sinusoidal

(adjective)

In the form of a wave, especially one whose amplitude varies in proportion to the sine of some variable (such as time).

Examples of sinusoidal in the following topics:

  • Sinusoidal Nature of Simple Harmonic Motion

    • The solutions to the equations of motion of simple harmonic oscillators are always sinusoidal, i.e., sines and cosines.
    • These are all sinusoidal solutions .
    • Uniform circular motion is therefore also sinusoidal, as you can see from .
    • Only the top graph is sinusoidal.
    • Review factors responsible for the sinusoidal behavior of uniform circular motion

  • Phasors

    • The sinusoidal function $A cos( \omega t + \phi)$ can be written as a phasor: $Ae^{j \theta}$ .
    • This can be particularly useful because the frequency factor (which includes the time-dependence of the sinusoid) is often common to all the components of a linear combination of sinusoids.
    • Sinusoids can be represented mathematically as the sum of two complex-valued functions:
    • Sinusoidal Steady State and the Series RLC CircuitPhasors may be used to analyze the behavior of electrical and mechanical systems that have reached a kind of equilibrium called sinusoidal steady state.
    • In the sinusoidal steady state, every voltage and current (or force and velocity) in a system is sinusoidal with angular frequency ω.

  • Motivation

    • For instance, when we studied the forced harmonic oscillator, we first solved the problem by assuming the forcing function was a sinusoid (or complex exponential).
    • We then argued that since the equations were linear this was enough to let us build the solution for an arbitrary forcing function if only we could represent this forcing function as a sum of sinusoids.
    • Later, when we derived the continuum limit of the coupled spring/mass system we saw that separation of variables led us to a solution, but only if we could somehow represent general initial conditions as a sum of sinusoids.

  • Damped transient motion

    • This looks like the equation of a damped sinusoid.
    • But the second term may or may not be a sinusoid, depending on whether the square root is positive.
    • First if $\frac{\gamma}{2\omega_0} < 1$ , corresponding to small damping, then the argument of the square root is positive and indeed we have a damped sinusoid.

  • Aliasing

    • The extrema (peaks and troughs) of a sinusoid of frequency $f_s$ will lie exactly $1/2f_s$ apart.
    • Figure 4.9: A sinusoid sampled at less than the Nyquist frequency gives rise to spurious periodicities.

  • Root Mean Square Values

    • ., sinusoids.The RMS value of a set of values (or a continuous-time function such as a sinusoid) is the square root of the arithmetic mean of the squares of the original values (or the square of the function).
    • We can see from the above equations that we can express the average power as a function of the peak voltage and current (in the case of sinusoidally varying current and voltage):
    • The RMS values are also useful if the voltage varies by some waveform other than sinusoids, such as with a square, triangular or sawtooth waves .
    • The voltage and current are sinusoidal and are in phase for a simple resistance circuit.

  • Driven Oscillations and Resonance

    • This equation can be solved exactly for any driving force, using the solutions z(t) which satisfy the unforced equation: $\frac{\mathrm{d}^2z}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}z}{\mathrm{d}t} + \omega_0^2 z = 0$, and which can be expressed as damped sinusoidal oscillations $z(t) = A \mathrm{e}^{-\zeta \omega_0 t} \ \sin \left( \sqrt{1-\zeta^2} \ \omega_0 t + \varphi \right)$in the case where ζ ≤ 1.
    • In the case of a sinusoidal driving force: $\frac{\mathrm{d}^2x}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}x}{\mathrm{d}t} + \omega_0^2 x = \frac{1}{m} F_0 \sin(\omega t)$, where $\!
    • \omega$ is the driving frequency for a sinusoidal driving mechanism.

  • Period and Frequency

    • ., in rotation) or the rate of change of the phase of a sinusoidal waveform (e.g., in oscillations and waves), or as the rate of change of the argument of the sine function.
    • Sinusoidal waves of various frequencies; the bottom waves have higher frequencies than those above.

  • Resistors in AC Circuits

    • If the source varies periodically, particularly sinusoidally, the circuit is known as an alternating-current circuit.
    • The voltage and current are sinusoidal and are in phase for a simple resistance circuit.

  • Convergence Theorems

    • Figure 4.12: The top left plot shows an input time series consisting of a single sinusoid.
    • In the middle we show the same input sinusoid contaminated with some uniformly distributed pseudo-random noise and its DFT.