ACTIVITY ON OSCILLATION SYNTHESIS (ANALYTICAL SOLUTION) (2nd PROBLEM: SAME WIDTHS, DIFFERENT FREQUENCIES)     

We are studying the synthesis of the two oscillations:  x1(t) = A1 sin(w1 t),  x2(t) = A2 sin(w2 t + f), f = 0, A1 = A2 = A.
The posistion of the body that performs the composite oscillation can be calculated as: x(t) = x1(t) + x2(t)  

1) We can observe that analytically:
y(t) = x1(t) + x2(t) = A sin(w1 t) + A sin(w2 t) = A  ( sin(w1 t) + sin(w2 t) ) =  2 A cos (w1 - w2)/2t sin (w1 + w2)/2t
If the two frequencies slightly differ, the factor  A' = 2 A cos (w1 - w2)/2t changes in a slower pace than sin (w1 + w2)/2t that changes with frequency w' = (w1 + w2)/2.
That means that  y(t) = A' sin (w' t), where w' = (w1 + w2)/2 and A' = 2 A cos (w1 - w2)/2t .

A = A1

y[t_] := 2 * A * Cos[(w1 - w2)/2 * t] * Sin[(w1 + w2)/2 * t]

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2) We are calculating the values of y(t), t = 0 (0.2) T1+T2 and we are plotting the corresponding points (t, x(t)), t = 0 (0.2) T1+T2

TableForm[Table[{t, N[y[t]]}, {t, 0, T1 + T2, 0.2}], TableHeadings→ {None, {"t", "y(t)"}}, TableAlignments→Center, TableDirections→Row]

t 0 0.2 0.4 0.6 0.8 1. 1.2 1.4 1.6 1.8 2. 2.2 2.4 2.6 2.8 3. 3.2 3.4 3.6 3.8 4. 4.2 4.4 4.6 4.8 5. 5.2 5.4 5.6 5.8 6. 6.2 6.4 6.6 6.8 7. 7.2 7.4 7.6 7.8 8. 8.2 8.4 8.6 8.8 9. 9.2 9.4 9.6 9.8 10. 10.2 10.4 10.6 10.8 11. 11.2 11.4 11.6 11.8 12. 12.2 12.4 12.6 12.8 13. 13.2 13.4 13.6 13.8 14.
y(t) 0. 7.28692 14.3151 20.8355 26.6186 31.4626 35.2015 37.7106 38.9116 38.7749 37.3205 34.6167 30.7768 25.9549 20.3386 14.1421 7.59747 0.945077 -5.57537 -11.7342 -17.3205 -22.1498 -26.0708 -28.9702 -30.7768 -31.4626 -31.0432 -29.5758 -27.1559 -23.912 -20. -15.5955 -10.8864 -6.06443 -1.31744 3.17837 7.26543 10.8106 13.7101 15.8924 17.3205 17.9916 17.936 17.2145 15.9139 14.1421 12.0221 9.6854 7.26543 4.89087 2.67949 0.732636 -0.869308 -2.07031 -2.84079 -3.17837 -3.10719 -2.6759 -1.95439 -1.02954 -2.44929*10^^-15 1.02954 1.95439 2.6759 3.10719 3.17837 2.84079 2.07031 0.869308 -0.732636 -2.67949

ListPlot[Table[{t, N[y[t]]}, {t, 0, T1 + T2, 0.2}], AspectRatio→1, PlotStyle→ {AbsolutePointSize[5], RGBColor[1, 0, 0]}, AxesLabel→ {"t", "y(t)"}]

[Graphics:../HTMLFiles/Physics, Oscillations_148.gif]

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