The 3+1D wave equation for spherically symmetric waves is
$$\frac{\partial^2 u}{\partial t^2} = c^2 \left( \frac{\partial^2 u}{\partial r^2} + \frac{2}{r} \frac{\partial u}{\partial r} \right) $$
where $u=u(t,r)$ and the initial conditions are
$$u(0,r)=g(r)$$ $$u_t(0,r)=h(r) . $$
If we let $v=ru$, we get
$$\frac{\partial^2 v}{\partial t^2} = c^2 \frac{\partial^2 v}{\partial r^2} $$
which is the one spatial dimensional wave equation, where the initial conditions are
$$v(0,r)=rg(r)$$ $$v_t(0,r)=rh(r) . $$
References:
"The Mathematical Theory of Huygens' Principle", B.B. Baker and E. T. Copson, 1987, p9
"The Mathematical Theory of Wave Motion", G. R. Baldock and T.Bridgeman, 1981, p31
https://en.wikipedia.org/wiki/Wave_equation#Spherical_waves
PART B) If $\phi(t,x)$ is a solution to the 1+1D wave equation and if the initial conditions $\phi(0,x)$ and $\phi_t(0,x)$ are given, then D'Alembert's Formula for the 1+1D wave equation gives
$$\phi(t,x)= \frac 12[ \phi(0,x-ct)+ \phi(0,x+ct) ]+ \frac1{2c} \int_{x-ct}^{x+ct} \phi_t(0,y)dy . $$
Letting $g(x)=\phi(0,x)$ and $h(x)= \phi_t(0,x)$ (with $c=1$, so $ct=t$) this is commonly written
$$\phi(t,x)= \frac 12[ g(x-t)+ g(x+t) ]+ \frac12 \int_{x-t}^{x+t} h(y)dy . $$
References:
http://mathworld.wolfram.com/dAlembertsSolution.html
http://en.wikipedia.org/wiki/D%27Alembert%27s_formula
http://www.jirka.org/diffyqs/htmlver/diffyqsse32.html
http://people.uncw.edu/hermanr/pde1/dAlembert/dAlembert.htm
THE QUESTIONS:
1) Then can we use D'Alembert's Formula for the 1+1D wave equation with initial conditions to get the solution to the spherically symmetric 3+1D wave equation with initial conditions?
2) If so, what is the physical (or intuitive) meaning of the integral term in D'Alemberts solution in the three spatial dimensional case?
UPDATE: The following references may indicate the answer to question 1) is yes.
math.ualberta.ca 337week0405.pdf after equation 180: "Thus the equation for $U$ can be solved using the formula we have discussed" {here $U$ is the 3D solution in $rU$ and the formula is D'Alemberts}.
Stanford Univ waveequation3.pdf page 4 Lemma 3 and page 5.
