Diagonalization of 2D wave equation

[malasti]

Homework Statement

I've just derived the 1D wave equation for a continuous 1D medium from a classical Hamiltonian. I simply wrote Hamilton's equations, where the derivatives here must be functional derivatives (e.g. δ/δu(x)) since p and u are functions of x, and I got the wave equation (see "Relevant equations").

Now I'm trying to get the wave equation(s) in 2D (which should be a 1D wave equation for each coordinate) starting from a general 2D continuous medium Hamiltonian. I write Hamilton's equations and I get the mixed derivative form seen in the underlined equation below. (I hope this is correct, it at least looks reasonable.)

My professor said that this could somehow be diagonalized, and my guess is the result should be similar to the 1D wave equation for each component (i.e. we would have a subscript "i" for the component).

I know how to diagonalize a matrix, but it is not at all clear to me how to even write this in matrix form. The summation is over three indices, and it is not just multiplication, but derivation.

So, how could I diagonalize/change basis or whatever to get nice, 1D-ish wave equations from my mixed derivative form below?

Homework Equations

Both the displacement field u and the momentum p are functions of x and t, u(x,t), p(x,t).

1D continuous medium Hamiltonian:
H=∫dx p(x,t)²/m+½*a*(∂u(x,t)/∂x)²
(m is the constant mass density)

1D equations of motion:
∂p(x,t)/∂t=-δH(u,p)/δu(x,t)=a ∂²u(x,t)/∂x²
∂u(x,t)/∂t=δH(u,p)/δp(x,t)=p(x,t)/m

1D wave equation:
∂²u(x,t)/∂t²=a/m*∂²u(x,t)/∂x²

2D continuous medium Hamiltonian (bold: 2D vector, u/p subscript: vector component, coordinates x₁, x₂, summation is over both components/coordinates):
H=∫∫dx₁ dx₂ p(x)²/m+Ʃ_{ijαβ} ½*a__ijαβ*(∂u_i(x,t)/∂x_j)*(∂u_α(x,t)/∂x_β)

2D equations of motion, i=1,2
dp_i(x,t)/dt=-δH(u,p)/δu_i(x)=Ʃ_αβγ a_iαβγ/m * ∂²u_α(x,t)/∂x_β ∂x_γ
du_i(x,t)/dt=δH(u,p)/δp_i(x)=p_i(x,t)/m

2D "wave equation", i=1,2
∂²u_i(x,t)/∂t² = Ʃ_αβγ a_iαβγ/m * ∂²u_α(x,t)/∂x_β ∂x_γ

The Attempt at a Solution

I have tried to write the summation in the mixed-derivative equation as a combined summation and matrix multiplication, but nothing good came out of it.

I have also tried assuming "a" to be independent of indices (this should correspond to an isotropic medium) to simplify the problem, but I am still left with all the mixed derivatives. I think I have to make some assumptions on "a" (i.e. on its symmetries) to be able to diagonalize.

I have tried writing the derivations as a linear operator, but I just end up with having to write a matrix of linear operators, and I have no idea how to diagonalize that.

 

[malasti]

I'm beginning to think that this should be viewed as a tensor equation. I could use that "a diagonal tensor can always be diagonalized at any point by changing coordinate axes to the tensors principal axes" (found that in a book).

However, I'm not quite sure how exactly to do this. The double differentiation ∂²/∂x_β∂x_γ is clearly diagonal in the indices. If I can, by argument from the Hamiltonian (I guess?), assume some similar symmetries in the constant "a", then the right-hand side would be a diagonalizable tensor. Although I would still be left with one summation index.

Also, I can't really see how to do formally do this. Also, would it be a problem that it would only be diagonalized at "a point", or could I argue for spatial homegeneity and say that it's the same everywhere??

 

[voko]

In your problem, you have two equations of the second order. In order for them to be a wave equation each, as you seem to imply, each should be reducible to this form:

$\frac{\partial^{2} u}{\partial t^{2}} = c^{2}\nabla^{2}u$

This implies certain restrictions on the coefficients "a" of your problem. Very severe restrictions, I should say.

Or is it something else you are after?

 

[malasti]

No, I think that's pretty much it. I want to derive the 2D wave equation (which should be possible by diagonalizing it to get the polarization vectors).

I know "a" will need some hefty restrictions, yes. Preferably I would like to be able to motivate those. Looking at the 2D Hamiltonian, and noting that ordinary multiplication commutes, it seems pretty evident that a_ijαβ=a_αβij, so at least there's that.

 

[voko]

Basically you are looking for a coordinate transformation that would yield the system in the form you desire. That will have some "natural" restrictions on the parameters of the problem. However, if you have to allow for anisotropy, you won't get wave equations in the form of the 1D wave-equation, there will be a tensor, rather than an a scalar, that describes the properties of the media.

 

[malasti]

I think it would be okay to assume that the medium is isotropic in my case. Then λ is just a number and we drop all the indices. Also, I think I might have messed up some indices, not sure, however I currently arrive at

(λ/2)*Ʃ_mkl ∂²u_m/∂x_k∂x^l=m ∂²_tu_i, i=1,2 (or x,y, whatever)

Now I can look for harmonic solutions (waves) of the form

exp(i(Ʃ_jk_ju_j-ωt)

then the equation becomes

(λ/2)*Ʃ_mkl k_k k_l [/SUP]u_m=m ω²u_i

which is symmetric in k and l (which I feel is important). Now I just want to switch basis so this looks like the 1D wave equation, i.e. something like

Ʃ_j k²_j u_j=∇²u=∂²_tu

(left out some constants there).

 

[malasti]

Okay, I think I figured it out. I misunderstood the problem, at least somewhat. What was needed was to realize that
1. I should restrict myself to an isotropic medium
2. the elastic modulus tensor ("a" here) is VERY restricted for an isotropic medium (of course)
3. I should not precisely get the 1D wave equation, instead I should get the equation of motion for displacements in an isotropic medium (found it in Landay & Lifgarbagez) which can then be converted to two wave equations, on for longitudinal waves and one for transverse waves, by expressing the displacements as a sum of longitudinal (defined as having curl zero vector) and transverse (defined as having zero divergence, i.e. no volume change) displacements.

I did that and then it worked out. I get the transverse and longitudinal wave equations, which look like the 1D one "but 2D" (i.e. scalar->vectors, derivative->nabla).