Welcome back. Last week I promised you that we are

going to look into simulations in two dimensions.

So far, we've only looked at one dimension.

Usually, that is not enough to actually get

interesting results for scientific problems.

Actually, in two-dimensions, often it's already possible

to understand the physical phenomena in more detail,

which is why I think that's an important issue.

So, to understand some of the peculiarities of two-dimensions,

it's actually useful to start with the full three dimensional wave equation.

Now, it's written again here.

Now, on the left hand side,

we have the second time derivative of p. Now,

here again, with all dependencies in space x, y, z,

and t. On the right hand side,

we have the propagation velocity c square that depend also on space, only x,

y, and z, multiplying the Laplace operator of the pressure field.

That, of course, also depends on space x,

y, z, and time.

Plus we, again, have the source term s which also depends on space x, y, z,

and t. Now, what happens if we want

to turn this three-dimensional equation into a two-dimensional problem?

Well, what we do is we say all fields that we have

here in the wave equation do not depend on one particular direction.

So, let's say this is the y direction,

so that means all partial derivatives with respect to y and zero.

So, let's express that again in mathematical terms.

So we have the pressure field that is no longer depending on y.

So, it will be only a function of x, z,

and t. The propagation velocity c will only depend on x and z.

Also, the source term will no longer be dependent on x,

y, z and t but now is only a function of x,

z, and t. Now,

let's try and see what this means geometrically.

First, let's look at a coordinate system.

Actually, in earth sciences, particularly in exploration we actually,

people often take the z coordinate downwards.

So, we have here x, y and z.

Now, what does it mean if, for example,

a wavefield or an earth model is independent of y.

Well, this is called then a two-dimensional model and here's an example.

Now, let us assume we have a velocity field.

This is colored.

So, the color corresponds to a propagation velocity,

and it only depends on x and z.

So, basically, what that means we can prolong it,

it's translationally invariant in that third direction,

in the y direction.

That's expressive of its two-dimensionality.

What about the source term?

Now, this is very important.

Now, if you consider, for example,

a point source somewhere inside that medium,

this source, this point in the x,

z plane is also invariant in y.

So what does that mean? It's no longer a point source,

it's actually a line source.

That actually creates a very particular waveform and

very specific effects that are very important to remember.

Similar concepts apply to

all other simulation types that you do in two dimensions.