Damped sinusoids are sinusoidal signals that are multiplied by decaying exponentials. These signals are commonly encountered in various fields, such as electrical engineering and physics. Understanding damped sinusoids is important as they are used to describe the behavior of many physical systems, including electrical circuits, mechanical oscillators, and electromagnetic waves.

Damped sinusoids can be mathematically represented as:

x(t) = A * sin(ωt + φ) * e^(-αt)

Where:
- x(t) is the damped sinusoidal signal.
- A is the amplitude of the sinusoid.
- ω is the angular frequency of the sinusoid.
- φ is the phase angle of the sinusoid.
- α is the damping factor, which determines the rate at which the exponential decays over time.
- t is the time variable.

Now, let's discuss why damped sinusoids are referred to as "damped" and why they are represented as a multiplication of sinusoids and decaying exponentials.

Damping refers to the process of reducing or suppressing the oscillations of a system. In the context of sinusoidal signals, damping occurs when there is a dissipation of energy over time. This dissipation can be caused by various factors, such as resistive losses in electrical circuits or friction in mechanical systems.

The decaying exponential term, e^(-αt), represents the damping factor α, which determines the rate at which the amplitude of the sinusoid decreases over time. As time increases, this exponential term decreases exponentially, causing the amplitude of the sinusoid to decrease as well. Therefore, the sinusoidal part of the signal is "damped" or attenuated.

The sinusoidal term, A * sin(ωt + φ), represents the oscillatory behavior of the signal. It describes the periodic variation of the signal with time. The amplitude A, angular frequency ω, and phase angle φ determine the specific shape and characteristics of the sinusoidal waveform.

By multiplying the sinusoidal term with the decaying exponential term, we obtain a damped sinusoidal signal that captures both the oscillatory behavior and the damping effect. The exponential term allows us to model the gradual decay of the sinusoid's amplitude, simulating the dissipation of energy in a system.

In summary, damped sinusoids are sinusoidal signals multiplied by decaying exponentials. They are referred to as "damped" because they represent the attenuation or damping of the sinusoidal waveform over time. The multiplication of sinusoids and decaying exponentials allows us to capture both the oscillatory behavior and the damping effect in various physical systems.