y = e^{-x}\sin x

Graph of the Damped Oscillation $y = e^{-x}\sin x$

$y = e^{-x}\sin x$ is the curve of a damped oscillation, a trigonometric function weighted by an exponential¹. It is the most basic form for a phenomenon that oscillates while its amplitude shrinks.

Domain and zeros

The domain is all real numbers. Since $e^{-x}$ never vanishes, $y = 0$ only where $\sin x = 0$ , that is at $x = n\pi$ . Even with the damping the zeros stay evenly spaced; only the amplitude changes, and that is the characteristic feature.

The envelope

From $|\sin x| \leq 1$ the curve is caught between two exponential curves.

-e^{-x} \leq y \leq e^{-x}

Those two are the envelope, and the curve touches them at $x = \dfrac{\pi}{2} + n\pi$ , where $|\sin x| = 1$ . The squeeze theorem then gives $y \to 0$ as $x \to +\infty$ .

Three characteristic positions

The derivative is $y' = e^{-x}(\cos x - \sin x)$ . Since $e^{-x} > 0$ , the extrema sit where $\cos x = \sin x$ , that is at $x = \dfrac{\pi}{4} + n\pi$ . The second derivative is $y'' = -2e^{-x}\cos x$ , whose sign changes where $\cos x = 0$ .

Position	$x$
Extremum	$\dfrac{\pi}{4} + n\pi$
Contact with the envelope	$\dfrac{\pi}{2} + n\pi$
Inflection point	$\dfrac{\pi}{2} + n\pi$
Zero	$n\pi$

Each extremum comes $\dfrac{\pi}{4}$ before the point of contact. Climbing while decaying, the curve passes its peak before $\sin x$ reaches its maximum. The inflection points, by contrast, coincide exactly with the points of contact. The first maximum is $\dfrac{\sqrt{2}}{2}e^{-\pi/4} \approx 0.322$ , at $x = \dfrac{\pi}{4}$ .

How fast it decays

Shifting by a half period gives $y(x + \pi) = -e^{-\pi}y(x)$ , so the ratio of the absolute values of consecutive extrema is constant.

Shift	Ratio
Half period $\pi$	$e^{-\pi} \approx 0.0432$
Full period $2\pi$	$e^{-2\pi} \approx 0.00187$

The oscillation dies away extremely quickly. The logarithm of that constant ratio is called the logarithmic decrement and is used as a measure of how strongly a system is damped.

Behavior on the left

As $x \to -\infty$ the factor $e^{-x}$ diverges, so the oscillation grows exponentially in amplitude. The same formula shows two completely different faces on the left and the right halves of the graph.

Integral

\int_0^{\infty} e^{-ax}\sin(bx)\,dx = \frac{b}{a^2+b^2}

Taking $a = b = 1$ gives $\dfrac{1}{2}$ . The formula is one of the basic entries in a table of Laplace transforms.

Applications

The transient response of a vibrating system with a spring and a damper
The damped oscillation of an $RLC$ circuit
The way the swaying of a building or a bridge settles down

It is the picture of what happens when damping is weak: the underdamped case, where the motion swings back and forth a few times before dying out.

Damping, Wikipedia

Graph of the Damped Oscillation y=e−xsin⁡xy = e^{-x}\sin xy=e−xsinx