Subsections

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 must be cleaned up. I do not want bars or step functions in my answer.

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??