Solutions to the Partial Differential Equation of Standing Waves
Definition
A wave that satisfies the following condition is referred to as a stationary wave.
$$ \begin{cases} u_{t} = 0 & , t>0 \\ u(t,x) = f(x) & , t=0 \end{cases} $$
Explanation
A stationary wave is a wave whose shape does not change over time. Here, $t$ represents time, $x$ represents position, and $u(t,x)$ represents the waveform at position $x$ when the time is $t$. $f$ represents an initial condition, specifically the waveform when $t=0$.
$$ f(x) = u(0, x) $$
If a solution exists to the partial differential equation of a stationary wave, the solution is as follows.
Solution
Take the definite integral from $0$ to $t$ on both sides.
$$ \int_{0}^{t} {{\partial u} \over { \partial t }} ( s , x ) ds = \int_{0}^{t} 0 ds $$
$$ \implies u(t,x) - u(0,x) = 0 $$
Regardless of $t$, $u(t,x) = u(0,x)$ always holds, therefore $u(t,x) = f(x)$ is the case given the initial condition.
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