}, {-1.82, 0.12, 0.72}, {-1.82, 0.08, 0.72}, {-1.82, 0.12,   0.64}, {-1.82, 0.12, 0.64}, {-1.81, 0.12, 0.64}, {-1.81, 0.12,   0.64}, {-1.81, 0.12, 0.56}, {-1.81, 0.12, 0.56}, {-1.8, 0.12,   0.56}, {-1.8, 0.12, 0.56}, {-1.8, 0.16, 0.56}, {-1.8, 0.12,   0.56}, {-1.79, 0.12, 0.56}, {-1.79, 0.12, 0.48}, {-1.79, 0.16,   0.56}, {-1.79, 0.12, 0.48}, {-1.78, 0.12, 0.48}, {-1.78, 0.16,   0.48}, {-1.78, 0.12, 0.4}, {-1.78, 0.12, 0.4}, {-1.77, 0.12,   0.4}, {-1.77, 0.16, 0.4}, {-1.77, 0.16, 0.32}, {-1.77, 0.16,   0.32}, {-1.76, 0.16, 0.32}, {-1.76, 0.16, 0.32}, {-1.76, 0.12,   0.24}, {-1.76, 0.16, 0.32}, {-1.75, 0.16, 0.24}, {-1.75, 0.16,   0.24}, {-1.75, 0.16, 0.24}, {-1.75, 0.16, 0.24}, {-1.75, 0.16,   0.16}, {-1.74, 0.16, 0.08}, {-1.74, 0.16, 0.16}, {-1.74, 0.12,   0.08}, {-1.74, 0.12, 0.08}, {-1.73, 0.16, 0.08}, {-1.73, 0.16,   0.08}, {-1.73, 0.16, 0.08}, {-1.72, 0.12, 0.08}, {-1.72, 0.16,   0.}, {-1.72, 0.16, 0.}, {-1.72, 0.16, 0.}, {-1.71,   0.16, -0.08}, {-1.71, 0.16, -0.08}, {-1.71, 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Posted 11 years ago
 Dear William, I would recommend you keep the following link handyhttp://mathematica.stackexchange.com/questions/18/where-can-i-find-examples-of-good-mathematica-programming-practiceSpecifically regarding the symbols @,&,/@,#,&http://mathematica.stackexchange.com/questions/18393/what-are-the-most-common-pitfalls-awaiting-new-users/25616#25616Please see the section on What the @#%^&*?! do all those funny signs mean?Regards.
Posted 11 years ago
 Thanks so much, that was really helpful! Could you by chance go a little bit deeper into the operation:yMean = Mean@Cases[data, {#, _, c_} -> c] & /@ xAxis; (*Calculate the mean of the column 3 for each specific value of column 1*) and designate how each peace of syntax is used? Such as the @,&, and # symbols. Simply for demonstrative purposes and to help me understand the core language a bit better.
Posted 11 years ago
 xAxis= Union[data[[All,1]]]; (*Extract unique values of independent variable*)  yMean = Mean@Cases[data, {#, _, c_} -> c] & /@ xAxis; (*Calculate the mean of the column 3 for each specific value of column 1*)  avg = Transpose[{xAxis,yMean}];  ListPlot[avg]By checking the chart we can see that this looks like a Sin Curve with an amplitude about 1.2 and frequency >3.sol = FindFit[avg, a Sin[b x ], {{a, 1}, {b, 3}}, x](*{a -> 1.2645, b -> 5.44369}*)f[x_] := a Sin[b x ] /. solShow[ListPlot[Transpose[{data[[All, 1]], data[[All, 3]]}]], Plot[f[x], {x, -3, 3}, PlotStyle -> {Red, Thick}]]