Periodic Function

• A function $F(t)$ is said to be periodic if there exists a nonzero number $T$ such that $F(t+T)=F(t)$ for all $t$ in the domain of $F$.

  • A nonzero constant $T$ for which this is the case is called a period of the function
  • The smallest such $T$ (if it exists) is called the fundamental period (or the period) of the function

• If $F(t)$ is periodic with period $T$, then:

  • $\forall n\in \mathbb{N},F(t+nT)=F(t)$
  • The function $G(t)=F(at)$ is periodic with period $\frac{T}{a}$ (for any nonzero constant $a$)
  • The amplitude of $F$ is $A=\frac{1}{2}(\text{max}(F)-\text{min}(F))$
  • When $F$ is a function of time $t$, then:
    • The period $T$ is the time it takes to complete one full cycle
    • The frequency of $F$ is $f=\frac{1}{T}$ (in ${\mathsf{s}}^{-1}=\mathsf{Hz}$)
    • The angular frequency of $F$ is $\omega =2\pi f$ (in $\mathsf{rad}\cdot {\mathsf{s}}^{-1}$)
  • $F(t)=\frac{A_0}{2}+\sum _{n=1}^{\infty }\left[A_n\sin (2\pi nft)+B_n\cos (2\pi nft)\right]$
    • $f$ is the fundamental frequency (or fundamental harmonic) of $F(t)$
      • $nf$ are the harmonics (or overtones) of $F(t)$
    • $A_n$ and $B_n$ are the sine and cosine amplitudes of the $n$th harmonic
      • $A_n=\frac{2}{T}{\int }_0^{T}F(t)\sin (2\pi nft)dt$
      • $B_n=\frac{2}{T}{\int }_0^{T}F(t)\cos (2\pi nft)dt$
    • $c=\frac{2}{T}{\int }_0^{T}F(t)dt$

• A sine wave (or sinusoid) (symbol: ∿) is any function of the form $y(t)=A\sin (\omega t+\varphi )$

  • A sine wave is a periodic function with period $T=\frac{2\pi }{\omega }$, amplitude $A$, and phase shift $\varphi$.
  • $y(x,t)=A\sin \left(\frac{2\pi }{\lambda }x-\omega t+\varphi \right)=A\sin \left(\frac{2\pi }{\lambda }(x-vt)+\varphi \right)$
  • is the equation of a traveling wave (to the right. if it is to the left, the minus sign is replaced by a plus sign)
    • $x$ is the position of the wave we are considering
    • $vt$ is the distance the wave has traveled from the origin at time $t$