one-dimensional wave equation

The one-dimensional wave equation is a partial differential equation that describes how waves behave in a one-dimensional medium. It can be written as:

$$ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} $$

Where:

• $u(x,t)$ is the displacement of the wave at position $x$ and time $t$
• $c$ is the speed of the wave

The general solution to the one-dimensional wave equation can be written as a superposition of two traveling waves:

$$ u(x,t) = f(x-ct) + g(x+ct) $$

Where $f(x)$ and $g(x)$ are arbitrary functions that determine the shape of the wave.

For a given initial displacement $u(x,0)$ and initial velocity $\frac{\partial u}{\partial t}(x,0)$, the functions $f(x)$ and $g(x)$ can be determined using the initial conditions.

For example, if the initial displacement is given by $u(x,0) = f(x) + g(x) = \sin(x)$, and the initial velocity is zero, $\frac{\partial u}{\partial t}(x,0) = 0$, then we can solve for $f(x)$ and $g(x)$:

$$ \begin{aligned} u(x,0) &= f(x) + g(x) \ &= \sin(x) \ \ \frac{\partial u}{\partial t}(x,0) &= -c \cos(x) + c \cos(x) \ &= 0 \end{aligned} $$

Therefore, the solution to the one-dimensional wave equation is:

$$ u(x,t) = \sin(x-ct) + \sin(x+ct) $$

This represents a wave that is traveling to the left with speed $c$, and a wave that is traveling to the right with speed $c$. The sum of these two waves is called a standing wave.