Week of 9/4
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(→when is it ok to assume physical expressions are complex?) |
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Consider the forced, damped, simple harmonic oscillator. | Consider the forced, damped, simple harmonic oscillator. | ||
| − | <math>\ddot{x} + \gamma \dot{x} + \omega_0 ^2 = F/m</math> | + | <math>\ddot{x} + \gamma \dot{x} + \omega_0 ^2 x_r = F/m</math> |
Let's pretend that both <math>x</math> and <math>F</math> are actually complex variables. | Let's pretend that both <math>x</math> and <math>F</math> are actually complex variables. | ||
| Line 20: | Line 20: | ||
<math>x = x_r + i x_i \ \ \ \ F = F_r + i F_i \,</math> | <math>x = x_r + i x_i \ \ \ \ F = F_r + i F_i \,</math> | ||
| + | |||
| + | Plugging these into the equation above we get | ||
| + | |||
| + | <math>\ddot{x_r} + \gamma \dot{x_r} + \omega_0 ^2 x_r + | ||
| + | i\left[\ddot{x_i} + \gamma \dot{x_i} + \omega_0 ^2 x_i\right] = F_r/m + i F_i /m </math> | ||
Revision as of 03:07, 8 September 2006
Try the following notebook
 Download manipulating complex numbers in Mathematica.
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Also, take a look at Mathematica Tips and Tricks.
A real amplitude and phase measurement. Top figure shows the amplitude of the transmitted and reflected electric fields. Bottom are the corresponding amplitudes.
when is it ok to assume physical expressions are complex?
Consider the forced, damped, simple harmonic oscillator.
Let's pretend that both x and F are actually complex variables. So we would write
Plugging these into the equation above we get
![\ddot{x_r} + \gamma \dot{x_r} + \omega_0 ^2 x_r +
i\left[\ddot{x_i} + \gamma \dot{x_i} + \omega_0 ^2 x_i\right] = F_r/m + i F_i /m](/csm/wiki/images/math/2/0/0/200ca3de1b5126a171fd0ec143de9e02.png)
