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# 数学代写|偏微分方程作业代写Partial Differential Equations代考|Consequences of D’Alembert’s Formula: Causality

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## 数学代写|偏微分方程作业代写Partial Differential Equations代考|The Domain of Dependence and Influence

The most general way to view causality (what causes what) is via the domain of dependence and domain of influence. Both are regions in the $x$ – vs. $t$-plane (space-time), one based upon the past and the other on the future.

The first is the domain of dependence, illustrated in Figure 3.3: Here we fix a point $\left(x_{1}, t_{1}\right)$ in space-time and look at the past. To this end, consider the dependence of the solution $u\left(x_{1}, t_{1}\right)$ on the initial data. Using D’Alembert’s formula, the solution $u\left(x_{1}, t_{1}\right)$ depends on the initial displacements at positions $x_{1}-c t_{1}$ and $x_{1}+c t_{1}$ and all the initial velocities between them. Hence, it depends on what happened initially (displacement and velocity) in the interval $\left[x_{1}-c t_{1}, x_{1}+c t_{1}\right]$. Now let us consider any earlier time $t_{2}$, i.e., $0<t_{2}<t_{1}$, starting our stopwatch here. We can think of $t_{2}$ as the initial time and ask what positions are relevant at time $t_{2}$ for determining $u\left(x_{1}, t_{1}\right)$. This is a shorter interval as less time has elapsed from $t_{2}$ to $t_{1}$ than from 0 to $t_{1}$. In fact the interval is exactly $\left[x_{1}-c\left(t_{1}-t_{2}\right), x_{1}+c\left(t_{1}-t_{2}\right)\right]$. Thus, the entire domain of dependence for a point $\left(x_{1}, t_{1}\right)$ in space-time is the shaded triangular region shown on the left of Figure 3.3.

## 数学代写|偏微分方程作业代写Partial Differential Equations代考|Two Examples: A Plucked String and a Hammer Blow

We present two examples wherein one of the data functions is identically 0 . They correspond, respectively, to the situations of a plucked string and a hammer blow.

Warning: In each case, the illustrated dynamics given by the wave equation will not coincide with your intuition of plucking a guitar string nor hitting a piano string. This is because of our assumption of an infinitely long string. An actual guitar, or piano, string is finite with fixed ends (no displacement at the end points), and as we shall see later, these boundary conditions greatly affect the wave dynamics.

The Plucked String. Let $c=1$,
$$\phi(x)= \begin{cases}1-|x,| & |x| \leq 1 \ 0, & |x|>1\end{cases}$$
and $\psi \equiv 0$. By D’Alembert’s formula, the solution is
$$u(x, t)=\frac{1}{2}[\phi(x+t)+\phi(x-t)] .$$
Thus, at any time $t$, the solution is simply the sum of two waves with the same initial shape but half the amplitude, one which has moved to the right by $t$ and one which has moved to the left by $t$. We plot some of the profiles in Figure 3.4.

## 数学代写|偏微分方程作业代写PARTIAL DIFFERENTIAL EQUATIONS代考|TWO EXAMPLES: A PLUCKED STRING AND A HAMMER BLOW

φ(X)={1−|X,||X|≤1 0,|X|>1