Complex oscillations

# Complex oscillations - Complex oscillations From Physics...

• 9

This preview shows pages 1–3. Sign up to view the full content.

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:

This preview has intentionally blurred sections. Sign up to view the full version.

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
This is the end of the preview. Sign up to access the rest of the document.
• Fall '15
• Complex number, damped harmonic oscillator, harmonic oscillator equation

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern