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Harmonic Oscillators

Methods for finding general solution to the linear inhomogeneous second-order ODE

$$a \ensuremath{\frac{d^2{y(t)}}{d{t}^2}} + b \ensuremath{\frac{d{y(t)}}{d{t}}} + c y(t) = F(t)$$ (22-21)

have been developed and worked out in MATHEMATICA$^{\text{\scriptsize {\textregistered }}}$ examples.

Eq. 22-21 arises frequently in physical models, among the most common are:

$$\begin{split}\text{Electrical circuits: \hspace{0.25in}}& L \... ...math{\frac{d{y(t)}}{d{t}}} + K_{s}y(t) = F_{app}(t) \end{split}$$ (22-22)

where:

	Mechanical	Electrical
Second Order	Mass $M$ : Physical measure of the ratio of momentum field to velocity	Inductance $L$ : Physical measure of the ratio of stored magnetic field to current
First Order	Drag Coefficient $c = \eta l_o$ ($\eta$ is viscosity $l_o$ is a unit displacement): Physical measure of the ratio environmental resisting forces to velocity--or proportionality constant for energy dissipation with square of velocity	Resistance $R = \rho l_o$ ($\rho$ is resistance per unit material length $l_o$ is a unit length): Physical measure of the ratio of voltage drop to current--or proportionality constant for power dissipated with square of the current.
Zeroth Order	Spring Constant $K_{s}$ : Physical measure of the ratio environmental force developed to displacement--or proportionality constant for energy stored with square of displacement	Inverse Capacitance $1/C$ : Physical measure of the ratio of voltage storage rate to current--or proportionality constant for energy storage rate dissipated with square of the current.
Forcing Term	Applied Voltage $V(t)$ : Voltage applied to circuit as a function of time.	Applied Force $F(t)$ : Force applied to oscillator as a function of time.

For the homogeneous equations (i.e. no applied forces or voltages) the solutions for physically allowable values of the coefficients can either be oscillatory, oscillatory with damped amplitudes, or, completely damped with no oscillations. (See Figure ). The homogeneous equations are sometimes called autonomous equations--or autonomous systems.

Simple Undamped Harmonic Oscillator

The simplest version of a homogeneous Eq. 22-21 with no damping coefficient ($b=0$ , $R=0$ , or $\eta=0$ ) appears in a remarkably wide variety of physical models. This simplest physical model is a simple harmonic oscillator--composed of a mass accelerating with a linear spring restoring force:

$$\begin{split}\text{Inertial Force} &= \text{Restoring Force}\... ...suremath{\frac{d^2{y(t)}}{d{t}^2}} + K_s y(t) & = 0 \end{split}$$ (22-23)

Here $y$ is the displacement from the equilibrium position-i.e., the position where the force, $F = -dU/dx = 0$ . Eq. 22-23 has solutions that oscillate in time with frequency $\omega$ :

$$\begin{split}y(t) & = A \cos \omega t + B \sin \omega t\\ y(t) & = C \sin ( \omega t + \phi ) \end{split}$$ (22-24)

where $\omega = \sqrt{K_s/M}$ is the natural frequency of oscillation, $A$ and $B$ are integration constants written as amplitudes; or, $C$ and $\phi$ are integration constants written as an amplitude and a phase shift.

The simple harmonic oscillator has an invariant, for the case of mass-spring system the invariant is the total energy:

$$\begin{split}\text{Kinetic Energy} + \text{Potential Energy} ... ...a t + \phi)& =\\ A^2 M \omega^2 & = \text{constant} \end{split}$$ (22-25)

There are a remarkable number of physical systems that can be reduced to a simple harmonic oscillator (i.e., the model can be reduced to Eq. 22-23). Each such system has an analog to a mass, to a spring constant, and thus to a natural frequency. Furthermore, every such system will have an invariant that is an analog to the total energy--an in many cases the invariant will, in fact, be the total energy.

The advantage of reducing a physical model to a harmonic oscillator is that all of the physics follows from the simple harmonic oscillator.

Here are a few examples of systems that can be reduced to simple harmonic oscillators:

Pendulum

By equating the rate of change of angular momentum equal to the torque, the equation for pendulum motion can be derived:

$$M R^2 \ensuremath{\frac{d^2{\theta}}{d{t}^2}} + M g R \sin{\theta} = 0$$ (22-26)

for small-amplitude pendulum oscillations, $\sin(\theta) \approx \theta$ , the equation is the same as a simple harmonic oscillator.

It is instructive to consider the invariant for the non-linear equation. Because

$$\ensuremath{\frac{d^2{\theta}}{d{t}^2}} = \ensuremath{\frac{d{\th... ... \ensuremath{\frac{d{ \ensuremath{\frac{d{\theta}}{d{t}}}}}{d{\theta}}} \right)$$ (22-27)

Eq. 22-26 can be written as:

$$M R^2 \ensuremath{\frac{d{\theta}}{d{t}}}\left( \ensuremath{\frac... ...nsuremath{\frac{d{\theta}}{d{t}}}}}{d{\theta}}} \right) + Mg R \sin(\theta) = 0$$ (22-28)

$$\ensuremath{\frac{d{}}{d{\theta}}}\left[\frac{M R^2}{2} \left( \ensuremath{\frac{d{\theta}}{d{t}}} \right)^2 - Mg R \cos(\theta)\right] = 0$$ (22-29)

which can be integrated with respect to $\theta$ :

$$\frac{M R^2}{2} \left(\ensuremath{\frac{d{\theta}}{d{t}}}\right)^2 - Mg R \cos(\theta) =$$ (22-30)

This equation will be used as a level-set equation to visualize pendulum motion.

Buoyant Object

Consider a buoyant object that is slightly displaced from its equilibrium floating position. The force (downwards) due to gravity of the buoy is $\rho_{bouy} g V_{bouy}$ The force (upwards) according to Archimedes is $\rho_{water} g V_{sub}$ where $V_{sub}$ is the volume of the buoy that is submerged. The equilibrium position must satisfy $V_{sub-eq}/V_{bouy} = \rho_{bouy}/\rho_{water}$ .

If the buoy is slightly perturbed at equilibrium by an amount $\delta x$ the force is:

$$\begin{split}F = & \rho_{water} g ( V_{sub-eq} + \delta x A_o... ...uoy} g V_{buoy}\\ F = & \rho_{water} g \delta x A_o \end{split}$$ (22-31)

where $A_o$ is the cross-sectional area at the equilibrium position. Newton's equation of motion for the buoy is:

$$M_{buoy} \ensuremath{\frac{d^2{y}}{d{t}^2}} - \rho_{water} g A_o y = 0$$ (22-32)

so the characteristic frequency of the buoy is $\omega = \sqrt{\rho_{water} g A_o/M_{bouy}}$ .

Single Electron Wave-function

The one-dimensional Schrödinger equation is:

$$\ensuremath{\frac{d^2{\psi}}{d{x}^2}} + \frac{2m}{\hbar^2} \left( E - U(x)\right) \psi = 0$$ (22-33)

where $U(x)$ is the potential energy at a position $x$ . If $U(x)$ is constant as in a free electron in a box, then the one-dimensional wave equation reduces to a simple harmonic oscillator.

In summation, just about any system that oscillates about an equilibrium state can be reduced to a harmonic oscillator.