# LaPlace Transforms

- How do we model a resonator when its forcing function is not continuous such as when a bell is struck?

#### Integral Transforms

#### The Laplace Transform

$$L\left\{f\left(t\right)\right\}=F\left(s\right)={\int}_{0}^{\infty}{e}^{-st}f\left(t\right)dt$$The laplace transform of a function only exists only if there exist numbers M and C such that: $$f\left(t\right)=<M{e}^{Ct}$$ For all values of t > 0.

Important Transforms | |
---|---|

f(t) | F(s) |

$\delta \left(t\right)$
(Dirac Delta Function) |
$1$ |

$u\left(t\right)$
(Heviside Step Function) |
$\frac{1}{s}$ |

t^{n} |
$\frac{n!}{{s}^{n+1}}$ |

e^{-at} |
$\frac{1}{s+a}$ |

sin(ωt) | $\frac{\omega}{{s}^{2}+{\omega}^{2}}$ |

cos(ωt) | $\frac{s}{{s}^{2}+{\omega}^{2}}$ |

Operational Transforms | |
---|---|

f(t) | F(s) |

C f(t) (multiplication by a constant) | C F(s) |

f(t)+g(t) | F(t)+G(t) |

$\frac{df\left(t\right)}{dx}$ | $sF\left(s\right)-f\left(0\right)$ |

$\frac{{d}^{2}f\left(t\right)}{{dx}^{2}}$ | ${s}^{2}F\left(s\right)-sf\left(0\right)-\frac{df\left(0\right)}{dx}$ |

$\frac{{d}^{3}f\left(t\right)}{{dx}^{3}}$ | ${s}^{3}F\left(s\right)-{s}^{2}f\left(0\right)-s\frac{df\left(0\right)}{dx}-\frac{{d}^{2}f\left(0\right)}{{dx}^{2}}$ |

f(t-a)u(t-a), a>0 (time shift) | ${e}^{-as}F\left(s\right)$ |

f(at), a>0 (time scale) | $\frac{1}{a}F\left(\frac{s}{a}\right)$ |

Integral Transforms

The Laplace Transform is an integral transform. It is useful for solving differential equations with non-continuous forcing functions such as modeling the response of a resonator after applying some pulse. s can be a complex value.

Shifts in the time domain

Shifts in the frequency domain

### Solving Differential Equations with Laplace Transforms

Transforming a differential equation into the s domain by means of the laplace transform makes its solution an algebraic problem. Then the solution must be transformed back into the t domain.