- 7.3.1 The physical problem
- 7.3.2 The mathematical problem
- 7.3.3 Transform the problem
- 7.3.4 Solve the transformed problem
- 7.3.5 Transform back
- 7.3.6 An alternate procedure

7.3 A hyperbolic example

This example illustrates Laplace transform solution for a hyperbolic partial differential equation.

It also illustrates that the transformed coordinate is not always a time.

7.3.1 The physical problem

Find the horizontal perturbation velocity in a supersonic flow above a membrane overlaying a compressible variable medium.

7.3.2 The mathematical problem

- Domain :
- Unknown horizontal perturbation velocity
- Hyperbolic
- Two homogeneous initial conditions
- One mixed boundary condition at and a regularity constraint at
- Constant , where is the Mach angle.

Try a Laplace transform. The physics and the fact that Laplace transforms like only initial conditions suggest that is the one to be transformed. Variable is our ``time-like'' coordinate.

7.3.3 Transform the problem

Transform the partial differential equation:

Transform the boundary condition:

7.3.4 Solve the transformed problem

Solve the partial differential equation, again effectively a constant coefficient ordinary differential equation:

Apply the boundary condition at :

Apply the boundary condition at :

Solving for and plugging it into the expression for
gives:

7.3.5 Transform back

We need to find the original to

Looking in the tables:

The other factor is a shifted function , restricted to the interval
that its argument is positive:

With the bar, I indicate that I only want the part of the function for which the argument is positive. This could be written instead as

where the Heaviside step function if is negative and 1 if it is positive.

Use convolution, Table 6.3, # 7. again to get the product.

This

I can do that by restricting the range of integration to only those values for which is nonzero. (Or is nonzero, if you prefer)

Two cases now exist:

It is neater if the integration variable is the argument of . So,
define
and convert:

This allows me to see which physical values I actually integrate over when finding the flow at an arbitrary point:

7.3.6 An alternate procedure

An alternate solution procedure is to define a new unknown:

You must derive the problem for v:

The boundary condition is simply:

To get the partial differential equation for , use

Similarly, for the initial conditions:

After finding , I still need to find from the definition of :

Where do you get the integration constant??