Shock waves

If a periodic disturbance exists in a fluid, moving at a very small velocity

compared with the wave celerity, the series of wavelets that form will be

practically symmetrical about their source. However, as the velocity of the

flow increases, the wavelets will move upstream less rapidly, and downstream

more rapidly, where the result of the pattern will become more and more

asymmetric. When the velocity of the fluid becomes equal to the celerity, the

upstream motion will be reduced to zero and the wavelets will all be tangent to

a normal line passing through the source. Further increase of the velocity of the

flow will cause all portions of the wavelets to be carried downstream. The

lines of tangency subtending an angle which becomes smaller and smaller than

1800 as the ratio of fluid velocity, the wave celerity increases

above unity, half this angle is known as the Mach angle, the reciprocal of this

sign is simply the Froude number in the case of gravity waves. In the case of

elastic waves its counterpart is known as the Mach number. Any body that is

involved in relative motion with a fluid at a velocity that is either a

fraction or a multiple of the wave celerity will produce a comparable flow

pattern, each element of a curved wall can be regarded as a disturbance

producing its own wavelets, the converging wavelets combining to form a surge

or shock wave which eventually moves upstream as the angle and as the wave

becomes too great. If the change in wall direction is abrupt rather than

gradual, the shock wave will begin right at the discontinuity, again starting

to move upstream as the angle of the wall becomes excessive.

The optical

method known as Schlieren produces a diffraction of light that varies with the

density or pressure of a flowing gas. In the picture, we notice that the shock

wave is well ahead of the boundary or the line of symmetry. This distance

decreases as the Mach number increases. The latter change being indicated by a

decrease in the Mach angle. The higher the Mach number of course, the greater

the pressure within the shock wave.

Hence, shock

waves are pressure waves that result from sudden and violent changes in

pressure in an elastic medium, these changes could occur due to many phenomena,

such as explosions, bullets, earthquakes, supersonic jets and even

extracorporeal lithotripsy, which is a medical technique used to shatter kidney

or gallbladder stones. Shock waves are different than acoustic waves in the

sense that at their wavelets, the temperature, stress and density of the medium

change violently, meaning that shock waves might alter the properties of a

material and hence can be used to study its equation of state. Also, shock

waves are non-linear waves due to this violent behavior. Compared to sound

waves, they travel much faster as their amplitude increases. However, they also

decay much faster than sound waves, meaning their intensities drop rapidly,

this is because a lot of the energy of the shock wave gets transferred into

heat in the surrounding system, so the amplitude drop is proportional to

the square of the displacement. At some point the shock wave will become a normal,

linear acoustic wave and can be studied analytically.

Solitary

waves and solitons

A solitary

wave is exactly what its name implies; a single pulse that travels at a

constant rate of speed along some medium with respect to time. The defining

characteristic of a solitary wave is that it comprises only one pulse, there is

therefore no frequency and it has no wavelength as such (there is only one

pulse). So there is no such thing as frequency, or period or wavelength in a

soliton, it is simply a defined pulse with a defined shape that moves along at

a constant speed in some medium. A good example of a soliton wave is a rope

tied to something and given a tug, a quick upward jerk, just once, and a

solitary wave emerges until it hits a barrier, then it either reflects back or

disappears.

In a medium,

if dispersive effects and nonlinear effects cancel out, meaning if a wave with

a speed dependent on the frequency suffers a nonlinear effect, a soliton will

be created. Solitons are merely the solution of a widespread class of weakly

nonlinear dispersive partial differential equations describing physical

systems.

Solitons have

some characteristics:

–

They are localized within a

region.

–

They retain their form over

long periods of time (permanent form).

–

If they interact with other

solitons, they would just pass through them and remain unchanged, except for a

phase shift.

–

They are stable, nonlinear

pulses that exhibit a fine balance between nonlinearity and dispersion.

Solitons can

be used to describe shallow water waves, nonlinear optics, electrical network

pulses and many other phenomena.

Solitons come

in different types; They can be humps that are bell-shaped curves, kinks that

have an s-type curve describing the change in their value, and breathers

(bions) which can be either stationary or travelling humps that oscillate.

Tsunami

Tsunami is a

Japanese term that means a “tidal wave”. It is used to describe a

series of travelling waves in water produced by the displacement of the sea

floor associated with submarine earthquakes, volcano eruptions or landslides.

Tsunami are long-wave phenomena and because the wavelengths of tsunami in the

ocean are long with respect to water depth, they can be considered shallow water

waves and can be modelled using the shallow water equations, where v=. However, as the wave encounters the

shore, the water depth decreases sharply resulting in a greatly increased surge

of water at the point where the wave strikes land. This requires a new modeling

technique, such as robust Riemann solvers or the level-set method which can

handle situations where dry regions become flooded and vise-versa.