Waves: Part 8 - Shock Waves

Shock waves are interesting phenomena that take place in a number of different media. For an arcane physical process, they have done quite well to be adopted by the mainstream media as one of their clichés, along with Mae West life jackets and aircraft black boxes (which are orange).

Shock waves are something of a misnomer. The shock part is correct, because something happens all of a sudden that was not anticipated, but the wave part is more troublesome, because shocks don’t have wavelength and are not periodic. The shock is an event, a singularity, a bit like a transient in audio.

Looked at from the standpoint of transform theory, a shock doesn’t have a wavelength because it has all wavelengths. All of those wavelengths happen to be at zero phase at the site of the shock and destructively interfere elsewhere.

Staying with the sonic aspect, at the speeds of most everyday objects, any disturbance due to motion is propagated ahead at the speed of sound, giving the air plenty of time to get out of the way. The pressure changes in the air are so small that calculations are simplified if the air is assumed to be incompressible. As speed increases, the air ahead has less time to move aside and the pressures involved are no longer negligible. This is known as the onset of compressibility.

There never was a “sound barrier”. That was an invention of the media. Exceeding the speed of sound required only sufficient thrust along with controls that would continue to work; the latter being harder to achieve. When an airplane approaches Mach 1, airflow over certain parts becomes supersonic and a shock wave will form. Downstream of the shock, the airflow detaches from the surface and if that is a control surface, it will lose effectiveness.

This was little understood at first and the Bell X-1 was a conventional airplane, albeit exceptionally strongly built. When Yeager took Glamorous Glennis beyond Mach 1 he was controlling it by adjusting the tailplane incidence, the elevators having ceased to function.

Later airplanes, such as the X-15, adopted tail airfoils that were wedge shaped, having maximum thickness at the trailing edge where separation would occur. The incidence of the entire surface would be changed for control purposes.

Fig.1 - The characteristic N-wave due to the passage of a supersonic airplane. The sudden pressure step is the bow shock, followed by falling pressure that becomes an under-pressure. The tail of the airplane causes another shock of the opposite sense because it has left a vacuum behind it.

Fig.1 - The characteristic N-wave due to the passage of a supersonic airplane. The sudden pressure step is the bow shock, followed by falling pressure that becomes an under-pressure. The tail of the airplane causes another shock of the opposite sense because it has left a vacuum behind it.

As the airplane is driven into air that has had no warning of its arrival, the air experiences a sudden pressure step, after which it leaks outward, falling in pressure, and gathering outward momentum. When the largest cross section of the airplane has passed, the air continues the outward movement because of that momentum. That results in an under-pressure. When the airplane has passed, it leaves a vacuum behind it that results in another shock as it is filled. Fig.1 shows that the passage of the airplane causes a characteristic N-wave. This is fixed relative to the plane, so those on board don’t hear it.

Fig.2 shows the geometry of the shock. The nose of the airplane is shown in two places. At A2, the spherical wave has not had time to propagate, but the wave emitted at A1 will have moved outward at the speed of sound. It follows that the sine of the half angle of the shock is one over the Mach number. The faster the airplane goes, the narrower the shock.

The shock surface is conical and moves with the airplane. On level ground the intersection of the conical shock creates a hyperbolic contact line. An observer on the ground will hear a boom as the contact line sweeps past. During the cold war, a Victor V-bomber, weighing about 90 tons, “accidentally” went supersonic over England and the resulting rather large sonic boom was heard far and wide.

Shocks are mostly invisible to the HVS, although in the right conditions of humidity the pressure step at the shock causes a small cloud to form, moving at the same speed as the airplane. These are known as vapor cones, or Mach diamonds. The best way of observing shocks is to appreciate that they cause steps in air density, which must affect the speed of light and the refractive index. Schlieren photography takes advantage of that by making the air density visible.

Fig.2 - Movement from right to left. At A1, corresponding to now, there has been time for the shock to radiate, whereas the shock from the earlier A2 has propagated at the speed of sound. All radiation in between will add coherently to create a shock cone. The half angle of the cone is a function of the Mach number, here about 2.8, corresponding to a half angle of 21 degrees.

Fig.2 - Movement from right to left. At A1, corresponding to now, there has been time for the shock to radiate, whereas the shock from the earlier A2 has propagated at the speed of sound. All radiation in between will add coherently to create a shock cone. The half angle of the cone is a function of the Mach number, here about 2.8, corresponding to a half angle of 21 degrees.

Compared to light and sound waves, that are quite simple, waves on a surface between two media, such as air and water are more complicated. Any object moving on such a surface, however slowly, will create a shock. Surface waves are massively dispersive, which means there is no characteristic velocity corresponding to the speed of sound or light.

Instead, surface waves have a phase velocity, which is equal to the speed of the excitation, and a group velocity which is half the phase velocity. If something turns up, be it a pond yacht or a super tanker, the shock will propagate forwards with one half the speed of movement. The resulting wavelengths will be those which travel at that speed.  The waves cannot move ahead of the vessel causing them so there is no warning of approach at any speed.

There is no relative motion at right angles to the direction of motion so the group velocity falls as a cosine function of the direction. Fig.3 shows what then happens. Motion is from right to left, where point A can be considered as now and point B corresponds to time that is earlier by an amount t. In the time t the radiation from point B will have propagated according to the cosine function. The propagation is circular, centered on a point 3/4t back from the initial shock.

If we mentally reduce t the result will be a succession of circular wavefronts whose diameter falls as the initial shock at A is reached. These wavefronts will re-enforce along a line from A that is tangent to the circular wavefronts.

The sine of the angle of that line is 1/3, corresponding to a half angle of about 19.5 degrees. The surprising result is that this angle does not change with speed, but remains constant over a wide range of speeds provided the water is deep enough. This was first explained by William Thompson, who subsequently became Lord Kelvin, in the 1880s.

Fig.3 - As the group velocity is one half the speed of the vessel and radiation suffers a cosine effect, falling to zero at right angles, the sine of the half angle of the bow wave is 1/3 and is independent of speed.

Fig.3 - As the group velocity is one half the speed of the vessel and radiation suffers a cosine effect, falling to zero at right angles, the sine of the half angle of the bow wave is 1/3 and is independent of speed.

Within the bow wave, transverse waves also form. One set are formed at the bow, another set at the stern. As before, the waves move with the vessel and the wavelength will be whatever wave can move at that speed. At low speeds, the wavelengths are short and the length of the hull may accommodate several complete waves. The front and rear waves interfere. The hull acts rather like a comb filter, where at some speeds the front and rear waves interfere coherently whereas at speeds in between they cancel one another.

As speed rises, the wavelength increases until one wavelength envelopes the hull. The hull is held up by its ends and there is an obvious depression in the water surface amidships. The vessel has reached hull speed and without huge increases in power it can go no faster. A keel yacht cannot usually climb out of its own depression, whereas a dinghy can make the transition between Archimedean buoyancy and dynamic lift by planing.

A stationary hovercraft that is on cushion depresses its own weight in the water below. Moving slowly, it creates a bow wave and transverse waves. However, when it reaches hull speed, known in hovercraft speak as hump speed, it usually has enough power to climb out of the displacement hole and accelerate. The water then has no time to move out of the way and no waves are generated, saving power.

Light has finite speed, but unlike the speed of sound, it cannot be exceeded. We should not be too surprised to find there is something like a Mach cone when we look at light.

Fig.4 shows a space-time diagram. The vertical line is the time axis, with time advancing upwards in units of seconds. The horizontal line is a section through a two dimensional plane that includes the origin O, which represents “here” in space and “now” in time. The units of distance across the plane containing O are light seconds; the distance light moves in a second.

Fig.4 - Point O represents here and now. What happens now at any other place cannot be known because it is outside the cone enclosing the affective past. See text.

Fig.4 - Point O represents here and now. What happens now at any other place cannot be known because it is outside the cone enclosing the affective past. See text.

Trailing back from O is a cone whose half angle is 45 degrees. Events within that cone can affect O, and so are known as the affective past. Event P on the time axis is in the direct past that O must have passed through. An event such as A can reach O without exceeding the speed of light.

Any event outside the cone can not reach O. The concept of here and now is tautological, because the only thing that can be known about the plane containing O “now” is at O. The size of “now” is zero. For example, the sun could be in the plane containing O, at S. If the sun went out “now”, we could not know, because S is outside the affective cone. S can only enter the affective cone after it has moved upwards for sufficient time, in this case about 8 minutes, the time taken for sunlight to reach the Earth.

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