The following function describes oscillations in some mechanical structures and electric circuits. In this function t is time, and is the oscillation frequency in

The following function describes oscillations in some mechanical structures and electric circuits.

 

In this function t is time, and ω is the oscillation frequency in radians per unit time. The oscillations have a period of 2π/ω, and their amplitudes decay in time at a rate determined by τ, which is called the time constant.

The smaller ' is, the faster the oscillations die out.

Suppose that 1 = 0, ω = 2, and τ can have values in the range 0.5 ≤ τ ≤ 10 sec. Then the preceding equation becomes

Obtain a surface plot and a contour plot of this function to help visualize the effect of ' for 0 ≤ t ≤ 15 sec. Let the x variable be time t and the y variable be τ.

z(t) = e sin(wt + 4) sin(ot + 4)