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Unit 10

Harmonious Math

10.4 Mathematics of Waves

PERIODIC FUNCTIONS

In the previous section, we looked at how to quantify different aspects of a sound wave mathematically. We saw that frequency, phase, and amplitude are the key quantifiable attributes that distinguish one wave from another. What of the actual wave itself? What is the mathematical function that represents a wave?

Obviously, we need a relationship that exhibits periodic behavior, returning to the same position or value with regularity. Remember that a sound wave causes air molecules to "vibrate" back and forth from their at-rest positions. Any function used to model waves should display the same output value for regularly repeated input values. If the function models air pressure, the input value is time, and we, therefore, would want a function that periodically returns to the same pressure value as time progresses. One such function is that old trigonometry favorite, the sine function.

triangle

NOTE: Throughout this discussion, we measure angles in units of radians. Recall that 2π radians are equivalent to 360°, a complete circle. Half that value, π radians, therefore corresponds to 180° (half a circle), equation radians to 90°, and so on.

Suppose that we have a right triangle. We can define a few quantities that relate the angles of such a triangle to the lengths of its sides. The most familiar of these relationships are the sine, cosine, and tangent of an angle. The sine of an angle is the ratio of the lengths of the opposite side and the hypotenuse. Similarly, the cosine of an angle is the ratio of the lengths of the adjacent side and the hypotenuse. The tangent, then, is the ratio of the length of the opposite side to the length of the adjacent side, or equivalently, the ratio of sine to cosine.

triangles

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WHEEL IN THE SKY

For simplicity's sake, let's focus on the sine function. Notice that in a triangle, the larger the angle, the longer the opposite side becomes. This fact is a natural correspondence of triangles: side lengths increase in proportion to their opposite angles.

triangles

In right triangles, the longest side will always be the hypotenuse. To find the maximum value of sine, we can investigate a series of right triangles and see exactly how large the side opposite our angle of interest can get. If we let the angle get close to equation radians, we see that the length of the opposite side approaches the length of the hypotenuse. If we let the angle equal equation (note that this is purely a mental exercise—the triangle we have been imagining disappears at this point), we interpret the opposite side and hypotenuse to have the same length and, thus, their ratio, the sine of the angle, is 1.

angles

As the angle increases further, beyond equation radians and we shift our perspective to look at the triangle formed by the angle's complement, the opposite side begins to shrink.

angles

The length of the opposite side diminishes toward zero as the angle approaches π radians. Notice that an angle of π radians and an angle of zero radians have the same sine value—0.

radians

So far, we've seen that the sine function starts at zero, increases to 1, then decreases back to zero as the angle steadily gets larger. This is somewhat reminiscent of how the waves we studied in the previous section behave. If we were to look at the air pressure of a particular region as a sound wave passed through it, we would observe the sequence of events depicted in the following images:

rarefaction

compression

baseline

rarefaction

In our investigation of the sine function so far, we have modeled the first two of these steps, the compression and return to baseline. In the following diagram we see that the sine also models the rarefaction of a sound wave by diminishing to the value of -1 and then returning to zero, where we started.

rarefaction

We have now seen that the sine of an angle oscillates between 0, 1, and -1 smoothly. Because the sine function exhibits this periodic behavior, it can serve as a rough model of a simple sound wave. Although there are really no natural sounds that are exactly modeled by a sine wave, we can create such an ideal, pure tone using a synthesizer. A synthesizer can produce such a sound through the exact control of the voltage that drives a loudspeaker.

There is an important difference, however, between the function that we use to model the air pressure changes brought about by the passing of a sound wave and the sine function, as we just described it. The sound wave pressure function is a function of time. The standard sine wave is a function of angle. We can reconcile this by establishing a relationship between angle and time.

How triangles relate to sin waves.

If we imagine the hypotenuse of the triangle that we just examined to have a fixed length of 1 unit, and we allow this line segment to rotate freely around the origin of the coordinate plane like the spoke of a suspended wheel, we can begin to reconcile the angle vs. time problem. The rotational speed with which the spoke rotates can be thought of as a frequency, because the spoke periodically returns to the same position. As the spoke rotates, the angle it makes with the positive horizontal axis at any given point in time can be found by looking at how fast the spoke is rotating and how long it has been rotating. Multiplying these two quantities results in an angle. So, instead of the sine of an angle, we can now consider the sine of the rotational speed times time. Graphing the value of sin (ωt) on the vertical and time on the horizontal produces the familiar sine curve.

This is how we connect triangles and unit circles with time-dependent sine waves.

How triangles relate to sin waves.

Now we have a good mathematical model of a simple sound wave:
F(t) = sin (ωt) where t is time and ω is related to the frequency of the wave.

trictly speaking, because ω is a rotational speed, it is measured in units of radians per second. If we can somehow get rid of the radians in this expression, we will be left with a quantity that has units of inverse seconds—the same as frequency! If we divide ω by 2π radians, we will have corrected for the radians and, thus, we will have found the frequency of the wave.

The amplitude of the wave will correspond to the maximum value that our function can output. Because the sine function normally oscillates between -1 and 1, any coefficient attached to the function will directly affect the amplitude of the wave. For example, the amplitude of the sine wave 4sin (ωt) is 4.

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THE WAVE EQUATION

Finally, it is important to realize that sound waves are not solely functions of time; as we have seen, they are actually pressure distributions in space that vary with time. In order to model this situation mathematically and completely, we need some formal expression of a wave's behavior in both time and space. We can think of its temporal behavior as related to frequency, but its spatial behavior is better thought of in terms of how the amplitude at a given time varies with the wave's position. To be clear, the spatial dependence we are talking about is not the height above or below baseline but rather is concerned with the distance perpendicular to that—the direction in which the wave travels.

We can illustrate this space/time dependence by imagining first what a wave would look like, were we to somehow stop time. If you've ever seen ripples frozen in a pond in winter, "frozen in time," so to speak, you have some idea of what this would look like.

Looking at a cross-section of the frozen surface, we can visualize the spatial dependence in one dimension, namely x, the horizontal dimension. We can see that the height of a wave depends on position. Measuring at a trough produces anegative height, whereas measuring at a crest produces a positive height.

graph Item 1871 / Oregon Public Broadcasting, created for Mathematics Illuminated, TWO DIFFERENT HEIGHTS AT TWO DIFFERENT X-VALUES (2008). Courtesy of Oregon Public Broadcasting.


This tells us that any function that we wish to use to model the height of a wave must somehow depend on position. We'll use x to represent this spatial coordinate. We'll see a little bit later that waves in the real world are rarely one-dimensional, in which case it becomes necessary to use additional coordinates to represent spatial distribution in more dimensions.

We saw in the previous section how a wave depends on time. We used the analogy of a steadily rotating spoke to express this dependence. With both spatial and temporal dependence in hand, we can create a function, u, that represents the height of a wave at any given point and time. We express this dependence by making u a function of both position, x and time, t, or u(x,t).

To express how u changes with both position, x, and time, t, we are going to need calculus, the mathematics of change. The calculus concept of a derivative, a generalized notion of slope, represents the instantaneous rate of change at a given point in time (or space). In this case, since u depends on both x and t, we will have to use partial derivatives to express how u changes. Partial derivatives enable us to talk about how u changes in regard to each of the quantities, x and t, separately.

u/∂x represents how u changes with respect to x.
u/∂t represents how u changes with respect to t.

It's also important to notice that the height of a wave changes at a non-constant rate. We can see this in the fact that a particle at a particular x, such as the fishing bob from a few sections back, moves more slowly at the top of a crest or bottom of a trough than it does when in transit between the two.

fish bob

To account for this changing speed, or changing rate of change, we must use second derivatives. The expression equation then represents the acceleration (positive or negative) of u and equation represents the spatial analogy of acceleration. By relating these two functions, we derive the one-dimensional wave equation.

equation =c2 equation

Here, c is a constant of proportionality. In some cases, c is the speed of the wave. Our wave function, u(x,t), is the function that solves this wave equation, a second-order partial differential equation. We can see that the sine and cosine functions from the previous sections will indeed satisfy this equation. For example, if the reader is curious and familiar with calculus, let u(x,t) = sin (cx + t), differentiate with respect to x twice, with respect to t twice, and verify that these expressions are proportional by c2.

Continuing any further in this direction of discussion will take us too far away from our main objective, the exposition of the music-mathematics relationship. The wave equation is important, however, in that it demonstrates how it is possible to represent a physical phenomenon, sound, using the language of mathematics. That is, we have now seen how we can express wave behavior mathematically. Specifically, we've seen that sines and cosines of triangles are periodic functions that can model the compression and rarefaction of groups of air molecules. We shall now return to our quest to understand exactly how it is that sounds become combined. With a solid mathematical understanding of sound waves in hand, we will be able to combine multiple waves mathematically using the power of Fourier analysis and synthesis.

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Next: 10.5 Fourier


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