## On supersonic passenger jets

I was recently watching this vox video and BBC documentary on the Concorde and supersonic travel. The fluid dynamics of air at mach speeds greater than 1 is a fascinating subject on its own accord.

• #### Stagnation enthalpy

• Stagnation enthalpy is defined as the ‘Enthalpy of a fluid when it is brought to rest from velocity v isentropically’. Stagnation properties such as Stagnation enthalphy, pressure, temperature are often used to quantify the fast moving air around a flight moving at supersonic speeds.
• “For very low Mach numbers, the density of the air is a constant. But as the Mach number increases into the supersonic regime, some of the energy associated with the motion of the object compresses the gas and changes the density from its static value….. As the Mach number increases into the low hypersonic regime, some of the energy of the flow excites the vibrational modes of the diatomic molecules.”

• ### How to photograph shock waves ?

• Imaging shock waves in a laboratory can be achieved through specialized tools, but NASA regularly posts images of shock waves around jets flying way up in the sky. How does one begin to visualize shock waves from an around an aircraft at several thousands of feet in the air. This post runs through a few techniques that are commonly employed

• Vapor cones and condensation
• Slowing down to Transonic speeds, a visible cloud of condensed water form around a fast moving aircraft or object known as a ‘vapor cone’.

## Leslie speaker, Doppler effect and the Nobel Prize in Physics 2019

The Nobel Prize in Physics 2019 was awarded “for contributions to our understanding of the evolution of the universe and Earth’s place in the cosmos” with one half to James Peebles “for theoretical discoveries in physical cosmology”, the other half jointly to Michel Mayor and Didier Queloz “for the discovery of an exoplanet orbiting a solar-type star.”

In this sub-section we will try to understand how Michel Mayor and Didier Queloz discovered the first ever exoplanet – 51 Pegasi b . Let’s first take the example of a Leslie speaker.

## Leslie Speaker

This speaker has two horns from which the sound emerges out.

The two horns are placed on a rotating platform which can spun at high speeds.

Therefore, if you play a tone at frequency ‘f’ and begin to spin the horns,  you can make the listener hear a higher frequency(f1) and a  lower frequency tone(f2) instead of ‘f’.

If the horns stop spinning, the listener will only hear frequency ‘f’ .

This is due to the Doppler Effect and leads to some really cool sound effects. This video offers a great demo around the 7:30 mark:

## Planet or no planet?

In our solar system the Sun, Earth, and all of the planets in the solar system orbit around a point called the barycenter. This is where the center of the mass of the solar system lies at :

This means that the motion of the sun and jupiter looks like so:

Top and side view (exaggerated for more clarity)

This same ‘wobbling’ idea applies to planets revolving other stars as well (called ‘Exoplanets’).

The star moves around in a circle like the horns of a Leslie speaker.

The spectrum of the star when it is moving towards us would be doppler shifted to a higher frequency and when the star is moving away would be doppler shifted to a lower frequency!

Measuring this wobble is one way to find whether a planet is orbiting the star or not.

Michel Mayor and Didier Queloz were awarded the Nobel prize for their discovery of 51-Pegasi b, an ‘exoplanet’ orbiting a sun-like star 51-Pegasi using this technique.

When they published their results in 1995 it was the first exoplanet to be discovered.

Today more than 4,000 exoplanets are confirmed to be in orbit around other stars but their research definitely stands as the cornerstone in what has now become a field of its own.

Source of gifs: NASA , UOregon

* Check out other techniques to find exoplanets here.

## Visualizing Doppler Effect using ripple tanks

Ripple tanks are really cool ways to explore the way a wave behaves under the influence of a perturbation.

They are fairly simple to make, and are usually available in college and school laboratories to render better understanding of the wave phenomenon.

## How does it work ?

There is a usually an oscillating paddle( above– used to produce plane waves) or a point source/s ( below – used to produce circular waves ) which are actuated by eccentric motors, solenoids, etc + a shallow tank of water.

Here are some of my favorite renditions of physical phenomenon on a ripple tank. Check sources for more. Enjoy!

1. Diffraction

2.Double slit experiment

3. Reflection

4. Refraction

5. Parabolic Reflectors

6. Doppler Effect

Doppler effect is the increase (or decrease) in the frequency of sound, light, or other waves as the source and observer move toward (or away from) each other.

If we have a speaker that is moving to the right (see animation above), if you are standing infront of the speaker you will hear a shorter wavelength- higher frequency sound and if you are standing behind the moving speaker, you will hear a longer wavelength – lower frequency sound.

We experience the Doppler effect everyday whenever a car whizzes past us. Here’s a demo:

** Source videos : Educational Services Inc-1964  and Aerodynamic generation of sound

## Even and Odd Harmonics of a vibrating string

In the previous section we took a look at the vibrating string fixed at both ends and found that in order for the boundary condition to be satisfied, the following are the only solutions possible:

The solutions on the left of the image are often termed as ‘Odd Harmonics’ because they have odd number of anti-nodes and the ones of the right have even number of anti-nodes hence ‘Even Harmonics’

If you pluck a string right at the center, you are essentially exciting only the odd harmonic components.

You can test this out by taking a string instrument and plucking it at the center. Download a spectrum app on your phone and take a look at the spectrum

You will see peaks in the spectrum only at the frequencies which correspond to the odd harmonics:

String instruments like the violin or guitar are plucked off-centered, so you get both the odd and even harmonics. But what differentiates one instrument from the other is the amplitude at which these harmonics are expressed when you play them.

Do watch the following video if you would like to know more:

## Standing Waves

If you have a solid understanding of what Traveling waves are (click here if you need a refresher) then when you add up a sine wave moving to the right with a wave moving to the left, you get a standing wave.

$y(x,t) = \sin(kx-\omega t) + \sin(kx + \omega t)$

Using $\sin(a) + \sin(b) = 2 \sin((a+b)/2) \cos((a-b)/2)$ and simplifying the above equation we get :

$y(x,t) = 2\sin(kx)\cos(\omega t)$

A plot of this looks like the following:

For the most part when one is referring to standing waves, it is customary to just talk about the resultant wave that you see above.

But one should understand that the way you form this is by taking a right moving wave and adding it up with a left-moving wave: