So I'm trying to solve the following wave equation $$u_{tt} - c^2u_{xx}=g$$ where $g$ is the gravitational constant, modelling a semi - infinite strong which is fixed at one end, can support wave-like motions and is allowed to fall under gravity, with initial conditions
$$u(0,t) = u(x,0) = u_{t}(x,0) = 0 \ \ (1)$$
$$\lim_{x \to\ \infty} u(x,t) \rightarrow \frac{gt^2}{2} \ \ (2)$$
By taking the Laplace transform of both sides and rearranging to yield $$\hat{u}_{xx}-\frac{s^2}{c^2}\hat{u}=-\frac{g}{sc^2}$$
This has solution $$\hat{u}(x,s) = A(s) e^{\frac{x}{c}s} +B(s)e^{-\frac{x}{c}s}+\frac{g}{s^3}$$
Using condition $(2)$ gives us $A(s)=0$ and $u(0,t) = 0$ gives us $B(s) = -\frac{g}{s^3}$
We can then find $$u(x,t) = \mathcal{L}^{-1}(\frac{g}{s^3} - \frac{g}{s^3}e^{-\frac{x}{c}s})$$
Writing $$\frac{g}{s^3}e^{-\frac{x}{c}s} = \frac{g}{s^2}\frac{e^{-\frac{x}{c}s}}{s}$$ and using the convolution theorem gives us $$u(x,t) = \frac{gt^2}{2} - \int^t_0 g\tau H(t-\frac{x}{c}-\tau) d\tau = \frac{gt^2}{2} - g \int^{t-\frac{x}{c}}_0 \tau d\tau =g\frac{x}{c} - \frac{1}{2}g\frac{x^2}{c^2}$$
which contradicts the earlier limiting condition. Clearly I have gone wrong somewhere, (I think it must be in my application of the boundary conditions), but I am unsure where. Any help whatsoever will be much appreciated.