Maximum order of essential and natural conditions

Is there a physical explanation for why when given a differential equation with spatial derivative of spatial order [itex]2m[/itex] the maximum order of the essential (displacement) boundary condition is [itex]m[/itex], and the maximum order of the natural (force) boundary condition is [itex]2m-1[/itex]

for example, the equaiton governing the dynamic behavior of a 1D bar is:
[itex] \frac{\partial ^{2}u}{\partial x^2} = \frac{1}{c^2} \frac{\partial ^{2}u}{\partial t^2} [/itex]

the maximum order of the essential boundary condition is 0 :
[itex] u(0,t)=0[/itex]
and the maximal order of the natural boundary condition is 1:
[itex] EA\frac{\partial u}{\partial x}(L,t)=R(t) [/itex]