Complex oscillations - Complex oscillations From Physics...

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Complex oscillations From Physics Notes The most common use of complex numbers in physics is for analyzing oscillations and waves. We will illustrate this with a simple but crucially important model, the damped harmonic oscillator . Contents 1 The harmonic oscillator equation 2 Complex solution 3 Complex frequencies 4 General solution for the damped harmonic oscillator 4.1 Under-damped motion 4.2 Over-damped motion 4.3 Critical damping 5 Specific solutions for the initial-value problem 6 Exercises The harmonic oscillator equation The damped harmonic oscillator describes a particle of mass , which can move along one dimension. Let denote its displacement from the origin. The particle is subject to a damping force with damping coefficient , and a spring force with spring constant . The parameters , , and are all positive real numbers. (The quantity is called the "natural frequency of oscillation", because in the absence of the damping force this system would act as a simple harmonic oscillator with frequency .) The motion of the particle is described by Newton's second law: Dividing by the common factor of , and bringing everything to one side, gives We call this ordinary differential equation the "damped harmonic oscillator equation". Since it's a second-order ODE, the general solution must contain two independent parameters. If we state the initial displacement and velocity, and , there is a unique specific solution. Note Sometimes, we write the damped harmonic oscillator equation a bit differently:
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The quantity in the square brackets is regarded as an operator acting on . This operator consists of the sum of three terms: a second-derivative operator, a constant times a first derivative, and multiplication by a constant. This kind of notation is quite common when dealing with differential equations. Complex solution The variable stands for the displacement of a particle, which is a real quantity. But in order to solve the damped harmonic oscillator equation, it's useful if we generalize to complex values. In other words, let's treat the harmonic oscillator equation as a complex ODE: The same parameter-counting rule applies to complex differential equations as real differential equations, except that we use complex parameters in place of real parameters. Since the complex damped harmonic oscillator equation is second-order, its general solution should contain two independent complex parameters—which is equivalent to four real parameters. Once we have that general solution, we can do one of two things: 1. Plug in a complete set of (real) boundary conditions, which will give a real specific solution (see the discussion below), or 2. Take the real part of the complex general solution, which will give the general solution to the real differential equation. (As noted previously, this works because the damped harmonic oscillator equation is linear.) So how do we find a complex general solution to the damped harmonic oscillator equation? First, note that the equation is linear . This means that if we have two solutions and
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