There are all sorts of resonances around us, in the world, in our culture, and in our technology. A tidal resonance causes the 55 foot tides in the Bay of Fundy. Mechanical and acoustical resonances and their control are at the center of practically every musical instrument that ever existed. Even our voices and speech are based on controlling the resonances in our throat and mouth. Technology is also a heavy user of resonance. All clocks, radios, televisions, and gps navigating systems use electronic resonators at their very core. Doctors use magnetic resonance imaging or MRI to sense the resonances in atomic nuclei to map the insides of their patients. In spite of the great diversity of resonators, they all share many common properties. In this blog, we will delve into their various aspects. It is hoped that this will serve both the students and professionals who would like to understand more about resonators. I hope all will enjoy the animations.

For a list of all topics discussed, scroll down to the very bottom of the blog, or click here.

Origins of Newton's laws of motion

Non-mathematical introduction to relativity

Three types of waves: traveling waves, standing waves and rotating waves new

History of mechanical clocks with animations
Understanding a mechanical clock with animations
includes pendulum, balance wheel, and quartz clocks

Water waves, Fourier analysis



Monday, October 29, 2007

Details on Infinite Series

In the previous chapter we were considering the following infinite series equations for the functions ex, sinx, and cosx.

infinite series for exponential function infinite series genie
infinite series for sine function
infinite series for cosine function

Since these were central to Euler's inventing imaginary numbers, perhaps we should examine the series in more detail. Here we look at the terms in the series, one by one. Since the expression for ex has more terms, we will start with the terms in that series.

The figures below are graphs of these terms versus x. The last image is that of all the terms or graphs superimposed together. In the last graph, I have colored the region of each where the particular term dominates, i.e. where it is larger than the other terms, that is, has a greater y value.


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graph of constant term in infinite series of exponential function graph of linear (first order) term in infinite series graph of second order term in infinite series
graph of third order term in infinite series of exponential function graph of fourth order term in infinite series of exponential function graph containing 0th through 4th order terms of infinite series of exponential function
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Looking at the last graph above, we see that the first term is largest for small x's, since it starts off at 1 whereas the other graphs start at zero. For a little larger value of x the second terms, the linear term, that grows at a constant rate, dominates. At still a larger x the parabolic curve, i.e. the third terms is largest. It starts off at zero and even has an initial slope of zero, but its higher power law dependence allows it to eventually become the largest. The same is also true for the third and four terms, the higher the power, the larger value of x is required for that term to dominate, but it eventually does. It is also true that we are dividing the higher power terms by a larger constant, sort of handicapping them, to initially stunt them, but eventually their higher power overcomes this effect and they dominate.

When we use an infinite series to represent a function like a sine, cosine, or exponential, we are using this feature, that different power terms can be made to dominate over different parts of the graph. We can choose the signs (the +'s or −'s) and the constants in front of the terms to adjust the effect of each term to make the series behave just like the function we are "fitting".




Fitting the Exp Function




We are first going to use this method to "fit" the exponential function y = ex shown below. This function is famous for eventually reaching greater values than any power law function can. "Exponential growth" of a population or consumption has come to mean an ever increasing rate growth, named after the properties of this function.

graph of exponential function illustration, rocket girl




In the series of graphs below, we see the effect of using one term, then two terms, then three terms, etc. to "fit" the exponential function, which is also shown on each graph.

graph, comparing exponential function with constant term graph, comparing exponential function with sum of first two terms graph, comparing exponential function with sum of first three terms
graph, comparing exponential function with sum of first four terms graph, comparing exponential function with sum of first five terms graph, containing curves of all five previous graphs


In the first graph, only the initial value is matched by the fitting curve, or constant expression. In the second curve, the initial value and the initial slope are both matched and the fitting function now slopes upwards. In the third graph, the initial value, the initial slope, and the initial curvature are are matched and the fitting curve now curves upward in loose approximation to the exponential function. Adding two more terms additionally fit the initial rate of change of the curvature and the rate of change of the rate of change of the curvature. These are actually calculus methods, and the resulting series is called a Taylor series. From the above graphs, we see, that the more terms, the better the fit and the longer the region over which the series fits the function. Adding even more terms to the series would make the fit still better.

The last graph above superimposes all the other graphs, so that you can better compare them.

Calculators and computers often use infinite series as a way to calculate functions, such as sines, cosines, other trigonometric functions, exponentials, logarithms, and square roots.




Fitting the Sine Function




We next turn our attention towards the sine function. Whereas the exponential function was ever increasing, much like most of the terms in the series, the sine function does not resemble the terms, but instead simple undulates between +1 and −1, much like a wave or a snake might do. Below we see the short section of a sine function. As we have seen before, over a much larger range of x values, the repeating, wave-like nature of the sine function is more evident, however here we shall focus on this shorter section.


sine function, angle in decimal radians illustration, snake




Below we see the fit to the sine function of one term in the series, two terms, three terms, four, and five terms. The series

equation, infinite series for sine function

is a little different from the exponential series above in two obvious ways.

  • First, only odd powers are in the series, i.e. only powers of 1, 3, 5, 7, etc.
  • Secondly, the signs (the +'s and −'s) between the terms change for each consecutive term, as opposed to the use of only +'s in the exponential series above.

The use of odd powers is a result of the sine function having odd symmetry about the y axis. That is, it is a mirror image of itself flipped over. As a result, the series will only use terms that also have this odd symmetry, i.e. those having the powers of 1, 3, 5, 7, etc.

The alternating signs make this series an "alternating series". It means consecutive terms partially cancel each other and prevent the sum from becoming too large. The use of both positive and negative terms allows us to fit the undulating sine function.



graph, comparing sine function with first term of infinite series graph, comparing sine function with first two terms of infinite series graph, comparing sine function with first three terms of infinite series
graph, comparing sine function with first four terms of infinite series graph, comparing curves in the previous four graphs waves


  • The explanation mark in some of the equations on the graphs is a short hand notation for factorials which are a set of consecutively decreasing numbers multiplied together. That is 3! ≡ 3 × 2 × 1 = 6 and 6! ≡ 6 × 5 × 4 × 3 × 2 × 1 = 720, and so on.
  • As before, as more and more terms are included, the series fits the sine function better and better, and over a longer span of x values.
  • In a computer or calculator program calculating the sine of a number, the series would probably only be used in the range of 0 to 45 ° and other steps in the program would reduce the actual angle to the equivalent angle in this range, over which it is easy to use 5 to 10 terms in the series to obtain outstanding accuracy. For angle in the range of 45 to 90° the program would use the relation that sin x = cos(90° − x) and use the series for cosine discussed next. Because of the use of an alternating series, we have an easy indicator of the error in the approximation. The error will be less than the next term evaluated at the value of x at which we want to know the error in the fit.





Fitting the Cosine Function




We next fit the cosine function, which is very much like the sine function, except that it is shifted over by 90 degrees, i.e π/2 radians. Because of this shift, the function has even symmetry (is a mirror refection of itself on either side of the x axis, and the resulting Taylor series for the cosine is composed only on even powers of x. As listed at the start of this discussion the series is:

infinite series expansion of cosine function

Like the sine expansion above, this expansion is an alternating series, where the signs reverse for each consecutive term.


graph of cosine function, angle in decimal radians illustration, turtle




Below we see a series of graphs of the fit to the cosine function of one term in the series, two terms, three terms, and so on. As before, the more terms, the better the fit and the long the range of x values over which the fit is reasonably good.



graph, comparison of cosine function and constant term of infinite series graph, comparison of cosine function and first two terms of infinite series graph, comparison of cosine function and first three terms of infinite series
graph, comparison of cosine function and first four terms of infinite series graph, comparison of cosine function and first five terms of infinite series graph comparing the curves of the five previous graphs




Next: the technology of oscillations and waves demands new mathematical methods.

© P. Ceperley, 2007.


Good references on WAVES Good general references on resonators, waves, and fields
Scroll down farther for a list of the various related topics covered in postings on this blog.