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Let the wave equation: $u_{tt}-c^2u_{xx}=0 \;\;\forall x \in \mathbb R,\;t\gt 0$

Show that:

  1. Huygens Principle holds if $u(x,0)=f(x)\;,\;u_t(x,0)=0$
  2. Huygens Principle doesn't hold if $u(x,0)=0\;,\;u_t(x,0)=g(x)$

For both parts, I used D'Alembert's formula , so I concluded to these:

  1. $u(x,t)=\frac{1}{2} [f(x+ct)+f(x-ct)] \;\;$
  2. $u(x,t)=\frac{1}{2c} \int_{x-ct}^{x+ct} g(y) dy$

My question is if it's enough to say from the above that:

  1. $u\;$ depends only on the value of $f\;$ at $x+ct,\;x-ct\;$ which are the boundary of 'domain of dependence' so the Principle holds
  2. .$u\;$ depends on the value of $g\;$ in the closed interval $[x-ct,x+ct]$ and so the principle does not hold.

I assume that this is not enough. It seems quite easy to be this the answer. Should I show something more? What am I missing?

I would appreciate any help. I 've recently started to study about Huygens Principle so it may be something elementary what I'm asking.. Thanks in advance!!

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    Seems like enough to me, unless you have some very precise definition of Huygen's principle that needs to be verified. You could give an example where $g(x+ct)=g(x-ct)=0$ but $u(x,t)\neq 0$. – 2017-01-20
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    @Jeff That was quick! No, the definition we have is a bit general, nothing specific! Thanks a lot for your comment – 2017-01-20
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    What is the definition you are using? – 2017-02-08

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