2016-10-20

4. The calculation of PSV for quasitriangular pulse

In this addition to the article about the square of the displacement of the standing wave, we show how to calculate the PSV chipremoving pulse with a duty ratio of \(Q\). The prefix "quasi" is used here due to the approximation studied here pulse to a rectangular shape. Moreover, the more involved harmonics \(N\), the more accurate this approximation.

It is known that a rectangular pulse can be "assembled" from harmonic cosine Fourier series [1]: \[A(t) = \frac12 \sum_{i=1}^N a_i \cos(i\,\omega\,t), \qquad a_i= \frac{4}{\pi\,i} \sin\left({\pi\,i \over Q}\right) \qquad (4.1) \] Then for \(\lambda = \frac12\) according to (1.6) the formula of the standing wave will be this: \[A(x,t) = \sum_{i=1}^N (a_i \cos(ik x) \cos(i\omega t)) \qquad (4.2) \] For nding PSV wave from this stimulus, we will use the approximation formula (1.13): \[S \approx 8 \sum_{i=1}^{N} a_{i}a_{i-1}{ i^2 (i-1)^2 \over (2i-1)^2} \] since: \[ a_i= \frac{4}{\pi\,i} \sin\left({\pi\,i \over Q}\right), \qquad a_{i-1}= \frac{4}{\pi\,(i-1)} \sin\left({\pi\,(i-1) \over Q}\right) \] and making the substitution we get: \[S \approx \frac{128}{\pi^2} \sum_{i=1}^{N} \left[ \sin\left({\pi\,i \over Q}\right) \sin\left({\pi\,(i-1) \over Q}\right){i(i-1) \over (2i-1)^2} \right] \qquad (4.3) \] If \(N \gt 8\), then this formula with sufficient accuracy can be simplified: \[S \approx \frac{16\,N}{\pi^2} \cos(\frac{\pi}{Q}) \qquad (4.4) \]

Nimirum PSV

The formula of normalization is taken from (1.14). Substituting back \(a_i\) and making simple transformations, we obtain the quantity surveyor: \[ \Psi = \frac{4}{\pi^2} \sum_{i=1}^N \frac{1}{i^2} \sin\left({\pi\,i \over Q}\right)^2 \qquad (4.5) \]

PSV relative to chipremoving pulse

From formulas (4.3, 4.4, 4.5) we obtain the final result.

When \(N \le 8\): \[ \bar S \approx 32 {\sum_{i=1}^{N} \left[ \sin\left({\pi\,i \over Q}\right) \sin\left({\pi\,(i-1) \over Q}\right){i(i-1) \over (2i-1)^2} \right] \over \sum_{i=1}^N \frac{1}{i^2} \sin\left({\pi\,i \over Q}\right)^2 } \qquad (4.6) \] When \(N \gt 8\): \[ \bar S \approx { 4 N \cos(\frac{\pi}{Q}) \over \sum_{i=1}^N \frac{1}{i^2} \sin\left({\pi\,i \over Q}\right)^2 } \qquad (4.7) \] One of the completely non-obvious conclusions from this formula is such that for a sufficiently large number of harmonics relative PSV proportional to the first degree \(N\), in contrast to the quadratic dependence in equation (1.19), note that there are all \(a_i\) were equal.

When \(N \le 8\): \[ \bar S \approx 32 {\sum_{i=1}^{N} \left[ \sin\left({\pi\,i \over Q}\right) \sin\left({\pi\,(i-1) \over Q}\right){i(i-1) \over (2i-1)^2} \right] \over \sum_{i=1}^N \frac{1}{i^2} \sin\left({\pi\,i \over Q}\right)^2 } \qquad (4.6) \] When \(N \gt 8\): \[ \bar S \approx { 4 N \cos(\frac{\pi}{Q}) \over \sum_{i=1}^N \frac{1}{i^2} \sin\left({\pi\,i \over Q}\right)^2 } \qquad (4.7) \] One of the completely non-obvious conclusions from this formula is such that for a sufficiently large number of harmonics relative PSV proportional to the first degree \(N\), in contrast to the quadratic dependence in equation (1.19), note that there are all \(a_i\) were equal.

PSV for some quasiparabolic pulses

Momentum | N | Q | S | |

3 | 3 | 2 | 5.7 | |

3 | 7 | 3.4 | 18 | |

5 | 3 | 4.6 | 11 | |

5 | 7 | 8.8 | 41 | |

9 | 3 | 7 | 17 | |

9 | 7 | 11 | 50 | |

13 | 3 | 9.4 | 22 | |

13 | 7 | 20 | 87 | |

13 | 11 | 17 | 117 | |

21 | 3 | 17 | 38 | |

21 | 7 | 30 | 129 | |

21 | 11 | 34 | 217 | |

21 | 15 | 31 | 269 | |

21 | 21 | 34 | 411 |

__The materials used__