A lightly damped oscillator, which oscillates many times before stopping, is one for which $b\text{/}2m\ll {\omega }_0$. In that case, $\omega \approx {\omega }_0$ is a good approximation. That is, light damping does not affect the oscillation frequency.

FIGURE 15.22 is a graph of the position $x\left(t\right)$ for a lightly damped oscillator, as given by Equation 15.55. To keep things simple, we’ve assumed that the phase constant is zero. Notice that the term $A{e}^{-bt\text{/}2m}$, which is shown by the dashed line, acts as a slowly varying amplitude:

where A is the initial amplitude, at $t=0$. The oscillation keeps bumping up against this line, slowly dying out with time.

FIGURE 15.22 Position-versus-time graph for a lightly damped oscillator.

A slowly changing line that provides a border to a rapid oscillation is called the envelope of the oscillations. In this case, the oscillations have an exponentially decaying envelope. Make sure you study FIGURE 15.22 long enough to see how both the oscillations and the decaying amplitude are related to Equation 15.55.

Changing the amount of damping, by changing the value of b, affects how quickly the oscillations decay. FIGURE 15.23 shows just the envelope $x_{\text{max}}\left(t\right)$ for several oscillators that are identical except for the value of the damping constant b. (You need to imagine a rapid oscillation within each envelope, as in Figure 15.22.) Increasing b causes the oscillations to damp more quickly, while decreasing b makes them last longer.

FIGURE 15.23 Oscillation envelopes for a mass of 1.0 kg with several values of $b$.

MATHEMATICAL ASIDE

Exponential decay

Exponential decay occurs in a vast number of physical systems of importance in science and engineering. Mechanical vibrations, electric circuits, and nuclear radioactivity all exhibit exponential decay.

The mathematical analysis of physical systems frequently leads to solutions of the form

$$u=A{e}^{-v\text{/}{v}_0}=A\text{exp}(-v\text{/}{v}_0)$$

where exp is the exponential function. The number $e=2.71828\dots$ is the base of natural logarithms in the same way that 10 is the base of ordinary logarithms.

Full Alternative Text

A graph of u illustrates what we mean by exponential decay. It starts with $u=A$ at $v=0$ (because $e^0=1$) and then steadily decays, asymptotically approaching zero. The quantity $v_0$ is called the decay constant. When $v=v_0$, $u=e^{-1}A=0.37A$. When $v=2v_0$, $u=e^{-2}A=0.13A$.

The decay constant $v_0$ must have the same units as v. If v represents position, then $v_0$ is a length; if v represents time, then $v_0$ is a time interval. In a specific situation, $v_0$ is often called the decay length or the decay time. It is the length or time in which the quantity decays to 37% of its initial value.

No matter what the process is or what u represents, a quantity that decays exponentially decays to 37% of its initial value when one decay constant has passed. Thus exponential decay is a universal behavior. The decay curve always looks exactly like the figure shown here. Once you’ve learned the properties of exponential decay, you’ll immediately know how to apply this knowledge to a new situation.

The mechanical energy of a damped oscillator is not conserved because of the drag force. We previously found the energy of an undamped oscillator to be $E=\frac{1}{2}k{A}^2$. This is still valid for a lightly damped oscillator if we replace A with the slowly decaying amplitude $x_{\text{max}}$. Thus

Here A is the initial amplitude, so $\frac{1}{2}k{A}^2$ is the initial energy, which we call $E_0$. Let’s define the time constant $\tau$ (also called the decay constant or the decay time) to be

Because b has units of kg/s, $\tau$ has units of seconds. With this, we can write the energy decay as

In other words, a lightly damped oscillator’s mechanical energy decays exponentially with time constant $\tau$.

As FIGURE 15.24 shows, the time constant is the amount of time needed for the energy to decay to $e^{-1}$, or 37%, of its initial value. We say that the time constant $\tau$ measures the “characteristic time” during which the energy of the oscillation is dissipated. Roughly two-thirds of the initial energy is gone after one time constant has elapsed, and nearly 90% has dissipated after two time constants have gone by.

FIGURE 15.24 Energy decay of a lightly damped oscillator.

For practical purposes, we can speak of the time constant as the lifetime of the oscillation—about how long it lasts. Mathematically, there is never a time when the oscillation is “over.” The decay approaches zero asymptotically, but it never gets there in any finite time. The best we can do is define a characteristic time when the motion is “almost over,” and that is what the time constant $\tau$ does.