We’ve seen how sinusoidal waves arise, these curves that are the graphs of sine and cosine functions. It turns out that they occur very frequently in nature and in many mathematical applications– for example, if you attach a mass to a spring and you set it vibrating, but also in an electricity lab if you look on the oscilloscope and measure the voltage and the current or sound waves or heart beats and so forth. Let’s have a formal definition of the kind of movement or phenomenon that is represented by these sinusoidal waves. You might be measuring the displacement of an object, let’s say by a real number y.
And it varies with time t according to a law of the form y of t equals A sinus [sine] omega t or possibly a cosinus [cosine] omega t. Now the omega there is the letter that looks a bit like a W. But it’s not. It’s another one of these Greek letters we love. It’s omega. So the A and the omega are fixed parameters, and you have a function of this type. That kind of behaviour is called simple harmonic motion, also called harmonic oscillation. If you were to graph the function y, what would its graph look like? Well, we know that it would be a sinusoidal curve somewhat like this.
Little jargon– the amplitude here would be the absolute value of A. This sinusoidal curve would have extreme values of minus absolute value of A and plus absolute value of A. The period of this harmonic motion would be 2 pi over omega. Why? Because if you look at a piece of the graph of length 2 pi over omega, that’s a basic piece which you can then repeat to get the whole graph, both left and right. The period is 2 pi over omega. And the frequency of this harmonic motion is omega over 2 pi. It’s the number of complete cycles that you have in a unit time. The higher the frequency, the lower the period, and vise versa.
Now it turns out– and this is an extremely important fact in many physical applications– that if you have sine or cosine waves of the same frequency, they can be combined. They will combine if you physically combine them to produce another such wave but shifted in its so-called phase. Here is a precise example of what I mean. Suppose you have the sum of two functions, A sine x plus B cos x. A and B are not both 0. It turns out that a sum of two functions like that can be expressed in the form k cosine of x minus phi for the right choice of k and phi. What is the right choice?
Well, the right choice is to define k to be the square root of A squared plus B squared. And then you define phi by saying that its sine and its cosine are as indicated. Let’s proceed to prove this fact. It will use standard trigonometric facts that we know. The first step is to know that from the definition of k, if you take B over k squared and add A over k squared, you get 1, Pythagoras. So the point B over k A over k is a point on the unit circle. Therefore it does correspond to the cosine and the sine of some angle. Let’s call that angle phi.
And therefore you see that, by definition, cosine of phi and the sine of phi are as indicated in the statement of the proposition we’re trying to prove. Once you have these facts– that is that k and phi have been chosen to satisfy these equations– let us now apply the cosine difference formula, again an identity, to the cosine of x minus phi. You get the sum of two expressions. You get k cosine phi occurring in front of cos x, and you get k sine phi occurring in front of sine x. But you recognize from the way k and cosine are defined that k cosine phi is B. And similarly k sine phi is A.
So you get the expression that you started with in the statement of the proposition. So you’ve proved the proposition. Now a remark, that the proposition can be extended to somewhat more general situations. For example, to prove that A sine omega x plus c plus B cos omega x plus d can be combined in an expression of the same type, as in the proposition. The important thing here is that the frequency is the same, the omega that comes before the x.
Let’s have a definition. We’ll say that one of these resulting expressions that we’ve gotten from carrying out this combination, an expression of the form A cosine omega t minus phi, we’ll say that that’s a harmonic motion that has phase phi. Notice that the phi is something that was 0 initially but now will be different from 0 when we combine two other sinusoidal curves. I’m going to take here A positive just to be a little simpler. We say that two such curves, two curves of that type that have the same frequency now– that’s important– are in phase when their phases, the numbers phi differ by a multiple of 2 pi.
It will happen in that case that the peaks and the troughs of their sinusoidal waves will coincide. Let’s illustrate that with a simple example. Here’s cosine of t. We know its graph looks like that. What if we look at the graph of cosine 2 t ? Well, it’s going to be something that has a frequency that is doubled, a period that is halved. It’s not going to be in phase with cosine t because its frequency is different. However if we start with the cosine of 2 t and we now consider 3 times the cosine of 2 t, what its graph look like?
Answer– it will have the same frequency and period, but it will have an amplitude that will be 3 times bigger. But you notice that it is in phase. The peaks, the points where it reaches its maximum amplitude, and the troughs where it reaches its minimal amplitude and negative amplitude, those will coincide in the two functions. So they are in phase. If I start with 3 cosine 2 t and I now consider replacing the t, the 2 t in the cosine by 2 t minus 2, I get the following function. You can see that it’s not in phase. The peaks and troughs don’t occur in the same places.
In fact if you superimpose them on the graph, they’re definitely not in phase. And in physical terms they will interfere with one another. What is the shift in these two? Well, you can see that there is one unit of shift between them. Why? Because the formula has a t minus 1 where the original one had a t– so a shift one unit to the right, which is not a multiple of the period. Hence they’re not in phase.
Decaying amplitude is a very frequent phenomenon that one sees, notably in the study of electricity. It gives rise to functions f of the following form, k times an exponential term– that’s going to be positive– multiplied by sinus [sine], for example, of omega t. Now the k and a here are positive constants. It turns out that these functions are oscillatory, but they have an amplitude that decays, that gets smaller over time. And it’s frequently the case that we want to understand the graph of such a function. Here’s an example. We want to sketch the graph of the function 8 exponential of minus t over 5 times sine 3t. We’re interested in positive time, t greater than 0.
How can we do it? Well, one way to argue is the following. We know that the sine part of it, that factor, is always going to be between minus 1 and plus 1. When you multiply that by the exponential term, 8 times the exponential term, you’ll see that then the function f that we’re considering will always be between minus 8 times the exponential and plus 8 times the exponential. In graphical terms, that means that plus the exponential gives you an upper envelope for our function f. And minus that same term gives you a lower envelope.
And you will have contact with these envelopes whenever the sine term is either equal plus 1– that’s the upper envelope contact– or minus 1– lower envelope contact. In other words, your function will oscillate between these two envelopes. And there is the graph of your function.