# In 1D wave propagation problem, how to find the curl of a given source function?

I am trying to solve 1-D wave equation by calculating potentials ϕandψxϕandψx for displacement field ux=ϕ+×ψux=∇ϕ+∇×ψ. I am trying to decompose my source function fxfx (such that fx=b+×hxfx=∇b+∇×hx) in terms of potentials bandhxbandhx, using which I can can compute ϕandψxϕandψx. While decomposing source term fxfx, trying to calculate bandhxbandhx, I have a problem/confusion in finding the curlcurl of hxhx. Please see the manual hand-written picture attached here

I am solving for ww via the Poisson's equation 2wx=fx∇2wx=fx, in order to evaluate the potentials bandhxbandhx. So that the scalar potential b=wxb=∇⋅wx and vector potential hx=×wx.hx=−∇×wx. But obviously, the curlcurl of wxwx (hx=×wxhx=−∇×wx), has components in jandkjandk directions. The wave propagation is considered only in 1-D (ii-direction). So, should I ignore the values in other directions (jandkjandk)?. I am pretty sure that I am missing some mathematics in understanding the curl of a 1D vector field.

What will be the components of the potentials bandhxbandhx looks like? Any references to this problem or curl operators of 1D vector is much appreciable.Please throw some light on my question. Many thanks and much appreciable for help.

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