u(t,x)=v(√(t^2-x^2))

∂^2/∂t^2 u(t,x)=∂/∂t (v' t/√(t^2-x^2))
=v'' t^2/(t^2-x^2)+v' √(t^2-x^2)/(t^2-x^2)
 -t^2v' √(t^2-x^2)/(t^2-x^2)^2
=v'' t^2/(t^2-x^2)-x^2v' √(t^2-x^2)/(t^2-x^2)^2

∂^2/∂x^2 u(t,x)=∂/∂x (-v' x/√(t^2-x^2))
=v'' x^2/(t^2-x^2)-v' √(t^2-x^2)/(t^2-x^2)
 -x^2v' √(t^2-x^2)/(t^2-x^2)^2
=v'' x^2/(t^2-x^2)-t^2v' √(t^2-x^2)/(t^2-x^2)^2

v'' +v' /√(t^2-x^2)=0
v''=-v' /√(t^2-x^2)
v''/v'=-1/√(t^2-x^2)
v''(p)/v'(p)=-1/p
ln(|v'(p)|)=-ln(|p|)+c
|v'(p)p|=e^c
v'(p)=±(1/p)e^c
v(p)=±log(|p|)*e^c+c2
pを改めてx, ±e^c=c1とおくと
v(x)=c1*log(|x|)+c2

v'(x)=c1/x
v''(x)=-c1/x^2
v''(x)/v'(x)=-1/x
v(√(t^2-x^2)=c1*log(√(t^2-x^2))+c2=u(t,x)