Shock celerity, the series of wavelets that form

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.

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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.