problem 6.1

One-dimensional finite amplitude waves

We consider one-dimensional finite amplitude waves on the basis of the continuity equation (6.47) and the equation of motion (6.48).

1. Using the isentropic relation (6.55) and rewriting (6.47) in terms of pressure p and velocity u, derive the following system of equations:

   
   
   
   

2. The two equations of (i) can be written as

   
   
   
   

   Determine the two functions J ₊ and J _?.

3. The functions J are called the Riemann invariants. Explain why and how J are invariant.

problem 6.2

Burgers equation

The following Burgers equation for u( x, t) can represent a structure of weak shock wave (Fig. 6.8):

Figure 6.8: A weak shock wave.

Assuming a solution of the steady progressing wave u( x ? c ₀ t) with a constant c ₀, determine a solution which satisfies the conditions: u ? c ₁ as x ? ?? and u ? c ₂ as x ? + ?, where c ₁, c ₂ are constants and c ₁ > c ₂ > 0.

problem 6.3

Wave packet and group velocity c _g

Suppose that we have a linear system of waves characterized by a dispersion relation ? = ?( k) for the frequency ? and wavenumber k (see Sec. 6.3.5), and that there is a traveling wave solution of...