How Does the Wave Equation Derive and What Solutions Exist?

[Small bugs]

$$\frac{\partial^2}{\partial t^2}u(x,t)=c^2\Delta u(\vec{x},t)\qquad \vec{x}\in \mathbb{R}^n$$
is known as the wave equation. It seems not very trivial, so is there any derivations or inspirations of it?

To solve this equation, we have to know the initial value and boundary conditions:
\begin{equation*}
\begin{cases}
u(0,t)=u(\vec{l},t)=0\\
u(\vec{x},0)=f(\vec{x})\\
u_t(\vec{x},0)=g(\vec{x})\\
\end{cases}
\end{equation*}
This above can be solved uniquely, with separation of variables.
And also see these conditions:
\begin{equation*}
\begin{cases}
u(\vec{x},0)=f(\vec{x})\\
u_t(\vec{x},0)=g(\vec{x})\\
\end{cases}
\end{equation*}
Why this above can be also solved uniquely with d'Alembert or Kirchhoff's method? Why the boundary conditions can be removed easily? So it seems that it has no influence?

 

[SteamKing]

$$\frac{\partial^2}{\partial t^2}u(x,t)=c^2\Delta u(\vec{x},t)\qquad \vec{x}\in \mathbb{R}^n$$
is known as the wave equation. It seems not very trivial, so is there any derivations or inspirations of it?

Obviously, there are. Have you tried looking for them?

Google "wave equation"

 

[mathman]

If you look at the one dimensional version it become more transparent that sine or cosine of the appropriate arguments satisfy the equation.

 

[Small bugs]

If you look at the one dimensional version it become more transparent that sine or cosine of the appropriate arguments satisfy the equation.

Im just wondering why the different initial conditions as i posted give both an unique solution?

 

[Jilang]

They first of the three is telling you that the ends are fixed, so it is relevant.

 

[vanhees71]

To make the solution unique you need all three conditions (the first is called a boundary and the 2nd and 3rd initial contitions). The (1+1)D case is indeed most simple. First you should show that the general solution of the wave equation in this case reads
$$u(t,x)=u_1(x-c t)+u_2(x+c t),$$
where $u_1$ and $u_2$ are arbitrary functions that are at least twice differentiable.

Hint: Introduce the new independent variables $\xi=x-c t$ and $\eta=x+c t$ and show that the wave equation is equivalent to
$$\frac{\partial^2}{\partial \xi \partial \eta} u=0.$$

Now think about the boundary conditions and how to work in the initial conditions.

Hint: You should start with the initial conditions, plugging in the above given general solution. What conclusions can you draw from them on the definition of the functions on the interval $[0,L]$? Then you should think about how to periodically continue the function to the entire real axis to fulfill also the boundary conditions.

That's called the d'Alembertian approach. Another very illuminating way is to use Fourier series, starting with the boundary condition, then using the wave equation to constrain the coefficients and finally use the initial conditions to fully determine them. Of course, both ways lead to the same result!