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| + | ==Complex Numbers and Waves== | ||
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Complex numbers can be used to represent waves and calculate their behavior. The simplest example of complex waves would be a simple mass, m, attached to a spring of stiffness S. | Complex numbers can be used to represent waves and calculate their behavior. The simplest example of complex waves would be a simple mass, m, attached to a spring of stiffness S. | ||
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[[image:ScottHamiltonspring.jpg]] | [[image:ScottHamiltonspring.jpg]] | ||
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The natural frequency of oscillation for this system is computed as <math> \omega_0 ^2 = \frac{S}{m} </math>. | The natural frequency of oscillation for this system is computed as <math> \omega_0 ^2 = \frac{S}{m} </math>. | ||
| − | The behavior of this spring-mass system can then be modeled using complex numbers. | + | The behavior of this spring-mass system can then be modeled using complex numbers where the real part of the solution represents it's physical, measurable properties. |
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| + | ==Example Formulas== | ||
<math> Displacement=A_1*e^{j * \omega_0 * t} + A_2*e^{-j * \omega_0 * t}</math> | <math> Displacement=A_1*e^{j * \omega_0 * t} + A_2*e^{-j * \omega_0 * t}</math> | ||
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<math>Acceleration=A*\omega_0^2*e^{j * \omega_0 * t} + B*\omega_0^2*e^{-j * \omega_0 * t} </math> | <math>Acceleration=A*\omega_0^2*e^{j * \omega_0 * t} + B*\omega_0^2*e^{-j * \omega_0 * t} </math> | ||
| − | Where ''A'' and ''B'' are constants and ''t'' is time. | + | Where ''A'' and ''B'' are determinable constants and ''t'' is time. |
Latest revision as of 16:42, 4 September 2008
Complex Numbers and Waves
Complex numbers can be used to represent waves and calculate their behavior. The simplest example of complex waves would be a simple mass, m, attached to a spring of stiffness S.
The natural frequency of oscillation for this system is computed as $ \omega_0 ^2 = \frac{S}{m} $.
The behavior of this spring-mass system can then be modeled using complex numbers where the real part of the solution represents it's physical, measurable properties.
Example Formulas
$ Displacement=A_1*e^{j * \omega_0 * t} + A_2*e^{-j * \omega_0 * t} $
$ Velocity=A_1*\omega_0*e^{j * \omega_0 * t} + A_2*\omega_0*e^{-j * \omega_0 * t} $
$ Acceleration=A*\omega_0^2*e^{j * \omega_0 * t} + B*\omega_0^2*e^{-j * \omega_0 * t} $
Where A and B are determinable constants and t is time.
