The 1D homogeneous wave equation is
$$u_{xx}(x,t)-\frac{1}{c^2} u_{tt}(x,t)=0$$
or more briefly, define the wave equation operator L as
$$L(u)\equiv u_{xx}(x,t)-\frac{1}{c^2} u_{tt}(x,t)$$
Then the wave equation is
$$L(u)=0 $$
Initial conditions are $$u(x,0)=f(x) $$ $$u_t(x,t)|_{t=0}=g(x) $$
Then the impulse response is normally defined by the non homogeneous equation
$$ L(u)=\delta $$
$$f=g=0$$
My question is Can the impulse response be defined in the homogeneous equation for each of the initial conditions?
For example--
for the displacement impulse response:
$$L(u)=0$$ $$f=\delta$$ $$g=0$$
for the velocity impulse response:
$$L(u)=0$$ $$f=0$$ $$g=\delta$$
Then the output could be found for other initial conditions by convolution.