# Doppler effect, Mach cone and supersonic

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## Mach cone and supersonic

You will certainly have already made the following experience: You can hear an airplane in the open, look up and not see the airplane where you suspected it. If you search the sky for the plane, you will find that the plane has flown a long way further than you expected. The reason is that the sound waves need a certain amount of time until they are on the ground and the aircraft continues to fly during this time.

### Wavefronts in the aircraft

Work order

Watch the sound waves propagate as the plane flies faster and faster. What happens when the speed of sound is reached? Investigate the relationship between the envelope of the wave front and the speed of the jet, especially when the speed of sound is exceeded.

The faster the airplane gets, the closer the wave crests in front of it move together. If the speed of the aircraft is exactly the same as the speed of sound, the wave crests converge to a point in front of the aircraft.

If the airspeed exceeds the speed of sound, the aircraft moves faster than the sound waves of the aircraft noise. It leaves the wave behind. The individual spherical waves overlap. The rest forms a cone-shaped wave front, which is also called the Mach cone. Outside the cone, no signal is received from the aircraft. A very excessive short wave pulse (shock wave) forms on the surface of the cone, in which the sound deflection takes on a peak value, hence the bang when the "sound barrier" is broken.

In the same way, you can see a water wave front on ships because the ship is faster than the water waves.

Opening angle of the Mach cone
Half the opening angle of the Mach cone is calculated geometrically using the following sectional drawing, which shows the circular propagation of the sound over time $t0$ shows and the object that generated the wave at the time $t=0$ and $t=t0$ .
$sinβ=ctut=cu$ whereby $M.=uc$ called the Mach number.
Note, however, that this is only one half of the cone. The total opening angle of the Mach cone is then $α=2β$. With the Mach number it is then calculated as follows: $α=2arcsin1M.$
Opening angle of the Mach cone
As soon as the speed $u$ of the object is greater than the speed of sound $c$, the Mach cone is formed. With the help of Mach's number $M.=uc$ the opening angle of the Mach cone is calculated using the following formula:
$α=2arcsin1M..$

The Mach number and the Mach cone were named after the Austrian physicist and philosopher Ernst Mach (1838-1916) who dealt with these phenomena.

## Experimental baptism replacement ceremony

Since we don't have much to do with church, but still felt like having cake, coffee and cold drinks, we had one last weekend Baptism ceremony organized for our daughter and by the way the new terrace was inaugurated. And the weather was soooo good!

Instead of religion, there was science: I had prepared a few experiments and even organized an old microwave for them some time ago through a colleague (thanks, David!). I picked the topics from various podcasts, blogs and Google:

Fascinatingly, every experiment has to do with vibrations and the only formula that comes up comes three times. I explain that to myself that the whole of physics actually only describes vibrations, from the smallest atomic-physical effects up to the macroscopic, when black holes release gravitational waves -)

At this point, praise to the audience, they really had it: preliminary questions such as “why could that be so” were usually answered correctly, the unknown sensibly delimited (“if there are two glasses and we are not allowed to touch one, we have to to the other one ”) and asked the speaker himself (“ ​​And what is the name of this spread? ”- luckily I had researched that beforehand).

According to old school wisdom

Unfortunately, the individual experiments were not always successful:

Rayleigh scattering The effect was a bit puny, which was probably also due to the tiny wheat glass (0.3l). I should have prepared better: I would have got a 0.5l glass from somewhere and the milk mixing ratio was definitely in need of optimization. Cappuccino tones It didn't really work with our Café Crema from the Senseo :-( I think I can remember having done it successfully at home. Back then in the hospital with the real cappuccino it worked great. Wine glass -Resonanz I had already successfully tested that last year, but the devil is in the details: As I have now discovered, there is only one of our wine glasses a Pair that has the same natural frequency. Fortunately, that was found quickly. Supersonic in the sink Successful - but I don't think anything can go wrong. Only the mach cone was a bit difficult to see, maybe next time really use a needle or something else pointed ... In-ear spoon bell It worked too! That was probably due to the perfect preparation: the day before I had already knotted two spoons with ribbon and tried them out. Measure the speed of light Yikes, such a glass of fluff is empty quickly! The hotspots were already recognizable after a short time (sooooo much fearful) (not even scorched, just "opened"). However, our measurement showed a distance of 9cm, which then leads to a speed of light that is 30% too high. I imagine that I have stumbled across this range of errors somewhere on the Internet while preparing. On the other hand, Florian almost hit the spot with his chocolate bar measurement, there was nothing with a 30% deviation.

What is still unclear to me about the experiment is the spatial geometry: The magnetron should be viewed more as a point source, then we have to measure the distance between the hotspots on a spherical shell and not flat on a piece of cardboard, right? Calculate whether the 30% will then disappear -) Plasma in the microwave Since I was (unfounded?) Worried about the microwave, I didn't want to test it beforehand. Unfortunately, it was only enough for a few short "puffs", a permanent, beautiful plasma ball was unfortunately not created :-(

Maybe it helps to google the instructions for the microwave: The two of us stood confused by the device and weren't sure how to formulate the simple wish “full power until I press stop”.

All in all, it was fun and one or the other can definitely be repeated somewhere. And the microwave is still there, so I can use it again at some point.

## Sonic boom - this is how it is created

• The wave peaks of this sound front are so close together in time that they can no longer be resolved individually when they reach the ear. You hear a bang.
• The effects are well known, especially from the supersonic flights of military or combat machines over inhabited land when these were (still) allowed: As soon as their speed exceeded the speed of sound in air, extremely dense cones of sound formed which hit the surface of the earth
• It “rumbles”, more or less loudly depending on the distance. However, it is astonishing that one often hears the "breaking the sound barrier", as the phenomenon is called, twice in quick succession.
• The reason is that the aircraft itself acts like a double sound transmitter: in front of its bow it compresses the air extremely, at the rear there is extreme air dilution. Accordingly, two pressure waves are formed, the time interval between which is only a few hundredths of a second. So it “rumbles” twice.

However, there are the most adventurous ideas about the term “sound barrier”. Even if one speaks colloquially of “breaking through the sound barrier”, one should in no way imagine this situation as an objective, even tangible barrier that is broken through with a loud crack.

The bang is a shock wave made up of many individual sound waves, which a supersonic aircraft “tows” in the form of a cone and “sweeps” over the earth. When the cone hits the ground, there is initially a huge increase in pressure, followed by a serious pressure drop.

If you are at this point you will hear a bang. In the meantime, however, the shock wave propagates along the flight path so that it can reach another receiver some distance away. This then also hears a bang, but with a time delay.

By the way: A measure for speeds in this supersonic range is named after the physicist Mach. The Mach number represents the ratio of the speed of the transmitter to the speed of sound in the surrounding medium. It is used for the dimensionless specification of the speeds of fast aircraft: Mach 2 means that the aircraft flies at twice the speed of sound.

## 4 Faster than sound

If the sound source moves as fast as the sound itself (instead of the police car a jet as an example), then the circles shift until they all have a common point and the so-called sound barrier is reached. Sound waves propagate spherically in all spatial directions at the speed of sound, but they cannot break away forwards in an even faster flying jet, which then creates the "Mach cone". Fig. 4 shows the conical propagation of the pressure wave and the course of the hyperbolic ground contact.

Fig. 4: Mach cone [5]

When the speed of sound is reached, the air resistance increases sharply. This is where the name comes from, a sound barrier that used to be breached. This breakthrough of the sound barrier becomes visible through the cloud disc effect: In the negative pressure zone at the tail of the jet, the air cools down considerably and (if the air humidity is high enough) the water vapor condenses and forms a cone-shaped cloud.

Fig. 5: Cloud disc effect [5]

## Sound is a mechanical wave

Sound is caused by mechanical vibrations of bodies. Can swing z. B. a tuning fork, the strings of a guitar, the column of air in an organ pipe, the vocal cords in humans or the membrane of a tambourine.
For example, when you hit a tambourine (Fig. 2), the membrane is deformed and the surrounding air is compressed as a result. The air compresses at this point and the pressure increases. Since air is elastic, it then expands again, which leads to a compression at an adjacent point. A pressure wave is created that spreads in space. This can also be interpreted using the particle model: the air particles are stimulated to vibrate when the membrane hits them.
So they swing back and forth. Areas with a larger number of particles (greater pressure) and areas with a smaller number of particles (lower pressure) are formed.

In general:
Sound waves are the propagation of pressure fluctuations in space.

Since the direction of propagation and the direction of oscillation of the particles coincide, sound waves are longitudinal waves.
The human ear is able to detect fluctuations in pressure
around 0.000 02 Pa (hearing threshold) to around 20 Pa (pain threshold).

## Superluminal speed in cosmology

### Speed ​​above light due to the expansion of space

Most galaxies have a redshift light spectrum. Edwin P. Hubble initially interpreted this shift as a Doppler effect. This means that the respective galaxy is moving away from the earth at a considerable speed $v$. When comparing the redshift of galaxies with a known distance $d$ to Earth, Hubble found a linear relationship. This is the Hubble law $v = H_0 , d$ with the Hubble constant $H_0$. According to this law, galaxies should travel faster than the speed of light from Earth if they are far enough away.

The interpretation of the cosmological redshift attributes this to the increase in distances due to the expansion of the universe, not to the Doppler effect. In the context of relativistic cosmology, Hubble's law is valid for any distance, if $d$ is interpreted as the physical distance (distance to a fixed point in time) and $v$ as the change in this distance over time. $v = dot d$ can be greater than the speed of light, which is sometimes seen as a contradiction to the theory of relativity and used as a counter-argument to the big bang theory. Conceptually, however, the rate of change in distance $dot d$ must not be confused with a speed. Velocities are local quantities that are subject to the restrictions of the special theory of relativity. As global variables, changes in distance are not subject to these restrictions and can be of any size. Real superluminal velocities are therefore not available even in distant galaxies.

### Cosmological theories with variable speed of light

Various cosmological theories with a variable speed of light (VSL) have been proposed. In particular, a proposal by João Magueijo and Andreas Albrecht from 1999, & # 9117 & # 93, in which the horizon problem and the problem of the flatness of the universe, which are usually explained today in the context of the inflationary model of cosmology, instead by up to 60 orders of magnitude higher speed of light in the early universe can be explained. In this theory, the speed of light is a dynamic variable, i.e. it changes over time, but in a special way that does not modify the form of the field equations of general relativity too much. The Lorentz invariance of the theory is broken explicitly, there is an excellent reference system (which is given by the cosmological expansion). According to Magueijo and Albrecht, the problem of the cosmological constant is also solved in this way. & # 9118 & # 93 & # 9119 & # 93 Magueijo also wrote a popular science book about it. & # 9120 & # 93 The Canadian physicist John Moffat made a similar proposal in 1992, & # 9121 & # 93, also with the intention of solving cosmological problems. & # 9122 & # 93

The theory stands in the tradition of temporally variable fundamental (dimensionless) physical quantities that have been discussed since Dirac. It makes sense to only discuss the variability of dimensionless quantities, since the variability of dimensional quantities in physics depends on the units of measurement used and is therefore of no fundamental importance. In the case of the VSL theories, the fine structure constant is variable, which in principle should be observable as a function of the redshift for objects that are far away. & # 9123 & # 93

### Wormholes

A related effect is the crossing of so-called wormholes, which is often used in science fiction novels. A spaceship does not move faster than the speed of light locally, but it takes a shortcut in curved space so that in the end it arrives at its destination faster than the light. As a two-dimensional analogy, one can look at the path over a folded sheet of paper. Instead of staying on the paper, a traveler can simply drill a hole in the paper and use it to reach the other side that has been folded over. Time machines would also be conceivable with this technology. While such wormholes can be constructed theoretically in the theory of relativity, it seems that in practice they would be very unstable, so that not even information could be passed through them.

### Hyperspace

The idea of ​​an abbreviation through a hyperspace, in which our spacetime could be embedded, would have a comparable effect, which is also often used in science fiction. The idea is as follows: To shorten the path from the North Pole to the South Pole, travel across the earth instead of along the surface. The path through the earth (via the third dimension) is shorter than the path on the (two-dimensional) surface of the earth. In the same way, one could imagine that our spacetime is also embedded in a higher-dimensional hyperspace (like the earth's surface in space), and could therefore be shortened through hyperspace. Here, too, you would not have to fly faster than the speed of light (in hyperspace) to arrive at the destination faster than light in normal space.

## Propagation of sound in the atmosphere

If sound propagates through the atmosphere (e.g. traffic noise), it is influenced by the meteorological conditions and the acoustic properties (acoustic impedance) of the ground.

### Air absorption

Some of the sound energy is absorbed by molecular friction and other molecular properties as it travels through the atmosphere. The degree of air absorption, which is usually given in dB / 100 & # 160m, depends on the air temperature and humidity. Higher frequencies are absorbed much more strongly than lower frequencies. A recognized calculation method for the degree of air absorption is specified in ISO & # 1609613-1. Atmospheric absorption coefficient means translated atmospheric absorption coefficient.

temperature humidity 125 & # 160Hz 250 & # 160Hz 500 & # 160Hz 1000 & # 160Hz 2000 & # 160Hz 4000 & # 160Hz 8000 & # 160Hz Attenuation in dB / km at 10° 70 % 0,4 1,0 1,9 3,7 9,7 32,8 117 20° 70 % 0,3 1,1 2,8 5,0 9,0 22,9 76,6

### Refraction

Like any form of wave, sound waves also change their direction if the speed of propagation for different wave trains is different.

Analogous to light rays that are deflected ("refracted") in the direction of the area with lower propagation speed (the optically denser medium), sound waves are also refracted in the direction of the area with lower sound speed.

Such areas are generally meteorological, especially due to the microclimate along the way. Two weather conditions - in principle independent in this context - are decisive here: air temperature and wind direction.

If the temperature decreases with altitude, the speed of sound also decreases upwards and the sound is refracted upwards. In such situations, from a certain distance (with sound sources close to the ground from around 200 & # 160m) an acoustic shadow zone with reduced audibility forms. This is mainly the case during the day because of the warming of the soil by solar radiation. If, on the other hand, the temperature increases with altitude (inversion), this leads to a downward refraction of the sound waves and possibly multiple reflections on the ground. The result is good audibility over great distances. This is especially the case at night.

Wind also causes spatial differences in the speed of sound: Since the wind speed usually increases with altitude, the sound is refracted downwards in the direction of the wind, so it can be heard better over long distances. Conversely, sound propagation against the wind by refraction upwards leads to a shadow zone and reduced audibility.

The weather-related fluctuation of the sound level at a distance of 500 to 1000 & # 160m from a constant sound source can be between 20 and 30 & # 160dB.

### Scattering

Some of the sound energy is scattered when passing through turbulence in the atmosphere. Scattering is a mechanism by which sound energy can enter shadow areas, such as upward refraction. Sound waves are mainly scattered when their wavelength has the order of magnitude of the expansion of the turbulence elements (eddies).

### Diffraction

Diffraction is another mechanism with which sound energy can penetrate into shadow areas, for example into shaded areas behind a building or a noise barrier. Long, low-frequency waves are bent more than short, high-frequency waves.

### Reflection on the ground

If sound waves hit the ground, they are reflected. Depending on the acoustic properties of the floor (soft sound = low impedance or hard sound = high impedance), more or less sound energy is absorbed in the floor or the reflected wave is phase-shifted so that the floor has a more or less sound-absorbing effect. Loose, porous soil and freshly fallen snow are sound-soft and therefore have a strong dampening effect, while compacted soil, asphalt or concrete are hard-sound and therefore have little dampening. A high level of floor attenuation is achieved above all with a sound-soft floor and shallow sound incidence (source and receiver close to the floor), since the reflected wave is phase-shifted by almost half a wavelength and thus the direct wave arriving at the receiver is almost canceled out by destructive interference.

A sound shadowing or a sound shadow occurs when there are obstacles on the direct sound path from the sound source to the listener or the microphone.

### Speed ​​of propagation

T in ° C c in m / s t in ms
35 352,17 2,840
30 349,29 2,864
25 346,39 2,888
20 343,46 2,912
15 340,51 2,937
10 337,54 2,963
5 334,53 2,990
±0 331,50 3,017
−5 328,44 3,044
−10 325,35 3,073
−15 322,23 3,103
−20 319,09 3,134
−25 315,91 3,165

The speed of propagation c a sound wave in air is 343 & # 160m / s at 20 & # 160 ° C that is about 1235 & # 160km / h. It takes absolute temperature by the root T to. The time it takes to travel one meter t is specified in milliseconds (ms).

The wavefront therefore needs about 3 & # 160ms per meter. In a homogeneous medium, it spreads along a straight line. Assuming a point source of sound, the air particles are excited to vibrate evenly on all sides of the space filled with matter. This means that all particles that are the same distance from the sound source, i.e. & # 160h. lie on a spherical surface, the center of which is the sound source, are in the same state of excitation (compression or dilution) or in the same phase.

Sound waves that spread evenly in all directions are therefore called spherical waves.

## Photonics: Visibly wrong light

People standing in the water look more compact. This is because the light is deflected a little when it passes from water to air. But what if you could drastically change that distraction? Could sheer fabrics with radically different properties make objects invisible? As early as 1904 in an optics textbook, Arthur Schneider toyed with the idea that a material with very exotic properties would refract light differently. In 1968 Victor Veselago developed the theoretical concept for this idea, which researchers are now trying to put into practice: They want to construct a material that refracts light "incorrectly" and thus turns our visual experience upside down. Initially, however, this only succeeded outside of the visible spectrum.

By skillfully combining two physically tailored crystals, Yong Zhang and his colleagues from the National Renewable Energy Laboratory succeeded for the first time in 2003 in showing the "wrong" refraction in visible light [1]. The tricky thing about such crystals, however, is that their behavior is extremely dependent on the angle and wavelength of the incident light and cannot be varied at will. Therefore, researchers are looking for other ways to fabricate "negatively refractive" materials.

This artificial form of light deflection is referred to as negative refraction because the angle between incident light and an imaginary perpendicular to the surface not only becomes smaller, as usual, but negative: the light behaves as if it were bouncing back from the perpendicular . As a result, the refractive index & # 8211 a measure of this angle of refraction, which depends on the electrical and magnetic properties of the material & # 8211 is also negative.

Scientists have now entered the visible realm in two ways. On the one hand there is the "real" negative refraction, as Gunnar Dolling from the University of Karlsruhe calls it [2]. It uses tiny components & # 8211 smaller than the wavelength of the light used & # 8211 which are combined into a so-called metamaterial. The smallest oscillating circuits regulate the magnetic conductivity of the material, a fine wire mesh its permeability for electrical fields. As long as the components are significantly smaller than the light wavelength, they do not affect the light as individual components, but as a blurred whole. "The light does not see these components, it only feels their effect," says Dolling.

With a correspondingly large wavelength, it is not a problem to keep the technical segments small enough and thus adjust the electrical and magnetic properties of a metamaterial in such a way that a negative refractive index is created: microwaves allow components in the millimeter range. However, visible light is considerably shorter-wave. By miniaturizing the components down to the nano range, the researchers finally reached the longer-wave, red end of the visible spectrum.

However, the process presents some unavoidable difficulties. On the one hand, the magnetic control by resonance circuits is based on resonance, which inevitably leads to partial absorption of the light: the wave emits part of it and thus excites the resonance circuits. On the other hand, the geometry of the components severely restricts the negatively refracted angle and wavelength range for visible light.

Henri Lezec and his team at the California Institute of Technology avoid these problems a little by shifting negative refraction into the realm of plasmonics, where it is easier to handle [3]. Plasmons are density fluctuations of charge carriers that propagate as waves in the material. They can be controlled very precisely in waveguides through the layer thickness of selected metals and insulators. In the experiment, the researchers excited plasmons in a sandwich of waveguides with blue-green light. At the other end of the waveguide, the electron oscillations emitted the light again at an angle that corresponds to negative light refraction.

The detour via electrons has a number of advantages: Waveguides are much easier to manufacture and tailor than metamaterials that are constantly being reduced in size and function largely independently of the angle of the incident light. Because the technology is not based on resonance, radiation losses can also be restricted more strongly. The negatively refracted wavelength range can also be significantly expanded through the higher dielectric conductivity of the waveguide core.

The applications that are emerging on the distant horizon are very attractive. So far, light microscopy has reached a limit: just as light "does not see" the components of a metamaterial that are significantly below its wavelength, light-based microscopy cannot identify details that are below this wavelength limit. There are already methods to push this limit slightly, but approaching a negative refractive index of exactly minus one would infinitely increase the resolution of a light microscope in the theoretical limit value: the perfect microscope.

Stealth caps are also often mentioned as a futuristic application of negative refractive materials. Strictly speaking, however, this does not require negative refraction, only the skillful manipulation of the magnetic light component, which is driven by research on metamaterials. For long wavelengths it is even possible to make a simple body "invisible" by cleverly guiding the microwaves around it.

But until these technologies are ready for the market, we can look forward to the fact that scientists from time to time report new light phenomena that run counter to intuition: light that is not reflected or apparently runs backwards a Doppler effect in which (transferred to sound) a receding sound source sounds higher than an approaching one, or particles that are preceded by their Cherenkov "supersonic" cone instead of following.

## Physics teaching group for engineers

The teaching group "Physics for Engineers" offers courses in basic physical education for all engineering courses offered at the TU Ilmenau. The range of courses includes the physics module, which is divided into the subjects Physics 1 and Physics 2 as well as the basic physics internship.

### Physics 2 (summer 2021)

This course covers the following topics:

1. Introduction to Thermodynamics (Thermodynamic Basics, 1st Law of Thermodynamics)
2. Technical cycle processes (basic principle of thermal machines, Carnot cycle, Stirling engine, internal combustion engines, heat pumps and refrigeration machines - left-hand Carnot processes, 2nd and 3rd law of thermodynamics - reversibility of processes)
3. Real gases (extension of the equation of state, phase transformation, liquefaction of gases)
4. Periodic state changes - oscillations (free, undamped mechanical oscillation, damping, resonance and superposition of oscillations, forced oscillations and resonance, superposition of oscillations)
5. Waves (basics, mathematical description of waves, intensity, energy transport, superposition of waves, Doppler effect and supersonic)
6. Optics (geometric optics, wave optics, quantum optics - light as particles)
7. Quantum physics (wave-particle dualism of micro-objects, Heisenberg's uncertainty principle)

### Online exam physics 2 N + W WS20 / 21

This course serves to hold the Physics 2 catch-up exam on April 13, 2021.