40 Stationary oscillations

Before we consider waves, which are a pattern that propagates through space, let’s look at stationary oscillations. The most common system used to exemplify stationary oscillations is a mass suspended at the end of a spring.

At its resting location, z = 0, there is a balance between an upward force due to tension in the coiled spring that balances the downward force due to gravity.

Animation of a mass and spring, an example of an undamped harmonic oscillator (click for animation).

Now, imagine pulling the spring downward from its resting position by a small displacement to z=-\eta_0.  Because of increased tension in the coiled spring, there will be a net upward force, called a restoring force, and the mass will accelerate upward when released.  The weight will pass through z = 0 with maximum velocity as the net upward force goes to zero and then begin to decelerate as the sign of the net force (spring tension + gravity) reverses.  It will have zero velocity at the top of the oscillation z=+\eta_0 and then begin to accelerate downward.  In the absence of friction, this oscillation can continue indefinitely.

If the restoring force is proportional to the displacement of the mass from its rest position, the system is said to obey Hooke’s Law.  In this case F = -k\eta, where the constant of proportionality between force and displacement is, in this example, called the “spring constant”.  We can write Newton’s second law (F = ma) for this system, rearranging slightly as

    \[ a - \frac{F}{m}=0 \]

and then write both the acceleration and the force in terms of the displacement \eta

    \[ \frac{\partial^2 \eta}{\partial t^2} + \frac{k}{m}\eta=0 \]

Let’s substitute in \eta = -\eta_0 cos(\omega t), recognizing that the displacement starts at t = 0 at \eta = -\eta_0.  The first deriative of the cosine function will be \omega \eta_0 sin(\omega t) and the second derivative will be \omega^2 \eta_0 cos(\omega t). Thus we get:

    \[  \omega^2 cos(\omega t) - \frac{k}{m}cos(\omega t)=0 \]

which will be true as long as

    \[ \omega = \sqrt{\frac{k}{m}} \]

Omega is the frequency of the oscillation – how often in time the cosine function makes a complete cycle over its possible values. Its units are radians per second, often written as s-1.  It is related to the period (T) of the oscillation, T=(2 \pi)/\omega.  The period gives the length in time for a complete cycle.

A sine function could also be a solution to this equation if we added a constant -\pi/2 to its argument (i.e., inside the parentheses), and it is somewhat arbitrary whether we choose to use the sine or the cosine. The added constant is called a phase shift, and is used to match the initial condition in either case. Sines and cosines are called harmonic functions, and this system is called a harmonic oscillator.

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