PX275 - G7 - dirac delta function

PX275 - E2b - dirac delta

the motivation is to explore another condition
for a vibrating string, governed by the wave PDE, considering injecting a local impulse near $x = x_{0}$ , ie. hitting it with a hammer of width, $w$
so, the impulse acts on a range: $x_{0} - w / 2$ to $x_{0} + w / 2$

\frac{\partial u (x, 0)}{\partial t} = {\begin{cases} 0 & x < x_{0} - w / 2, \\ V_{0} & x_{0} - \frac{w}{2} < x < x_{0} + w / 2, \\ 0 & x > x_{0} + w / 2 \end{cases}

making the hammer narrower and narrower, ie: $w \to 0$ , for a constant $V_{0} :$

\int \frac{\partial u (x, 0)}{\partial t} d x \to 0

but if $V_{0} \propto 1 / w$ , $lim_{w \to 0} V_{0} \to \infty :$

\int \frac{\partial u (x, 0)}{\partial t} d x = const

the limit of a step function is the dirac delta function, $δ (x)$
it is normalized, ie:

\int_{- \infty}^{\infty} δ (x) d x = 1

in the limit, the function has zero width, and infinite height
the function is non-zero only at a single $x$ value

definition 1

only non-zero at a single $x$

δ (x) = {\begin{cases} 0 & | x | > w / 2, \\ 1 / w & | x | \leq w / 2 \end{cases}

definition 2

the limit of a gaussian:

lim_{w \to 0} δ (x) = N \exp (- \frac{x^{2}}{w^{2}})

this can be normalized
it is arbitrarily narrow

δ (x - x_{0}) = {\begin{cases} 0 & x \neq x_{0} \\ \infty & x = x_{0} \end{cases}

\int δ (x - x_{0}) = {\begin{cases} 1 & if integration range includes x_{0}, \\ 0 & otherwise \end{cases}

this can be used for:

\int f (x) δ (x - x_{0}) = {\begin{cases} f (x_{0}) & if integration range includes x_{0}, \\ 0 & otherwise \end{cases}

implementation

using the dirac delta function for the local injection of pulse

\begin{matrix} u (x, 0) = 0 & 0 < x < L \\ \frac{\partial u (x, 0)}{\partial t} & = V_{0} δ (x - x_{0}) \end{matrix}

for a clamped string:

u (x, t) = \sum_{n} \sin (\frac{n π x}{L}) (C_{n}^{'} \cos (\frac{n π t}{L}) + D_{n}^{'} \sin (\frac{n π t}{L}))

u (x, 0) = \sum \sin (\frac{n π x}{L}) C_{n}^{'} \forall x

⟹ C_{n}^{'} = 0 \forall n

\frac{\partial u (x, t)}{\partial t} = \sum_{n} \sin (\frac{n π x}{L}) D_{n}^{'} \frac{n π c}{L} \cos (\frac{n π c t}{L})

at $t = 0 :$

\sum_{n} \sin (\frac{n π x}{L}) D_{n}^{'} \frac{n π c}{L} = V_{0} δ (x - x_{0})