Triangulation surface block on Grisedale Pike

I'd seen one of these before on Robinson but hadn't known what it was, but after some reseraching about trig points, I found that this is called a SURFACE BLOCK. http://trigpointing.uk/info/types.php and http://trigpointing.uk/trig/7422 I followed the link on the second site to find out what a "third order" site means. It seems that they used the triangulation pillars to do the main survey, found some gaps and set up a second order set to fill the gaps. And then...

Watching the sound

There have been a lot of military planes training in the valleys this week. Several came thundering down the corridor over Keswick and over Bassenthwaite Lake. Their distance away was far enough to be able to tell that light travels much faster than sound. When the first one came through, I was looking at the sound and didn't see the plane. I saw the following plane and as it came past (circled in the picture) I started to count. I got to 5 seconds before the sound was directly in front of me. Sound travels at 330 metres per second so the plane was just over a mile away.

Knocking on the Gruffalo

We found the Graffalo at Whinlatter. I started knocking on the wood, first on the arm and then on the jaw. I got two distinct notes. The arm was higher than the jaw. I tried to estimate how much by do-re-mi.

I don't have perfect pitch so I couldn't tell you the note but was trying to get the relative jump. I found this: https://thatsmaths.com/2014/08/28/temperamental-tuning/ which doesn't quite agree with https://prgomez.com/why-do-re-mi/ but implies that the frequency of the arm is roughly 1.25x higher than the jaw.

Dendochronology: reliability and calibration

I found more tree rings in the Aber Valley in north Wales. Pete found this website for me https://www.environmentalscience.org/dendrochronology-tree-rings-tell-us - thank you! I think this tree will be oak which they say is the most reliable in terms of producing a ring every year. It works well on the first picture. I can count the correct number of rings between the dates. Even with the real tree in front of me, I was struggling with the first 100 years. I was interested in the idea that the growth cycle of one tree can be matched to records from other trees and the attempts to calibrate the dating against radiocarbon dating.

Standard deviation rule of thumb

I found this little link https://www.tutorialspoint.com/statistics/range_rule_of_thumb.htm. Now at school we say that 1/2 x range of repeat readings is the uncertainty. This means that an approximation is that the uncertainty is 2 x standard deviation. We only ever do 3 repeat readings. Thus the data doesn't really have statistical merit. This page https://en.wikipedia.org/wiki/Standard_deviation has a nice graph showing that roughly 95% of data lies within 2 standard deviations of the mean.

Integrating a Gaussian

The work on wave packets has led me to the Gaussian function exp(-ax^2). Here I plotted for a = 0.01. I was interested that a had to be so small to spread the curve out. Integrating this means finding the area under the curve. It tells you how to do the integration here http://www.umich.edu/~chem461/Gaussian%20Integrals.pdf The big two tricks seem to be to first square the integral and then use this to switch to polar coordinates. I was struggling with the area element dA - the small increase in area. In Cartesian coordinates that is dA = dxdy but in polar coordinates it becomes dA = rdrd(theta). This website has a picture and explains all http://citadel.sjfc.edu/faculty/kgreen/vector/block3/jacob/node4.html

Stationary phase approximation

My work on wave packets in quantum mechanics has led me to something called the stationary phase approximation. The idea is that if you are adding together a lot of sinusoidal waves to make a wave packet but you are doing this across a continuous range of frequencies instead of the discrete frequencies I tried the other day, then you get points where the frequencies change so quickly that they come in and out of phase adding up incoherently. To get a decent superposition, you want the response to be roughly the same over a range of frequencies instead of rapidly varying. I have been working from https://en.wikipedia.org/wiki/Stationary_phase_approximation I have gone for the section called "An example". So in the following picture, the integral means you are adding up waves across a continuous range of frequencies, w. (Because w=2pi.f) and expi(kx-wt) is a way fo writing sinusoidal waves using imaginary numbers.

kx-wt is the phase and k is actually a function of w, for examples perhaps k=2w.

If you differentiate the phase with respect to frequency, you find how fast it is varying. The stationary point is when the differential = 0.

Rearranging that leads to a relationship. Now x/t has units metres per second, so dw/dk is a velocity. It is called the group velocity of the wave packet.

Group velocity and phase velocity

Having been working on a wave packet that is made up of lots of different sinusoidal waves together, the result is a set of inner waves inside an outer envelope. The envelope moves at a different speed to the inner waves - hence the distinction between the outer group velocity and the phase velocity of the inner waves. I like the last animation on this page https://en.wikipedia.org/wiki/Phase_velocity This shows the shorter wavelength waves (in blue) travelling with a higher phase velocity then the longer wavelength ones. I hadn't realised that the different waves would go at different speeds - but I suppose that the different colours of light go different speeds in glass.

Making a wave packet

Inspired by this webpage http://hyperphysics.phy-astr.gsu.edu/hbase/Waves/wpack.html I set out to make my own wave packet. If you scroll down to the section called Wave Packets from Discrete Waves you'll see an illustration of putting together sinusoidal waves of slightly differing frequency. A single sine wave has constant amplitude everywhere - the wave is not localised. The superposition of slightly differing frequencies concentrates it with a peak in one place. I made a spreadsheet of sinwt + sin1.1wt + sin1.2wt + ... +sin1.8wt and here's what I got:

It would have looked more like theirs if I'd gone negative on the x-axis but it's not bad at showing localisation.

Working on Fourier Transforms

 I've been working on Fourier Transforms. This involves integrating the product of the function of my choice with exp(-iwt) for all times that are real numbers. So I chose to try a unit step function between -T and T. And it worked. I got the standard answer.

Then I set up a spreadsheet to produce a graph of the answer.

It's an odd shape. It what's the unit step looks like recast as a function of frequency. I'm still working on the significance of it. What does it actually mean?

Jellyfish and the Universe

When I was looking at the jellyfish at the aquarium, I was thinking about how they are a surface like curved space time. Then I noticed that their shadow was being projected onto the surface below. The idea of projecting a multi-dimensional object onto a space of lower dimension is a good mathematical trick employed in many areas of Physics. Then I noticed that the APOD was called the Jellyfish Nebula. What could be better? https://apod.nasa.gov/apod/ap190307.html

Eels and reflection

This eel at Maryport Aquarium spent a lot of time confronting its own reflection. It must seem convincingly like a rival eel. One of the rules of reflection is that the image appears to be the same distance behind the the glass as the object is in front. This can be shown by drawing two reflected rays coming to the eye at slightly different angles. The brain can't cope with the idea that the light bent and thinks that the light has come in a straight line from a point behind the glass. If you extend the virtual rays backwards as dotted lines, they cross at a point a distance behind the glass that is equal to the object distance in front. see below

MBBR water treatment process

We found this waste cleaning system at Maryport Aquarium. There is a biofilm on the plastic shapes and the microbes can process substances in the water. The plastic is less dense that the water so it should float. There must be something constantly pushing the plastic shapes downwards. This film shows little fans doing just that in two of the tanks https://www.youtube.com/watch?v=BduxDqwmBHg The upwards resultant force on a plastic piece would be upthrust - weight. By Archimedes' Principle, upthrust = weight of water displaced. Density of water = 1000 kg per cubic metre. Plastic is 950. So upwards resultant force = (volume of plastic x 1000 x g) - volume of plastic x 950 x g) = Volume of plastic x 50 x g. The closer the densities, the lower the resultant force and the smaller the acceleration with which the plastic rises to the surface.

Are these thermometers really more precise?

Students love digital thermometers. They are easier to read and they seem to be better because they read to 1 decimal place. Here are 3 in the same place. We say that the uncertainty in the reading is 1/2 x range and thus is 1/2 x (21.6-20.7) = 0.45 degree Celsius. So in other words, the true precision of these thermometers is to the nearest 0.5 degrees, which is the same as our old-fashioned analogue thermometers.

A misconception about the seasons

I've been marking some work and astute students know that the Earth moves in an elliptical orbit, thus being closer to the Sun at some times of the year than at other times. The suggestion is that it is summer when we are closest to the Sun. I've always known that this explanation doesn't hold water but never really bothered to look it up. Here's the stark reality - we are closest to the Sun (perihelion) in early January when us northern hemisphere people are in mid-winter. This year it was on 3 January https://earthsky.org/tonight/earth-comes-closest-to-sun-every-year-in-early-january This other site explains the analemma shape well - saying that when the Earth is closer to the Sun it is moving faster and moving slower when it is further away from the Sun. https://www.space.com/3304-earth-closest-sun-dead-winter.html I do need to do some more thinking about it.

Green power at Saltholme

This is the view from the platform at Saltholme RSPB on Teeside. I have been reading the book Drawdown on strategies for reducing the carbon dioxide level in the atmosphere. https://www.drawdown.org/ They see burning rubbish as an interim measure. Burning the waste produces an ash that needs to go into landfill and has concentrated the harmful products. It is popular here in Europe but has gone out of fashion in the US. Burning 1 tonne of waste can generate as much electricity as one-third of a tonne of coal - that's a plus. Also, the ash will need be decomposed by bacteria so the methane levels from the landfill will drop. However, they think that recycling is a much better way to reduce carbon dioxide levels long term.

Frogs and pressure

To celebrate the arrival of frog spawn in the garden pond and sightings of the resident frog, here are pictures of our frog toy. The way that the tongue uncurls as air pressure increases reminded me of a diagram of an aneroid barometer that I had to copy down and learn in Physics at school. It looked like the one in the picture on this website https://www.thoughtco.com/definition-of-barometer-604816 The idea was that a needle was attached to the tube. The tube uncoiled as the pressure went up so the needle went round the meter. However, it seems that this is not actually the case. See https://www.explainthatstuff.com/barometers.html for how one actually works. There is a partial-vacuum chamber that resembles the coiled tube idea but the spiral shape with the needle is probably a spring. I'd like to find that old text book and see if I had just misunderstood what it was saying.