Projectile Motion I: No Air Resistance
Projectile Motion II:Air Resistance Linear in Speed

General Equations of Motion

Using the same co-ordinate system as before the general equations of motion for projectile can be written:

Where:

b is a constant and represents the coefficient of air resistance

is the gravity vector. In our co-ordinate system gravity points downwards so the y component is negative.

is the velocity vector with x and y components.

is the magnitude of the velocity vector i.e. the speed.

is the unit vector in the direction of

In addition, we have the following boundary conditions (i.e. initial values at t=0):

at t=0

EOM x-Direction

For motion in the x-direction we can now write the equation of motion as:

          Equation (1)

with initial conditions

EOM y-direction

          Equation (2)

with initial conditions

The first thing to notice about equations (1) and (2) is that they are: (a) nonlinear and (b) coupled (i.e. the behaviour in the x-direction depends on properties of motion in the y-direction and vice versa). These properties mean that solutions can’t be written in closed form i.e. no analytical solution can be derived for the general situation that we’ve defined to this point. instead, the general equations defined by (1) and (2) need to be solved numerically with relevant boundary conditions on a case by case basis.

Solutions to EOM

The general equations of motion for the x and y directions are nonlinear and coupled and for solutions in the general case require numerical methods (e.g. Runge Kutta). Special case solutions can be found by studying the situation where the motion is constrained to be purely horizontal (vertical velocity=0) or purely vertical (free fall under gravity, horizontal velocity=0). These cases are interesting in their own right and can be studied by solving equations 1 or 2 by setting the relevant velocity component =0 and looking at the behaviour of the solutions to the EOM with ther elevant boundary conditions.

I recently found a paper addressing this situation that shows that it is actually possible to derive analytical results in closed form using the Lambert function in certain specific circumstances. The paper can be downloaded from this location, if you’re interested in this it’s well worth a read and working through the analysis:

Analytic Approximations of Projectile Motion with
Quadratic Air Resistance

The authors of the above paper consider 3 logical domains and solve the equations of motion (equations 1 and 2 above) in each of the domains. They provide comparisons between their solutions and the ideal case of zero air resistance along with comparisons with real flight data.

The 3 domains considered are:

LAT – Low Angle Trajectory

This is the region where i.e. where the motion of the projectile is mainly horizontal (low angles) and hence the horizontal velocity sifgnificantly exceeds the vertical velocity. Closed solutions are obtained using the Lambert function and useful algorithms are provided for calculating the Lambert function (these are useful in their own right for other applications).

HAT – High Angle Trajectory

In some ways this is the opposite of the LAT and has i.e. the vertical velocity greatly exceeds the horizontal velocity (high angles). again, closed solutions for the EOM is obtained.

SAT – Split Angle Trajectory

The so called split angle trajectory has i.e. middle angle trajectories.

The authors derive the solutions to the equations of motion in each of these ranges and compare the solutions with each other and with the ideal case (zero air resistance). Interesting discussion around the calculated trajectories and actual experimental flight data is provided.

If you’re interested in this sort of thing I highly recommend you grab a copy of the papere and have a read. Eork through the analysis, plot some trajectories using Excel, MATLAB, Mathematica, Java programming or whatever your particular weapon of choice happens to be.

That’s it on projectile motion. I had fun, if anyone reads this hopefully you’ll find it useful

TI-68 Handbook/UserGuide

In a previous post projectile motion was considered with no air resistance. What happens to the projectile motion when we include an air resistance factor? In reality including air resistance is very complex and in the general case leads to coupled nonlinear differential equations which don’t lend themselves to closed analytic solutions, instead they are usually solved using numerical integration methods on a case by case basis.

In this analysis a simple air resistance force that is linear in the speed of the projectile is considered. The force associated with this air resistance is opposite in direction to the velocity vector at any particular time. In addition to the force associated with the linear air resistance there is the force associated with uniform gravitational field acting on the projectile. The physical situation is shown in Figure 1.

Figure 1: Trajectory of Projectile

Forces Involved

The force due to gravity acting on the projectile is

The force due to the air resistance is

The total force acting on the projectile is

2. Equation of Motion (EOM)

The equation of motion for the projectile is:

This vector equation is actually 2 equations 1 for motion in the x-direction and 1 for motion in the y-direction as follows:

EOM x-direction

with initial conditions x(0)=0 and

EOM y-direction

with initial conditions y(0)=0 and

3. Solutions to EOM

The equations of motion above are ordinary second order linear differential equations which can be solved analytically using the boundary conditions to derive the constants of integration. Integrating each equation gives the following solutions

x position

     Equation (1)

Note that in the limit we can see

This is shown in Figure 1.

Figure 1:x(t) converging to

y position

     Equation (2)

Differentiating these expressions once gives the speed in the x and y directions as functions of time and differentiating twice gives the acceleration. We get:

Speed in x-direction

Speed in y-direction

Acceleration in x-direction

Note that unlike the case where there is no air resistance, acceleration in the horizontal direction in not zero.

Acceleration in y-direction

Equation of the Trajectory

An explicit equation for the trajectory can be derived by using equation (1) to eliminate the time “t” and substituting that value into equation (2). We find:

where      Equation (3)

     Equation (4)

This trajectory showing Y as a function of x is shown for representative values of the parameters in Figure 2.

Figure 2: Trajectory With Linear Air Resistance

The mathematical solutions to equation (2) are shown in Figure 3 but we are only interested in physical solutions i.e. those for which y is positive.

Figure 3: Mathematical Solutions to the Trajectory

Figure 4 shows the trajectories for 2 projectiles with the same parameters, 1 projectile experiencing zero air resistance and the other experiencing linear air resistance as described in this post. The projectile with linear air resistance is the lower curve and shows a significantly reduced range compared to the “no air resistance” projectile (not surprisingly). Not only is the range reduced when air resistance is present but the shape of the trajectory is also changed. Specifically, the trajectory with no air resistance is described by a parabola which is symetric about the point of maximum height. When air resistance is present the trajectory is not symetric and is not parabolic. Instead it is skewed towards the high-z end of the flight because of the diminishing horizontal speed (remember the horizontal speed was constant in the case of no air resistance).

Figure 4: Comparison of Trajectories when Air Resistance Present

4. Trajectory

We can calculate some features of the trajectory by looking at the behaviour of the solutions to the EOM in specific limiting circumstances.

Time to Maximum Height

We can calculate the time of maximum height by solving for t where is given by equation (2). We find:

From which      Equation (5)

Co-ordinates of Maximum Height

We can calculate the x position of the maximum height by solving for x where y(x) is given by equation (4). We find:

Which, when solved for x gives the horizontal co-ordinate of the maximum. We find:

     Equation (6)

We can calculate the y co-ordinate of the maximum (i.e. the maximum height reached) by evaluating equation (4) for where is given by equation (5). Substituting equation (5) into equation(4) and doing some algebra results in:

     Equation (7)

Taken together, equations (5),(6) and (7) define the maximum point in the trajectory. Notice that we couldn’t use the same convenient relationships as we did in the case of no air resistance (x velocity constant, trajectory symmetric, flight time= twice time to maximum height) because the trajectory with linear air resistance is not symmetric and the x velocity is not constant.

Horizontal Range

We can now look at the horizontal range. This is calculated by finding all the real solutions for x from equation (4) when we set y=0. In general we can’t solve this in closed form (i.e. write a general formula for the solution), instead it has to be solved numerically on a case by case basis.

However, we can make some progress if we restrict ourselves to instances where the air resistance is low. In our equations this means looking at the case where “b” is small. before proceeding with this analysis though, it gets messy with all the symbols and can make equations look more complex than they are, so concealing the essential points. To address this we can re-scale the variables so that the equations become dimensionless and (hopefully) simpler to write and manipulate.

For the x and y co-ordinates, a sensible quantity to use to scale the co-ordinates would be the maximum horizontal range in the absence of air resistance i.e. . We can therefore define and . Using these new variables we can re-write equation (4) as:

     Equation (8)

Where for convenient writing.

If we assume then we can expand the logarithm around zero and retain only terms linear in B (this means retaining terms up to the cubic term in B in the expansion). Writing the series expansion for the log around zero and doing some algebra results in the following quadratic:

The horizontal range is found by finding solutions to Y=0 i.e.

Solving the quadratic gives the following roots for X:

Physical solutions require X to be positive so we’re only interested in the positive root of the quadratic. Since B is small we can expand the square root and like before only keep terms that result in a linear dependence on B. This emans retaining expansion terms upto . Doing the algebra results in the solution:

     Equation (9)

At this point we can check if this makes sense by checking the case where there is no air resistance i.e. B=0. In this case which agrees with the results in the previous post where projectiles with no air resistance were analysed. So far so good.

For a fixed value of B the range is plotted in Figure X for various launch angles. As you can see there’s an optimum launch angle that gives a maximum range for the chosen value of B. We’ll calculate that optimum angle in the next section.

Figure 5: Horizontal Range v Launch Angle with Linear Air resistance Present

Angle of Maximum Horizontal Range

To calculate the angle giving maximum horizontal range we need to solve for , we can then calculate the actual range by evaluating X at this angle. So, first things first, let’s calculate the angle for maximum range by evaluating the derivative of equation (9) and setting it to zero.

     Equation (10)

We can check if this feels right by setting B=0 and confirming the solution is , the result for no air resistance. If we set B=0 then we have which is true for in the range of launch angles of relevance to us. So, looking good so far.

For the case B=0 we’ve just found that the solution for optimal angle is . For B non zero but small, this allows us to assume a solution of the form and to solve for .

With B small we can expand the trig entries in equation (10) to produce a simpler equation to solve. Calculating the series expansions for the trig terms in equation (10) we find:

Inserting these values into equation 910) and performing some algebra gives the result:

Giving the optimum angle for a specific value of B, which reduces to in the absence of air resistance as expected.

The importrant point to take away from this result is that in the rpesence of air resistance, the optimum angle for maximum range is less than the result when there is no air resistance.

We can now calculate the range at this optimum angle by substituting the above result for into equation (9) and evaluating. Remembering that B is small the trig terms in equation (9) reduce to:

So, from equation (9) we have:

In the case where B=0 this reduces to X=1 or in agreement with the previous result for no air resistance. In the case where B is non zero the range is reduced, as expected.

5. Comparison With No Air Resistance

So, in the presence of air resistance we find:

A point to remember in all this is that the detailed analysis only applies for small values of “b” i.e. the air resistance factor. The results derived from the formulae above will begin to diverge from those obtained from numerical modelling of the trajectory as “b” increases. Beyond the divergence threshold solutions to equation (4) where y=0 have to be found numerically.

So, that’s it for projectiles with linear air resistance. There’s more you could do with this, writing a Java program to solve equation (4) and comparing the results with the calculations for small “b” to establish when the two approaches diverge for example. For now, however, this post is complete. A final post on projectile motion with quadratic air resistance will follow this one. have fun

1. Equations of Motion (EOM)

The physical situation being considered is shown in Figure 1. We’re considering a projectile that hass mass “m” being launched at an arbitrary angle to the horizontal axis with an arbitrary initial velocity . There is no air resistance and the only force acting on the projectile is the force of gravity – which for now we’ll take as being constant, g.

Figure 1: Projectile Motion

We’ll constrain the projectile to move in only 2 dimensions, we’ll call them x and y with x along the horizontal, increasing left to right and y running vertically, increasing bottom to top. the position of the projectile at any time t can be written asa a position vector

The motion of the projectile is governed by Newtons laws of motion, for the discussion here, Newtons 2nd law of motion is the important one. This can me written in general as:

where F and a are both vector quantities

i.e. gravity acts vertically in the negative y direction i.e. straight downwards and the force due to gravity on a mass m is the weight given by mg.

Newtons 2nd law can now be written as follows in vector form:

This allows us to write 2 separate equations of motion, 1 for motion in the x direction and 1 for motion in the y direction as follows:

EOM x-direction

                  Equation (1)

EOM y-direction

                  Equation (2)

Straight away equation (1) shows that motion in the x direction is not subject to a force, therefore from Newtons 2nd Law there is no acceleration therefore the velocity in the x direction is constant (remember the velocity is a vector so has both magnitude and direction) which means the speed in the x-direction is constant.

Equation (2) shows that this is not the case in the y-direction where there is a force but the motion is independent of the mass of the projectile m. This might be surprising if you’ve never come across this subject before, you might think that heavier objects will fall faster. They don’t. In an ideal vacuum all objects fall at the same rate – see the video below from Apollo 15 where a hammer and feathere are relased at the same time and land at the same time:

2. Solutions to EOM

To find the equation of motion for the particle we need to solve equations (1) and (2). In mathematical terms these are called ordinary differential equations. An important point to note is that both are linear and they are uncoupled. These properties make exact solutions possible. To solve equations (1) and (2) we need to know some starting values (called initial conditions). Well, we do know some. From Figure 1 we can see that x(t=0) = 0 and y(t=0) = 0 and the velocity in the x-direction is with the velocity in the y-direction being

To find the solution to equation (1) and (2) with initial conditions we need to integrate (1) and (2) and use the initial conditions to calculate the constants of integration.

Integrating equation (1) gives:

                  Equation (3)

where the constant of integration is = 0

Integrating equation (2) gives:

                Equation (4)

where the constant of integration is = 0

Equations (3) and (4) provide parametric equations for x and y. We could use the ideas in one of the previous posts where parametric equations were covered (see: http://www.riotstories.co.uk/science/seashells/ to plot x and y as functions of t.

Instead wew can express y as a function of x by using equation (3) to eliminate “t” then substitute into (4) to get an explicit expression for y in terms of x alone. from equation (3) we find:

Substituting this into equation (4) and doing some algebra gives:

          Equation (5)

This curve is shown in Figure 2 and is parabolic.

Figure 2: Trajectory y(x)

From equation (5) we can find

          Equation (6)

and

          Equation (7)

3. Features of the Trajectory

We can now use the solutions of the EOM to determine some features of the trajectory.

Maximum Height

From equation (4) the maximum height () can be calculated by solving for t to determine the time at which the maximum occurs (call this then evaluating equation (4) for this value of .

We find:

Flight Time

The “flight time” of the projectile is twice the time to reach maximum height (it has to go up and come back down) i.e.

This gives:

Range

The range of the trajectory is the solution to equation (5) for y=0, or alternatively the solution to equation (3) for . In either case we find:

Clearly the range is maximised when giving

Radius of Curvature

From mathematics the radius of curvature, is defined to be:

So we can use equations (6) and (7) to calculate the radius of curvature of the trajectory at each point. However, at the top of the trajectory (i.e. point of maximum height) we already know that

so at this point

from equation (7).

We have already calculated so substituting the expressions we find:

From this expression we find that when

4. Inclined Plane

How is the range of the projectile affected if instead of a the flat horizontal plane used so far, the projectile is fired up an inclined plane like that in Figure 3 where the plane is inclined to the positive x-axis at an angle “a” and the projectile is fired at an angle to the plane.

Figure 3: Inclined Plane

From the diagram we can see that . Substituting these values into equation (5) and solving for R gives 2 roots as follows:

The first of these is just the origin and is of no physical interest. The second root is greates when from which we see that the maximum range up the inclined plane is achieved when i.e. an angle which is slightly greater than the case for a flat plane.

If we plug these values into the experssion for the range R we calculate

As a check, setting a=0 in all these results for the inclined plane reduces the expressions and values to the same as we get for the horizontal plane – a useful check.

5. The Monkey Hunter Experiment

I thought it might be interesting to bring all these ideas together into a single example. It’s commonly referred to as the “Monkey Hunter” and you’ll find lots of Flash and Java animations on the internet illustrating the idea. I’m no good with Flash and I’m only a beginner with Java so I’ll give the graphics a miss – a quick Google search will uncover lots.

The basic idea is this: A hunter spots a monkey in a tree and wants to hit it with a dart gun (some modern day variants have the hunter replaced with a naturalist who throws bananas at the monkey instead of firing darts). The monkey in question is a special breed. He drops vertically from the tree immediately when the projectile is fired at him (dart, bullet, banana). The question is how should the hunter aim in order to hit the monkey ?

To analyse this problem let’s suppose the monkey is up a tree, a height “H” above the ground, the hunter is a horizontal distance “d” away from the tree, the line of sight from hunter to monkey makes an angle with the horizontal and the projectile is launched with an initial velocity . There is no air resistance to worry about.

From previous work we can immediately write the EOM for the projectile and the monkey as follows:

EOM for Monkey

                Equation MH1

                Equation MH2

EOM for Projectile

                Equation MH3

                Equation MH4

Remember, when the projectile is fired, the monkey lets go the branch and drops vertically at the instant the projectile is fired. So, the first question is “will the projectile actually reach the monkey” i.e. travel far enough? Notice this is a different question from “will the projectile hit the monkey”. That comes next.

In order to reach the monkey the projectile must have a range, R, that is at least equal to teh distance to the vertical line down which the monkey falls from the tree i.e. R must be at least equal to “d”. Using the previous expression for the horizontal range R we can show that there is a condition on the initial speed of the projectile to esnure that it can at least reach the monkey, specifically requiring means . A speed less than this means the projectile simply wont have the range to reach the monkey.

Assuming that the projectile does have the range to reach the monkey, the two will impact when they have the same x and y co-ordinates. The x co-ordinate of the monkey is always “d” and because the horizontal velocity of the projectile is constant (no force) we can calculate that the bullet will reach the position x=d in a time .

At this time we can calculate the y position using MH2 or MH4 (since they will both be the same i.e. y position of monkey = y position of projectile since they impact). Evaluating MH4 for gives

                Equation MH5

This is just the previous equation (5) with x=d. This equation is shown in Figure 4 as a function of .

Figure 4: Impact y position as a function of speed

From Figure 4 the point where the curve crosses the horizontal axis is the point beyond which and gives the critical derived at the beginning of this analysis. Speeds below this value give no physical (i.e. positive) solutions for y meaning the projectile and monkey never impact. Notice that there is a positive solution for y for every speed above the threshold, meaning that as long as the projectile has enough energy (i.e. speed) to cover the required range it wil lalways hit the monkey.

So apart from the threshold speed, the speed of the projectile doesn’t determine whether or not the monkey gets hit – it always gets hit. The speed does, however, determine where in the vertical path the monkey gets hit. from Equation MH5 or Figure 4, you can see that for higher speeds the monkey and projectile impact at large values of y i.e. near the point y=H. for lower speeds they collide for low values of y. in the limiting cases of [v_{0}=v_{c}[/latex] the projectile and monkey impact at y=0 and for they impact at y=H. For intermediate they collide ay y positions that increase with increasing speed.

Since the y co-ordinates of the monkey and projectile are equal at impact equating equations MH2 and MH4 leads to the relationship meaning the hunter aims directly at the monkey.

CONCLUSION: The hunter should aim directly at the monkey and as long as the projectile has sifficient energy to cover the distance “d” the projectile will always hit the monkey. The speed of the projectile has no affect on whether the monkey is hit or not though it does determine where in the vertical path it gets hit – faster projectile hits the monkey near the top, slower projectile speed hits the monkey nearer the ground.

That’s the end of this post on projectiles with no air resitance. The next post will cover the case where there is air resistance that varies linearly with the speed and a subsequent post will consider air resitance that has a quadratic dependency on speed.

In the course of arriving at the final result we’ll cover some interesting physics – mainly the ideas behind standing waves, chromatic musical scales and the physics of vibrating strings. This isn’t a post about building musical instruments, it’s limited purely to understanding how the position of the frets are calculated.

There are many online fret position calculators that can be used to check the results that I’ll derive using the ideas developed in this article and if you’re building or maintaining a guitar I recommend using them because they provide many useful additional information e.g. the position and angle of the bridge on an acoustic, the position of the bridge on an electric.

There’s a good one on the StewMac site here

So, let’s make a start…

The Physics of Vibrating Strings

Some Rules Governing Vibration on Strings

A guitar string is an example of a string fixed at both ends that vibrates when plucked. In this section we’ll consider the physics that governs how such a plucked string can behave and see how those rules lead to an understanding of what is possible and what isn’t for such a string. Specifically we will establish and make use of the following 3 rules:

Rule 1: When a string fixed at both ends is subject to a constant tension, the frequencies that the string can vibrate at are inversely proportional to the length of the string. For example if are examples of frequencies that strings of length can vibrate at when both lengths are held at the same tension, the values are related as follows:

Rule 2: When a string fixed at both ends is of constant length, the frequencies that the string can vibrate at are directly proportional to the square root of the string tension. For example, if are examples of frequencies that a string of fixed length vibrates at when subject to tentions then the frequencies are related as follows:

Rule 3: For physically different strings having the same length and tension, the frequencies that each string can vibrate at are inversely proportional to the mass density. For example, if are examples of frequencies that strings of mass densities vibrate at when the strings have the same lengths and tensions then the frequencies are related as follows:

In wave motion, there is a relationship between the speed of the wave, the frequency of the wave and the wavelength of the wave as follows:

          Equation (1)

where v= speed, f = frequency and = wavelength

These rules can be used to infer some behaviour that we would expect of vibrating strings:

• A short string will have a higher pitch than a long one
• The tighter a string is stretched the higher its pitch will be
• A thick heavy string will have a lower natural pitch than a thin one

Speed of a Transverse Wave Along A String

The speed of a wave on a string depends on the properties of the string (mainly the tension, length and weight of the string). Wave motions along a string under tension are solutions of the so called “wave equation” and can be shown to be given by:

          Equation (2)

For anyone interested in the details of this, an online reference is item 3d in here: http://www.physnet.org/modules/pdf_modules/m202.pdf

This is useful because equations (1) and (2) relate the physical properties of the material the wave is travelling along to properties of the wave itself. Something we will use.

Standing Waves

We can step back from the maths for a moment and think about what’s going on with these standing waves.

With the ends of the string fixed, any wave motion must be zero at those end points (otherwise we would have a travelling wave). For this to happen the string of length L can’t support just any old frequency of wave, instead it can only support waves that allow the motion of the end points to be zero. In other words, only waves that have a wavelength that is related to the length of the string L in a particular way can be supported. We find that only waves for which a whole number of half-wavelengths fit into the length L can be supported. The longest such wave has a wavelength 2L so that the string of length L supports only a single half-wavelength and nodes at each endpoint (i.e. places where the wave is stationary). This maximum wavelength (or via Equation (1) minimum frequency) wave is referred to as the “fundamental”.

If we doubled the frequency (i.e. reduced the wavelength by half) we would find that a whole wave fitted into the same length L, but this time there would be 3 nodes (i.e. stationary points, the previous 2 end points of the string plus a new node in the middle). This behaviour is shown in the figure below and represented mathematically by Equation (3)

   n=1, 2, 3, etc     Equation (3)

The Fundamental Frequency on a String fixed At Both Ends

From equation (3) we see that for the fundamental,

Using this result for the fundamental wavelength and eliminating between equations (1) and (2) gives the frequency of the fundamental:

More generally

n=1, 2, 3, etc     Equation (4)

Equation (4) is at the heart of the design of most stringed musical instruments. The tension of the string is adjusted using tuning heads on the guitar. You can see the 3 rules quoted at the beginning of this article in equation (4).

Octaves

In music theory, an octave of a partuclar note is just the note at twice the frequency so a note having frequency has the first octave at frequency the second octave at , the third at and so on until we get the general rule with n=1, 2, 3, …

Musical Chromatic Scales

At this point we need to introduce the subject of chromatic scales in music. A great reference for this and other related topics is the book “Mathematics and Music” by David Wright, chapter 6 of which covers what we’re discussing in this section.

A “chromatic scale” generally divides the octave into a number of “chromatic divisions”. In Western music we tend to divide the octave into 12 divisions i.e. if we take a note and its upper octave as the two bounding frequencies the range is subdivided into 12 equally spaced divisions; each division is commonly referred to as a semi-tone in music. The previous relationship defines a geometric series for octaves i.e. successive terms are fixed multiples of preceding terms. If we split the octave range itself into 12 divisions the frequency of any particular note will be of the form where the parameter “r” is the multiplying factor for successive terms.

For the Western chromatic scale we have 12 divisions in an octave. Using that as the boundary condition we can see and from the definition of an octave we know that which allows us to write . This lets us calculate the ratio for the 12 note chromatic scale as or .

For the general chromatic series the octave is divided into n intervals with the ration being given by or . You can see how this reduces to the familiar chromatic scale when we let n=12.

Using this rule, we can define the frequency for each note in the chromatic scale (we’ll stick to the 12 note chromatic scale but you could experiment with more of fewer notes and it’s an interesting read to research chromatic scales with different numbers of notes). The notes in the chromatic scale, starting at are now given by:

where n=1, 2, 3, …

Fret Position

We can now use what we’ve learned so far to calculate the positions where we should place frets on the fretboard of an instrument to ensure we get the correct frequencies. Imagine we have a string of length L. We know that the fundamental frequency is given by

Suppose wew change the length of the vibrating portion of the string so that the vibration frequency is 1 semi-tone higher than previously. The frequency of this vibration is calculated to be:

Dividing these 2 equations gives

We have already established that frequencies follow a geometric progression with

so where

We can show that in general we can see that

Calculated Fret Positions

Using equation (5) we can now calculate the position of the frets for a given value of “L”. These calculated values are shown in the table below for L = 26, of course you can pick whatever value of L you want.

NOTES:
1. So far we’ve talked about a string having a particular length which we’ve called L and which is fixed at both ends. That’s all fine for modelling the physics of a vibrating string but how does it relate to an actual guitar? Well, the length of the vibrating string which we’ve called L is just the length of the string between the fixed end points. On a guitar the fixed end points are the “nut” and the “saddle”. The length L is the length between them and is referred to in guitar building and maintainance as the “scale length”. The scale length varies between manufacturers, models and even types of guitar e.g. resonators used in blues slide guitar will have different (probably shorted) scale length from say a Gibson SG.

2. The numbers in the table above can easily be checked using some of the online fret position calculators. Fret positions are sometimes quoted relative to the guitar nut instead of the saddle (as I’ve calculated here). It’s easy to convert between the 2 (fret position from nut = L – fret position from saddle). I’ve shown both positions in the table above for a scale length of 26. You can check the calculated values by visiting the fret calculator on the StewMac site and trying their calculator using a scale length of 26 as in my example.

Calculate Note Frequencies

I thought it would be fun to calculate the actual note frequencies in a 12 note chromatic scale for each of the strings on a guitar fretboard using equation (6) above. To do this I need to know the fundamental frequency for each string (i.e. the frequency of the string when plucked in the open position), this will give me the value in equation (6) for each string. I can then use equation (6) to calculate the successive frequencies (i.e. notes) in the chromatic scale.

I/ve shown the calculated frequencies in the table below. In the table you can see relationships between the notes and octaves. For a given string, n=0 shows the frequency of the open string. If you take any 2 successive frequencies on any string and calculate the ration (highest frequency/lower frequency) you will see the result is always which is the ratio in the geometric series we discussed earlier (watch the number of decimal places you use in the calculation or rounding errors will mess up the division – always use as many as you can).

You can check the frequency calculations by comparing them with the values on http://www.vaughns-1-pagers.com/music/musical-note-frequencies.htm which also shows note frequencies and octave groupings for piano.

That’s it – in this post I learned a bit about the physics of vibrating strings and how to use that understanding to calculate the frequencies of the notes on the guitar strings along with where to put the frets on the guitar fretboard. I thought it was quite interesting, hope anyone reading it does too – maybe even useful if you’re interested in playing, building or maintaining musical instruments

I thought it would be fun to investigate the shapes of seashells and try to understand the factors that shape them.

In this post I’ve recorded what I’ve learned with examples for future reference. I’ve tried to make this post sufficiently complete that it could be used in the future as a reasonably comprehensive refresher in case I decide to come back to the subject and try playing with it further. In the course of thinking about this topic other ideas become important and the first part of this post summarises them (e.g. parametric surfaces, polar co-ordinates, ideas from Euclidean geometry). Where a particular subject requires it I’ve added a separate post to go into more detail and included a link to it in this post.

Background ideas

This section just outlines some of the background ideas that are necessary for this particular investigation.

1. Cartesian Co-ordinates

Everything that follows in this post assumed Euclidean space i.e. the flat space that we’re all used to in everyday life where the rules of geometry we were taught in school apply. Nothing written in this post is relevant if you’re thinking about curved space where different rules of geometry apply.

In 2D geometry it’s usual to think of the co-ordinates of a point as being specified on a so called “Cartesian Co-ordinate System”. That’s just a fancy way of saying “imagine a flat plane where conventionally we call the direction stretching from left to right i.e. running horizontally from left to right the x-axis and the direction running vertically the y-axis”. The x and y axes define 1 Cartesian co-ordinate system. This is shown in Figure 1.

Figure 1: Axes in a 2D Cartesian Co-ordinate System

A point, call it “P”, in that co-ordinate system would be specified as (x,y). In this co-ordinate system the point “P” has 2 properties: a length measured from the origin, call it “r” and an angle with the positive x-axis, conventioanlly this angle is labelled (the letter “Theta” in the Greek alphabet (pronounced “Thee-ta” or “Thay-ta”). It’s conventional for science and maths to label variables with letters in the Greek alphabet. The point “P” and the properties are shown in Figure 2.

Figure 2: Properties r and of the point “P”

One of the interesting features of this geometry is that the properties r and can be calculated from the co-ordinates (x,y) using Pythagoras’s Theorem from which we find:

So far so good. We have learned how to specify the position of a point in a 2D Cartesian plan comprising a co-ordinate system denoted by an x-axis and a y-axis. Working in a 2D Cartesian plane like this is very useful as we will see but it is useful to extend this concept to a 3D co-ordinate system as follows

2. 3D Cartesian Co-ordinates

The 2D co-ordinate system described so far allows us to specify the position of a point on a flat plane. By introducing a 3rd co-ordinate axis we can specifiy the position of a point in 3D space. Conventionally this 3rd axis is denoted “z” and the position of our point “P” is then given as (x, y, z).

Much can be written about the co-ordinate systems discussed so far but it would distract too much from the point of this post – plenty of good reference material can be found online.

Our new z-axis is orthogonal (i.e. at right angles) to our existing x-y plane, but that means there are two possible orientations for the positive z-axis. How do we decide which one to choose?

Conventionally we choose a left-handed or a right-handed co-ordinate system. The direction of increasing z is determined by imagining you rotate the x-axis onto the y-axis using the curled fingers of the left hand for a left-handed co-ordinate system or the right hand for a right-handed co-ordinate system. In either case the thumb of the chosen hand will point in the direction of increasing z for that co-ordinate system. This is shown in Figure 3.

Figure 3: Left and Right Handed Cartesian Co-ordinate Systems

We now have a 3D Cartesian co-ordinate system like the one shown in Figure 4 (which happens to be a right handed system – I usually end up choosing a right handed system for whatever reason).

Figure 4: Right-Handed 3D Cartesian Co-ordinate System

3. Distance in 3D Co-ordinate Systems

In the 2D Co-ordinate system discussed above Pythagoras’s Theorem was used to show the length of the line from the origin (the point (0,0)) of the co-ordinate system to the point “P” (denoted by r) is related to the (x, y) co-ordinates of “P” through the relationship:

Imagine now that we have 2 points on the x-y plane with co-ordinates (x2, y2) and (x1, y1) instead of the origin (0, 0) and a single point (x, y). The distance between these 2 points is just an extension of result already quoted from Pythagoras’s Theorem and that distance is:

or

See the similarity between this and the previous result ? This is shown in Figure 5.

Figure 5: Distance Between (x2,y2) and (x1,y1) in the 2D Cartesian Plane

This idea can be extended into our 3D Cartesian co-ordinate system to give the following general result for 2 points (x1, y1, z1) and (x2, y2, z2).

or

Applying this result to the specific case where one of the points in our 3D system is located at the origin (0,0,0) and the other point “P” is at the general position (a,b,c) we can calculate the distance (call it “d”) from the origin to the point “P” as follows:

This is shown in Figurer 6.

Figure 6: Distance in 3D Co-ordinate System

The Story So Far…

So far we’ve established:

• How to define a 2D Cartesian co-ordinate System
• How to specifiy the location of a point in a 2D co-ordinate system
• How to calculate the distance between points in a 2D co-ordinate system
• How to define a 3D Cartesian co-ordinate system
• How to define the location of a point in a 3D Cartesian co-ordinate system
• How to calculate the distance between points in the 3D co-ordinate system

4. Polar Co-ordinate Systems

We’ve discussed how to represent a point on the x-y plane using Cartesian co-ordinates. Howevere, that’s just 1 co-ordinate system that is useful. There are others. For our purposes in this investigation into the shapes of seashells another co-ordinate system is equally useful – the “Polar Co-ordinate” system. We’ve already touched upon the conept of polar co-ordinates indirectly in the previous sections.

We showed in Figure 2 that a point in the x-y plan can be represented as a distance from the origin (we labelled the distance “r”) and an angle in a counterclockwise direction from the positive x-axis which we labelled

In the polar co-ordinate system of Figure 2 the point P has the polar co-ordinates (r, )

The point P(x,y) or P(r, ) is actually the same point and any fundamental behaviour associated with P must be the same regardless of co-ordinate system (this is “invariance” is a fundamental concept in relativity theory). this means there must be a transformation between the x-y and polar co-ordinate systems.

From Figure 2 we can establish the following relationship between (x,y) and (r, ):





allowing transformation from polar to Cartesian

and





allowing transformation from Cartesian to polar

This idea of using Cartesian and polar co-ordinates to represent a position on plane will be useful in what follows and leads naturally onto the next piece of background information that is necessary, that of parameteric curves.

The Story So Far…

So far we’ve established:

• The concept of 2D & 3D Cartesian co-ordinate Systems
• How to calculate the distance between points in a 2D & 3D co-ordinate system
• How to define the position of a point in a 2D polar co-ordinate system
• How to translate the co-ordinates of a point between 2D Cartesian and polar co-ordinate systems in either “direction”

5. Parametric Curves

Instead of defining y in terms of x i.e. y= f(x) or x in terms of y i.e. x= f(y) we define both x and y in terms of a third variable called a parameter as follows:

x=f(t)
y=g(t)

Here we have introduced a new variable (which is usally) denoted by “t” – naturally, it doesn’t have to be called “t”, you can call it anything you want.

Sometimes the values that “t” can take will be restricted, sometimes they won’t. Whether it is restricted or not depends on the situation and what we are trying to do. Most of the time it will be reasonably clearwhether “t” needs to be restricted or not. In what follows you will see instances where the parameter is restricted and others where it isn’t.

Each value of “t” defines a point in the x-y plane that can be plotted, each point x or y being the value of a function f(t) or g(t) evaluated for that particular value of “t”. The collection of (x,y) points that results from letting “t” take on all possible or allowed values is the graph of the parametric equations x=f(t) and y=g(t) and is called the parametric curve.

Here are some examples:

Example1: Parametric Equation of a Circle

If we set R=2 and allow t to take on values in the range we get the following curve:

I haven’t shown it on this diagram but the points on this diagram evolve in a counter-clockwise direction – to see this just evaluate the (x,y) pairs for various values of the parameter “t” in the allowed range, below I’ve shows the (x,y) pairs for a given value of “t” in the format t=value: (x,y)

If we wanted the curve to go in the opposite direction we would set different boundaries on the parameter “t” so that it was in the range: and we will find that the arithmetic signs of the “R” parameter in the above results reverses for the values of different from 0 or showing that the radial line sweeps out the circle in a clockwise direction starting in the horizontal position lying along the positive x-axis.

Example 2: parametric Equation of an Elipse

If we set R=2 and allow t to take on values in the range we get the following curve:

There are many other curves occuring in mathematics and in nature that govern physical behaviour and interactions, some alternatives that can exhibit interesting behaviour are:

Astroid

Astroid Pedal Curve

Bifoliate

2D Spirals

We can now use the ideas that we’ve learned about so far to take the first step towards modelling teh shape of seashells. Most people if asked would probably agree that seashells appear to exhibit a spiral type structure so that’s where I’ll start.

Spirals are an interesting geometric shape and it can be fun to study their properties for its own sake. There ar many different spiral structures but to get started I’ll look at a simple one and a more realistic one. at the end of this section I’ll list some others that you can have a play with if intrested.

Referring back to the previous discussion on polar co-ordinates the spirals have the property that the radial parameter r is a function of the angle so:

Archimedean Spiral

A common class of spiral is the so called “Archimedean spiral”, the general form of which is:

The form of this spiral that we will use here is the case where x=1 giving:

The parametric equations we will use can now be written for the spiral in the x-y plane as follows:

where
and initially

The parametric equations can now be re-written as follows by substituting for r:

with a, b constants and

Using the previous formula for calulating distances in a 2D cartesian co-ordinate system we can show that the distance between points on the successive turns of this spiral along a given angle is equal to (clue: a turn means that along a given angle the value of “t” has increased by ). Since “b” is a constant for any given instacne of this spiral the distance between successive turns is also a constant. We will see that this is different from the next spiral we look at (which is also more realistic in that it occurs frequently in nature).

The figure below shows this spiral plotted for a=2 and b=4 and

The parameter “a” determines the starting radius of the spiral on the x-axis (i.e. it’s the radius of the circle that would result if b=0) and is shown in the figure below where we have increased “a” from 2 in the previous diagram to 5 and b=4 as previously

As derived previously the separation between turns of this spiral is a constant and is proportional to “b” so bigger “b” means arcs that are further apart.

Logarithmic Spiral

Another spiral form that is important because it is a form that occurs in nature in many places is the so called “Logarithmic Spiral”. The polar form of the logarithmic spiral is:

where once again “a” and “b” are constants. An interesting property of this spiral is that it is self-similar, for example if the spiral is scaled by a factor of then is the same as the original without rotation, just bigger. It turns out many shells in nature follow growth patterns defined by logarithmic spirals which allows the animal to grow physically bigger but without changing shape.

The parametric equations for the logairthmic spiral in the x-y plane are given by:

In the same way as we calculated the distance between points on successive curls along a given angle for the Archimedes spiral we can do so for the logarithmic spiral. The result obtained is:

For increasing “n” these values form a geometric series with scale factor . Compare this with the equivalent for the Archimedes spiral where the successive differences form an arithmetic sequence with difference given by

The figure below shows a logarithmic spiral with “a”=0.15 and “b”=0.2 plotted over the range

3D Spirals

We’ve now got the details of 2 common spiral shapes in 2D (the x-y plane). To use these spirals to model seashell shapes we need to extend them into 3D.

Essentially, extending into 2D is a logical extension of what we’ve already covered for 2D, this time the z co-ordinate becomes a function on our angle so we have:

3D Archimedian Spiral

for the Archimedes Spiral used previously we already know .

Initially we assume a linear variation of z as a function of so we have

where c is a constant, giving

We can plot example spirals using these relationships and various values of a, b, c. An example of the Archimedes spiral with a=0.2, b=3, c=3 is shown below. It is interesting to experiment with different values of a,b,c to understand the effect these individual parametrs have on the shape of the spiral.

3D Logarithmic Spiral

We can adopt a similar approach when using the Logarithmic spiral, this time we can use giving the parameteric equations for the spiral as

Notice we’ve given the general form for and used c and d for the parameters. We could of course choose values so that c=a and d=b, in which case the z=r. Generally a,b,c,d are constants.

An example of a Logarithmic spiral with a=0.1, b=0.2, c=0.3, d=0.14

Model Seashells

Now we can use the spirals above to generate some sample seashell shapes. To do this we need to treat the spirals we’ve generated as a path in space. This gives a line not a 3D surface with extent which is what a physical seashell is. We call the 3D spiral a “helico spiral” and the way we create a 3D surface is to move a generating curve along the helico-spiral pathway – a bit like moving a hoop through the air by hand to create a pattern of soap bubbles. Part of the fun is in creating different generating curves. To start with I’ll use a simple semi-circle to illustrate the idea. The figure below is an attempt to show the geometry and illustrate how we use it.

From the image you can see that the generating curve has a radius “R” and that radius creates an angle

A point on the surface we will create by moving the generating curve along the helico spiral is defined in terms of 2 angles and . As before but

From the figure showing the generating curve you can see that the position on the new surface i.e. “r” is now increased as a result of the shape of the generating curve. You can do some straightforward trigonometry to show that

In addition, the z co-ordinate is also increased by the amount of the R vector on the generating curve that is projected onto the z-axis (i.e. R-the component of R on the baseline of the semi-circle for a given angle). Again some straightforward trig will show you that

Using these modified expressions for “r” and “z” we can write the parametric equations using an Archimedian spiral as the helico spiral as follows:

,

Here are some examples:

Increase the number of rotations of the spiral by increasing the upper limit of and make the shape of the generating curve a circle instead of a semi-circle by changing the uppere limit of from to to get something like this:

You can now play around with ideas. for example you could let the radius of the generating curve change as the curve rotates, make it a linear growth with angle along the lines

where f and g are constants and f=R when . With this modification to R, The parametric equtions become

and here’s an example of what you get

Instead of a semicircle for the generating curve, you could try an ellipse. In that situation you follow the same logic as before but this time you will find that the radius and z component get modified along the following lines:

Here’s an example of a surface using the parametric equations modified for an ellipse instead of a semi-circle generating curve and using a range of

Hopefully that gives you the general idea. The geometries covered in this article have been limited to spirals (which are very interesting in themselves and worth playing with) and circles, semi-circles and ellipses as the generating functions.

If you check out the examples I’ve used in this article you will notice that most of the shells generated have open tops. I think that’s because the generating curves I’ve used mostly had “R” = constant. When I used the top of the shell seemed to close over. I plan on experimenting further with this.

In addition, an interesting experiment would be to use some sort of rendering software to apply a finish to the generated shells and see how lifelike they can be made. However, I don’t have any experience (and only moderate interest to be honest) in that sort of thing. Maybe someday Have fun.

Another class of HTTP 404 errors seems fundamentally different in nature – they look like attempts to connect directly to plugins that have known vulnerabilities or attempts to connect direct to some of the important files on the site. I’ve taken different action on those and will continue to monitor carefully. An interesting learning opportunity those!

Ever wish you didn’t bother ?

What a performance. The WAMP stuff looked OK until I actually tried using it! Initially I ended up with a #1045 error on the root user when trying to use phpMyAdmin. Trying to sort that was a joke and a compelte waste of time. Thankfully it was my dev environment so I could easily trash the database and completely uninstal WAMP from the PC (registry, files, database the lot) followed by a quick re-instal of everything including creating a new dev database.

phpMyAdmin worked fine after that carry on – i.e. back to where I was an hour earlier!!

Now I had to re-install WordPress for my local development purposes and immediately encountered the “ERROR: PHP IS NOT RUNNING”. What!!!???? Yes it is! No matter what path direction I have I just couldn’t get WP to instal. I then had the idea of creating a virtual host for the site in the Apache httpd.conf file. This worked a treat and I installed and configured WordPress locally with a nice new version, copied allmy plugins, configured things the way I’ve got them on live and it’s looking good.

Only remaining problem I have is none of the pages in the menu or imported posts work on the local instal – I probably didn’t do something correctly related to “moving wordpress”. Permalinks are the same structure in my local dev environment as on the live box. I’ll worry about that later.

All in all this has been a major PITA and a lot of wasted time and effort.

1. Lost in Translation
2. Bladerunner
3. As Good As It Gets
4. Cast Away
5. Groundhog Day
6. Gorky Park

Here goes:

Adjust your “single.php” file

1. Download your single.php file from the default twentyeleven theme folder on your server to make edits locally

2. If you’re running a web server on your local PC with a WordPress instal you will be able to test your changes locally and just upload to your live server at the end when you’re finished (e.g. WAMP)

3. Open your downloaded single.php file and add the following code snippet:

<?php get_template_part( 'content', 'single' ); ?> // *** Original code in the template as a position marker ***

<!-- ****************************** -->
     <div id="authorbox"> <!--author box added by me-->
          <div class="authortext">
               <?php if (function_exists('get_avatar')) { echo get_avatar( get_the_author_email(), '80' ); }?>
               <h4>About <?php the_author_posts_link(); ?></h4>
               <p><?php the_author_description(); ?></p>
          </div>
      </div>
<!-- ****************************** -->	                    

<?php comments_template( '', true ); ?>           // *** Original code in the template as a position marker ***

4. Test this code insertion to make sure it’s appearing in a location on your page that you’re happy with. don’t worry about how it looks, you’ll be styling the content next. the mission for now is to make sure it appears in a place you’re happy with. When I did this I was using the default Twenty Eleven theme so the location above is relevant for that template. I placed my code after the “get_template_part” and before the “comments_template” tag. You can change the location to suit your own needs but most templates are similar at this level.

Style your new author biog box

5. Now that you have the author biog box where you want it it’s time to add some style to it. You can make this as simple or elaborate as you want to.

6. Add the following CSS rules to your theme stylesheet (if you’re using a child theme this will be the stylesheet in your child theme folder):

/*Style author box*/
#authorbox {background:#fcf8d7; border:1px solid #e2dede; max-width:620px; margin:0 auto; margin-bottom:20px; overflow:hidden; padding:10px;}
#authorbox h4 {font-size:16px; color:#FFF; margin:0; padding:0; clear:inherit !important;}
.authortext {padding-left:10px;}
#authorbox img {margin-right:10px; padding:0; float:left; border:5px solid #e2dede;}
#authorbox p {color: #FFF; margin:0; padding:0px; font-size:12px;}
#authorbox h4 > a {text-decoration:none; color: #FFF;}
#authorbox a {font-weight:bold;}
#authorbox {-moz-border-radius: 9px; border-radius: 9px;}
#authorbox {background: #03f;	background: -moz-linear-gradient( #03F, #036);	background: -o-linear-gradient(#0f0f0f, #2B2B2B); background: -webkit-gradient(linear, 0% 0%, 0% 100%, from(#0f0f0f), to(#2B2B2B)); /* Older webkit syntax */ background: -webkit-linear-gradient(#0f0f0f, #2B2B2B);}
div.authortext p a {color: #FF3;}

7. I mentioned in the intro that some of the CSS styling might have to change depending on the theme you’re using. On line 3 of the CSS snippet you will see a “clear” instruction. that’s one that is likely to need modification in your theme. I had to use it because of previous styling rules applied to the h4 tag in the Twenty Eleven theme (so I had to adjust those rules for this specific addition while leaving them in tact for the rest of the site). There might be similar little quirks in the theme you’re using. Just play around with the CSS to get what you want. I’ve used “max-width” to get a nice width on the biog box when the browser window is full size, when the browser window is reduced the biog box shrinks dynamically. This setting looks OK on iphones, ipads, tablets etc.

8. Change the colours, border radius, link colours etc to suite your needs. Strip the CSS right back to have a basic minimalist author box or make it quite elaborate with background images etc.

9. That’s it. You should now have an author biog box appearing after each post on your site in a location of your choice looking the way you want it to look.

Notes:

• The code used here assumes the post author has a gravitar set up and that they have completed the user biog section in their user setup with something sensible. If they haven’t this will be a bit of a waste of time
• You could replace the php “get_avatar” call with a simple image insert if you only have 1 author for your site and you don’t want to set up a gravatar
• The biog box will only appear at the end of the full post – not in excerpts or in the post preview (on the front page if you’re using the Twenty eleven theme). that’s because you added the code to the single.php template file. You could experiment with other files/locations if you wanted the information to display on other pages etc
• You can see the results of this in action if you open any of the posts on this site.
• You might find this a bit of overkill and repetition if you are the only poster on your site. I’d probably agree with that. I use this site to experiment with WordPress and learn how it works so I’m less concerned with that sort of consideration and more interested in how to achieve a particular result. If you’re running some sort of commercial site or more professional blog then you might be right in thinking twice about implementing this sort of thing.

NOTE: In the Twenty eleven theme there’s already an author block included when a post is made in a multi-user site and the author has entered their biog data. You will find the code for this in the Twenty eleven theme file “content.single.php”. the div involved is “author-info” and there are vaious classesand default styling rules in the default style.css file. Instead of adding new HTML etc to your single.php file as I’ve done in this post you could just re-style the default div. (I found this out after having done the work because this feature didn’t appear correctly – mainly I suspect because only a single publishing author had been sued in my blog until I made a mistake and published using a different user).