I am trying to solve the inhomogeneous wave equation
$$\frac{\partial ^2\phi(x,t)}{\partial x^2}-\frac{\partial ^2\phi(x,t)}{\partial t^2}=\delta(x)u(t)$$
on domain $-\infty < x < \infty$ and $t>0$ using two different methods. In a maths course, I learned that this equation can be solved via Greens functions
$$G(x,t,x,t')=-\frac{1}{4\pi}\frac{\delta(|x-x'|-t+t')}{|x-x'|}$$
The solution to the inhomogeneous wave equation is then given the convolution of the Greens function with the inhomogeneous term
$$\int dxdt\,G(x,t,x,t')\delta(x')u(t')=-\frac{1}{4\pi}\frac{u(t-|x|)}{|x|}$$
However I am trying to obtain the same solution via application of Laplace and Fourier Transform. One gets then
$$-k^2\tilde{\phi}(k,s)-s^2\tilde{\phi}(k,s)=\frac{1}{s}$$
where we have initial conditions given by $\phi(x,0)=0$ and $\dot{\phi}(x,0)=0$. Rearranging yields
$$\tilde{\phi}(k,s)=-\frac{1}{s(s^2+k^2)}$$
Now taking the inverse Laplace Transform with the help of a lookup table we get
$${\hat{\phi}}(k,t)=-\frac{1-\cos(kt)}{k^2}$$
However the inverse Fourier transform of this doesn't yield the same solution as above. What am I doing wrong?