### Dynamic Response » Response to Specific Input Functions » Low-frequency Sine Wave

A measurand that varies as a sine wave with frequency arbitrarily selected to be 0.29 Hz (i.e., a period of 3.4 s) is shown in the next figure. This period is long in comparison to the assumed first-order time constant (𝜏 = 1 s) but still below the natural frequency of the second-order system.

Depiction of a first (M1) and second order (M2) response to a sinusoidal input x, for which the responses show a phase lag and mild damping. Image from NCAR/EOL.

In this example, both response curves (M1 and M2) show attenuated response and a significant time lag between the input and response signals. The time lag leads to a phase lag; i.e., the peaks of the output signal (M1 and M2) waves occur after the peak of the input signal (x). For lower frequencies, the attenuation and phase lag are both reduced, but even at 0.1 Hz, important attenuation and phase lag remain evident in the response curves. There is some transient response at the start of this example because the calculation is started with the measurement and its derivative both set to zero, but the initial transient response no longer has much effect after a few cycles of the sine wave.