The equation you provided, f(t) = sin(ωt) + e^(-λt), is not a correct representation of a frequency function with damping.
In the presence of damping, the behavior of oscillatory systems is typically described by the damped harmonic oscillator equation, which includes both the oscillatory term and an exponential decay term. The equation for a damped harmonic oscillator can be written as:
x(t) = A * e^(-λt) * cos(ωt + φ),
where:
- x(t) represents the displacement of the oscillator at time t,
- A is the initial amplitude of the oscillation,
- λ is the damping constant (related to the damping coefficient),
- ω is the angular frequency of the undamped oscillator, and
- φ is the phase constant.
The term e^(-λt) represents the exponential decay, while the cos(ωt + φ) represents the oscillatory behavior.
It's important to note that the frequency of the damped oscillator is not constant but depends on the damping factor λ. As the damping increases, the oscillations become less pronounced, and the frequency decreases.
In summary, the equation you provided does not accurately represent the frequency function with damping. The correct equation for a damped harmonic oscillator includes both an exponential decay term and an oscillatory term.