First harmonic motion

Question

The vector Laplace's equation is expressed by:

$\nabla^2 F=0.$ A function $\phi$ which satisfies Laplace's equation is said to be harmonic.

I want to look at a vibrating string in the unit square that moves according to the first harmonic. So, let's say you have a string that is fixed at $(0,0)$ and also at $(1,1)$. How do I express this string moving according to the first harmonic?

I can't figure this out because the string is not normally defined as being tilted as such in the picture. Could I find the solution and then somehow rotate that solution? I really am stuck.

Answer 1

Express this as a half-cosine wave of length $\sqrt{2}$, fixed at the origin $(0,0)$, then rotate each $(x,y)$ around the origin by $\pi/4$.

Answer 2

Define $u=x+y, v=x-y$ as new coordinates. The diagonal of the square is the $u$ axis, running from $0$ to $2$. The origin stays at the same point. The displacement of the string is then $v=A\sin \frac \pi 2 u$ where $A$ is the half amplitude.