Theoretical description of shock waves

For a theoretical understanding of shock waves, the compressibility of the medium has to be taken into account. The one-dimensional linear wave equation for a pressure wave P with sound velocity c

can be derived from the equations of mass and momentum conservation under the assump-tion of small particle velocities u and small density changes |∆ρ| ¿ ρ0 of the medium.

These assumptions lead to a constant propagation velocity c of any part of the wave regardless of the local pressure or density. While these approximations are justified for small amplitude waves, they cannot be applied to the general analysis of shock waves.

The equations for the conservation of mass, momentum, and energy for a compressible fluid neglecting dissipation and viscous forces read

∂ρ

with the specific internal energyE being the sum of thermal and chemical energy per unit mass. To complete the set of equations an equation of state is needed. The Tait equation yields a simple relation in weakly compressible media:

P +B

B andnare parameters depending on the medium. For pressures up to 10 GPa in water at 20^◦C we can useB ≈305 MPa andn≈7.15 [79]. Note that equation (3.3) yields a direct relation between pressure and density P =P(ρ) ifB and n are taken as constants. Since dissipation was neglected changes of state are isentropic and the equation of energy (3.2) becomes superfluous for a pressure calculation. Another often used equation of state for weakly compressible fluids is the stiff gas equation

P = (n−1)ρE−nB ,

which produces similar results to the Tait equation for small pressure values [80].

Figure 3.3: Steepening of a sinusoidal pressure wave in water due to nonlinear propagation for eight different times. The amplitude of the pressure is color coded, red and blue displaying high and low pressures, respectively. The arrow indicates the steep front developed during propagation.

As mentioned before, shock waves can arise due to nonlinear propagation of a pressure wave. Figure 3.3 shows the steepening of a high amplitude sinusoidal wave after prop-agation along the x-axis at 8 different time steps. The results were obtained by solving the equations (3.1) and (3.3) in one dimension using a finite element method. Parameters have been chosen to simulate a pressure wave in water at 20^◦C with ρ₀ = 998 kg/m³ and P₀ = 0.1 MPa. At x = 0 a sinusoidal excitation pressure is applied as a boundary condition. The wave propagates from left to right and finally steepens so that the pressure front becomes almost discontinuous as indicated by the arrow in Fig. 3.3. The steepening occurs due to the increase of sound velocity with pressure. Thus parts of the wave with high pressure travel faster than regions with low pressure and the leading edge of each positive pressure pulse will become progressively steeper. At some point the leading edge will approach a discontinuity and dissipative factors have to be taken into account. How-ever, in experiments it has been shown that shock waves may indeed travel with an almost discontinuous front and dissipation effects are very small [64]. Kirkwood approximated the thickness of a shock front in water taking into account the dissipated enthalpy. For a pressure of 20 kilobars the calculations predict a shock front of about 6 nm [64]. In this regime the approximation of a discontinuous front is justified and leads to a set of simple

equations relating the speed of the shock front to the pressure. Let us denote the pressure, density, and normal particle velocity of the fluid immediately in front of the shock wave asP₀,ρ₀, and u₀ and behind the shock wave asP,ρ, and u. For an observer moving with the shock front of velocityU the fluid will apparently move towards him with a velocity of U−u₀. The mass per unit area entering the front in timedtwill therefore beρ₀(U−u₀)dt.

From conservation of mass this must be equal to the mass per unit area leaving the front and we get:

ρ0(U −u0) =ρ(U −u) . (3.4)

Another equation can be derived from the balance of force. The difference in momentum per unit area and time for the fluid entering and leaving the shock front is given by:

ρ₀(U−u₀)u₀−ρ(U −u)u=ρ₀(U −u₀)(u₀−u) .

Neglecting dissipation, the change in momentum must equal the pressure difference across the shock front, which yields:

P0−P =ρ0(U −u0)(u0−u) . (3.5) Furthermore, we know that the change in energy must be compensated by the work done by the pressures P and P0. With E0 and E being the specific internal energy per unit mass before and after the shock wave we get:

P u−P₀u₀ =ρ₀(U −u₀)

Hereby, the difference in kinetic energy is expressed as

∆E_kin =ρ₀(U−u₀)dt· 1 2

¡u²−u²₀¢ .

Equations (3.4) and (3.5) were first developed by Rankine. The equation of energy bal-ance (3.6) was later added by Hugoniot (see [64]). In the following chapters we will only regard shock waves which travel into a previously undisturbed medium. Therefore, we can set the initial velocity u0 = 0 and receive a simpler form of equations (3.4-3.6):

ρ(U −u) = ρ0U

With the first and second equation of eqs. (3.7) we can express the shock velocity in terms

Together with the equation of state (3.3) the shock velocity is a function of pressure alone and we get:

Similarly, we can express the particle velocity as a function of pressure:

u= ρ−ρ0

In Fig. 3.4 the speed of the shock front and particle velocity of eqs (3.9) and (3.10) are plotted for pressures up to 4 GPa. The local speed of soundc= p

dp/dρcan further be derived from the Tait equation (3.3)

c=

Upon reflection of a pressure wave at a low acoustic impedance boundary the pressure is inverted. Figures 3.5 and 3.6 depict such an event for a 1-dimensional pressure pulse incident from the left. The vertical and horizontal axes display time and space in the 2-dimensional plots, respectively. Amplitudes of the pressure (Fig. 3.5) and the parti-cle velocity (Fig. 3.6) are color coded. At the right boundary the pressure was fixed to maintain a constant ambient pressure value of 100 kPa. This boundary condition charac-terizes a low acoustic impedance boundary, for instance a water-air interface. It can be seen in Fig. 3.5 that the pressure pulse is turned into a tensile pulse after reflection at the boundary. This effect is particularly important for the generation of cavitation, which can arise when the pressure of the tensile pulse drops below the vapor pressure. When shock waves are applied in a human body reflections at low acousitic impedance boundaries (e.g.

the lungs) should therefore be minimized to avoid secondary bioeffects due to cavitation.

However, cavitation may also arise during lithotripsy in the tensile pulse following the initial shock wave. Delius et al. [81, 82, 78] have reported on tissue damage after shock wave application, which is probably attributed to cavitation generated in this way.

0 500 1000 1500 2000 2500 3000 3500 4000 0

500 1000 1500 2000 2500 3000 3500 4000

pressure (MPa)

velocity (m/s)

U u

Figure 3.4: Speed of the shock frontU and of the particle velocityuas a function of pressure as derived from eqs (3.9) and (3.10).

Figure 3.5: Reflection of a high amplitude pressure pulse at a low acoustic impedance boundary. Amplitudes of the pressure pulse are color coded, red and blue displaying high and low pressure values, respectively.

Figure 3.6: Particle velocity of the pressure pulse shown in Fig. 3.5. High particle velocities are colored red and low particle velocities are colored blue.