1. Physical and Mathematical Foundations of Sound Modeling

In order to have useful discussions about sound, we need a very simplistic, but practical, understanding of the physics and mathematics associated with sound.

1.1 Physics: What Is Sound?

For our purposes, sound is any kind of vibration that is detectable by the ear or devices analogous to the ear. Treatments of sound in physics books tend to focus attention on the transmission of sound vibrations through the air. We will focus instead on the vibrating systems that produce and detect sounds, and just assume that the air is capable of transmitting vibrations from sound producers to the detectors in the ear.

1.1.1 Vibrating Springs

The simplest sort of vibration to understand is that of a spring. To really simplify things, imagine an environment with no gravity, and with a mass (a solid chunk of something) moving along a frictionless track that is fixed so the track cannot move. The track constrains motion of the mass to a straight line, so we do not need to consider the three dimensions of space. Finally, imagine a spring attached at one end to the mass, and at the other end to some fixed point on the track. Be a bit liberal-minded, and imagine that the spring has length 0 when it is not stretched, and that the mass can move freely past the point where the spring is attached. A picture of our imaginary system is given in Figure 1.1.

Figure 1.1: Ideal spring resonator

At any moment in time, the state of the spring system can be described by two real numbers: the displacement $y$ of the mass to the right of the point on the track to which the spring is fixed, and the velocity $v$ of the mass to the right. Displacement to the left is represented by negative values of $y$, and motion to the left is represented by negative values of $v$. Now, imagine that we displace the mass to the right and hold it in a fixed position, stretching the spring. That is, we establish an initial condition where $y>0$ and $v=0$. When we release the mass, the spring pulls it to the left, causing a state where $y>0$ and $v<0$. Eventually the mass reaches the center of the track at $y=0$, but at this moment $v<0$ and inertia carries the mass beyond the center, to the left where $y<0$. Now, the spring pulls the mass to the right, canceling out the motion $v<0$ to the left. Eventually the mass stops with $v=0$, but at this moment the spring is stretched to the left with $y<0$, so the pull to the right continues and causes the mass to move right with $v>0$. This motion to the right eventually moves the mass past the center, so $y>0$. The leftward pull of the spring opposes the motion until $v=0$. So we return to a condition that is similar to the initial one: $y>0$ and $v=0$, and the cycle repeats. Figure 1.2 shows a schematic qualitative view of the vibration of the spring.

Figure 1.2: Qualitative states of spring

To complete the simplistic physics of a vibrating spring, we need to convert the qualitative observations above into quantitative information that we can use in a mathematical analysis. For this purpose, let $t$ be a real number representing the time that has passed since some arbitrary starting moment when $t=0$. For each quantity $q$ that depends on time, $dq/dt$ means the instantaneous rate of change of $q$ with respect to $t$--when the independent variable $t$ is understood from context, $dq/dt$ is often abbreviated $q'$. When no outside force acts on the spring and mass, its behavior is described by the following two equations:

$$y' = Av$$ (1.1)

$$v' = -By$$ (1.2)

$A$ and $B$ are positive real number constants (independent of time)--their actual values do not matter to us right now. Equation 1.1 holds because velocity is defined to be the change of location over time, and displacement is just a measure of location in the form of distance from a particular origin--the constant $A$ takes care of any conversion of units between $y'$ and $v$ (normally the units are the same and $A=1$). Equation 1.2 represents the fact that the force exerted by a spring increases in magnitude proportionally to the distance that the spring is stretched, and the force acts to pull the ends of the spring together; also the change in velocity (called acceleration) is proportional to the force. The value of $B$ is determined by the stiffness of the spring and the mass. Equation 1.2 is an approximation, because no real spring exerts a force precisely proportional to the stretching distance--in particular when a spring is stretched too far it changes radically, becoming stiffer, or becoming softer, or breaking, depending on its construction.

The right practical approach to understanding vibration is to do as much analysis as possible based on the simple approximate equations above, and then do the potentially complicated corrections only when greater accuracy is required. From one point of view, equations 1.1 and 1.2 are approximations to physical reality. From another point of view, they are engineering specifications, and physical systems are approximations to the equations.

Vibrating objects that produce sound, and others (such as the hairs in the cochlea of the ear) that detect sound, can be modeled fairly well by systems of vibrating springs connected together in various ways. Other vibrating systems have other physical parameters that measure the vibrating behavior, but in most cases there are two real numbers--for example pressure and flow of vibrating air, potential and current of vibrating electrical charge--that behave analogously to displacement and velocity in a vibrating spring.

1.2 Mathematics: How Do We Model Sound?

The key to understanding the mathematical analysis of sound is to visualize the mathematics using graphs and geometric diagrams. The right way to visualize the mathematics does not look like the physical system of vibrating springs or other objects that it is describing. The value of the mathematics is to give us a different way of visualizing sound, that is much more convenient for analytic reasoning than the actual physical configuration of vibrating objects. Mathematically, the important properties of a vibrating spring are just Equations 1.1 and 1.2. We can forget that they arose from the physical properties of a spring, and just consider the numerical behavior of two real numbers $x$ and $y$ as functions of $t$, when they satisfy the equations.

$$y' = Ax$$ (1.3)

$$x' = -By$$ (1.4)

From now on, lower case Roman variables, such as $x$ and $y$, stand for real numbers that are functions of a time parameter $t$. Upper case Roman variables, such as $A$ and $B$, stand for real number constants, which are the same as unvarying functions of time (but widely used notations, such as $e=2.71828\ldots$, are left alone). Occasionally, we will use the form $x(t)$ to denote the value of a function $x$ at a particular time $t$, but usually we will refer to entire functions rather than individual values. When an expression $\alpha(t)$ containing an independent variable, such as $t$, should refer to an entire function, rather than a single value of the function, we write $[t]\alpha(t)$.

1.2.1 Vibration as the Circular Movement of a Rotor

To visualize all the possible states of a vibrating system, consider a plane in which the horizontal axis gives the value of $x$ and the vertical axis gives the value of $y$--in this way each possible state of the system is a point in the plane.

First, consider the simple case where $B=A$, so Equations 1.3 and 1.4 specialize to

$$y' = Ax$$ (1.5)

$$x' = -Ay$$ (1.6)

Figure 1.3(a) shows an example point $\langle x,y\rangle$ and the corresponding point $\langle x',y'\rangle=\langle -Ay,Ax\rangle$ as vectors in the plane, when $A=3/8$. Notice that the angle between these two vectors is always a right angle.

Figure 1.3: State and change vectors for vibrating spring

Since $\langle x',y'\rangle$ represents a change in $\langle x,y\rangle$ it is useful to displace the origin of the vector representing $\langle x',y'\rangle$ to the end of the vector representing $\langle x,y\rangle$, as shown in Figure 1.3(b). Now, it is easy to see that the state of the system must trace out a circle in the plane centered about the origin, because the direction of change is always at right angles to the state vector. The size of the circle can be any nonnegative real number--setting the size corresponds to providing an initial displacement to the mass on the spring. Furthermore, since the magnitude of the state vector always stays the same, the magnitude of the change vector is also always the same ($A$ times the magnitude of the state vector), so the state moves around the circle at a constant speed. I call such a system with a point moving around a circle at a constant speed a rotor. The time required for one full rotation is the period $P$ of the rotor. The number of full rotations in a unit of time is the frequency of the rotor: its value is $1/P$. The magnitude of the state vector is the amplitude of the rotor. The angle of the state vector with respect to the $x$ axis ( $\langle 1,0\rangle$) at time $t=0$ is the phase of the rotor.

Take 5 minutes to visualize the relationship between the rotor and the vibrating spring system, as suggested in Figure 1.4. Notice that we have no interest in actual physical devices that look like rotors--the rotor is purely a mathematical concept that allows us to analyze the behavior of a vibrating system. Now forget about springs, and always visualize vibration in terms of rotors and similar mathematical systems that we investigate later.

Figure 1.4: Spring system vs. rotor

While the speed of a rotor state around its circular path is constant, the $x$ and $y$ components of the rotor state oscillate sinusoidally. Consider a rotor with amplitude $R$ (that is, the circular path has radius $R$) and frequency $F$ ($F$ full rotations per unit time), starting at time $t=0$ in state $\langle x,y\rangle=\langle R,0\rangle$. The values of $x$ and $y$ at any time are given by the trigonometric $\cos$ and $\sin$ functions.

$$x = R\cos(2\mathord{\mbox{\boldmath$\pi$}} Ft)$$ (1.7)

$$y = R\sin(2\mathord{\mbox{\boldmath$\pi$}} Ft)$$ (1.8)

The multiplication by $2\mathord{\mbox{\boldmath $\pi$}}$ is required because we measure angles in radians, and one full rotation is $2\mathord{\mbox{\boldmath $\pi$}}$ radians. Notice that the maximum (minimum) values for $x$ and $y$ are both $R$ ($-R$), and each reaches its maximum and minimum when the other is 0. Figure 1.5 shows $x$ (solid line) and $y$ (dashed line) as functions of time $t$ for a rotor with a frequency of $1/4$ rotation per unit time.

Figure 1.5: Rotor state parameters $x$ and $y$ as functions of time $t$

Figure 1.6 shows a three-dimensional plot of $x$, $y$, and $t$. The path of the state is a helix, circling about the $t$ axis. Think of the helix as the trace of a point running around the circle from Figure 1.3.

Figure 1.6: Rotor parameters $x$ and $y$ vs. $t$ as a helix in three dimensions

When $A\neq B$ in Equations 1.3 and 1.4, the state vector traces out an ellipse, whose aspect ratio is $\sqrt{A/B}$. The speed of the state vector around the ellipse is not constant (but the period and frequency are still well defined). Instead of figuring out a detailed description of an elliptical rotor, notice that we can always normalize a rotor to have circular motion, by changing the units in which $x$ and $y$ are measured. In an elliptical rotor with frequency $F$, starting at time $t=0$ in state $\langle x,y\rangle=\langle R_x,0\rangle$ and crossing the $y$ axis in state $\langle x,y\rangle=\langle 0,R_y\rangle$, the values of $x$ and $y$ at any time are still given by the $\cos$ and $\sin$ functions, but with different scaling factors for each.

$$x = R_x\cos(2\mathord{\mbox{\boldmath$\pi$}} Ft)$$ (1.9)

$$y = R_y\sin(2\mathord{\mbox{\boldmath$\pi$}} Ft)$$ (1.10)

In this case, the maximum (minimum) value for $x$ is $R_x$ ($-R_x$), and for $y$ it is $R_y$ ($-R_y$). As before, each parameter reaches its maximum and minimum when the other is 0.

1.2.2 Rotor State as a Complex Number

It is mathematically convenient to think of the two-dimensional rotor state vector $\langle x,y\rangle$ as a single complex number $x+\mathord{\mbox{\boldmath $i$}} y$, where $\mathord{\mbox{\boldmath $i$}}$ is the ``imaginary'' number defined to be the principal square root of $-1$ (if you read engineering books and articles, you may see this number written as $\mathbf{j}$ instead of $\mathord{\mbox{\boldmath $i$}}$). Do not look for deep significance in the names ``real number,'' ``imaginary number,'' ``complex number.'' These names are just tags made up by mathematicians--``real'' numbers are no more real than other numbers, ``imaginary'' numbers are no more imaginary, and ``complex'' numbers are used to simplify a lot of the analysis that we need to do. For our purposes, the complex number $x+\mathord{\mbox{\boldmath $i$}} y$ is just a particular notation for the vector $\langle x,y\rangle$, which is particularly convenient because the familiar operations of addition, multiplication, and exponentiation on the real numbers extend very naturally to operations on complex numbers that are just right for analyzing vibration.

1.2.2.1 Review of Complex Arithmetic

From now on, we use Greek letters $\alpha$, $\beta$, $\gamma$, etc. as variables ranging over complex number functions depending on the time variable $t$. Complex number constants independent of time are denoted by bold face Greek letters $\mathord{\mbox{\boldmath $\alpha$}}$, $\mathord{\mbox{\boldmath $\beta$}}$, $\mathord{\mbox{\boldmath $\gamma$}}$, etc. It is important to be fluent in the following facts about complex numbers, and to be able to do complex arithmetic and algebra just as easily as you learned to do real arithmetic and algebra in calculus class. Make sure that you visualize each of the facts below in terms of vectors in the plane.

1.2.2.1.1 Cartesian form of complex numbers.

$$x_1+\mathord{\mbox{\boldmath$i$}}y_1=x_2+\mathord{\mbox{\boldmath$i$}}y_2\text{ if and only if }x_1=x_2\text{ and }y_1=y_2$$ (1.11)

Addition and multiplication extend to complex numbers by using the commutative, associative, and distributive laws, and the fact that $\mathord{\mbox{\boldmath $i$}}\mathord{\mbox{\boldmath $i$}}=\mathord{\mbox{\boldmath $i$}}^2=-1$. Addition of complex numbers may be visualized in terms of the vectors represented by the two numbers: shift the origin of one vector to the head of the other vector as shown in Figure 1.7.

Figure 1.7: Adding two complex numbers

The conjugate of a complex number, written $\overline{\alpha}$, is the reflection of $\alpha$ through the real axis, as shown in Figure 1.8.

Figure 1.8: The conjugate of a complex number

$$(x_1+\mathord{\mbox{\boldmath$i$}}y_1) + (x_2+\mathord{\mbox{\boldmath$i$}}y_2) = (x_1+x_2) +\mathord{\mbox{\boldmath$i$}}(y_1+y_2)$$ (1.12)

$$(x_1+\mathord{\mbox{\boldmath$i$}}y_1)(x_2+\mathord{\mbox{\boldmath$i$}}y_2) = (x_1x_2-y_1y_2)+\mathord{\mbox{\boldmath$i$}}(x_1y_2+x_2y_1)$$ (1.13)

$$\overline{x_1+\mathord{\mbox{\boldmath$i$}}y_1} = x_1-\mathord{\mbox{\boldmath$i$}}y_1$$ (1.14)

$$\alpha+\beta = \beta+\alpha$$ (1.15)

$$(\alpha+\beta)+\gamma = \alpha+(\beta+\gamma)$$ (1.16)

$$0+\alpha = \alpha$$ (1.17)

$$\alpha\beta = \beta\alpha$$ (1.18)

$$(\alpha\beta)\gamma = \alpha(\beta\gamma)$$ (1.19)

$$1\alpha = \alpha$$ (1.20)

$$0\alpha =$$ 0 (1.21)

$$\mathord{\mbox{\boldmath$i$}}\mathord{\mbox{\boldmath$i$}}=-1$$ (1.22)

$$\alpha(\beta+\gamma) = \alpha\beta+\alpha\gamma$$ (1.23)

The real and imaginary parts of a complex number are defined to select out the two components of the vector.

$$\Re(x+\mathord{\mbox{\boldmath$i$}}y) = x$$ (1.24)

$$\Im(x+\mathord{\mbox{\boldmath$i$}}y) = y$$ (1.25)

$$\Re(\alpha+\beta) = \Re(\alpha)+\Re(\beta)$$ (1.26)

$$\Im(\alpha+\beta) = \Im(\alpha)+\Im(\beta)$$ (1.27)

$$\Re(\alpha\beta) = \Re(\alpha)\Re(\beta)-\Im(\alpha)\Im(\beta)$$ (1.28)

$$\Im(\alpha\beta) = \Re(\alpha)\Im(\beta)+\Im(\alpha)\Re(\beta)$$ (1.29)

$$\alpha = \Re(\alpha)+\mathord{\mbox{\boldmath$i$}}\Im(\alpha)$$ (1.30)

$$\alpha=\beta \Re(\alpha)=\Re(\beta) \Im(\alpha)=\Im(\beta)$$ (1.31)

$\Re(\alpha)$ and $\Im(\alpha)$ are called the Cartesian coordinates of the complex number $\alpha$.

1.2.2.1.2 Polar form of complex numbers.

The reason why complex numbers are particularly convenient for analyzing vibration is that they may be manipulated according to the magnitude (length of the vector) and argument (angle of the vector with respect to $1$) as well. A magnitude is just a real number $\geq 0$, representing the length of a vector. Angles are a bit trickier.

1.2.2.1.2.1 Rotational and directional angles.

There are really two connected but different concepts that are both called ``angles.'' First, there are rotational angles that measure an amount of rotation. A rotational angle may be any real number--positive numbers represent counterclockwise rotation, and negative numbers represent clockwise rotation. A rotational angle of $2\mathord{\mbox{\boldmath $\pi$}}$ represents a full rotation counterclockwise. Even though the direction that an object points after a full rotation is the same as before the rotation, $2\mathord{\mbox{\boldmath $\pi$}}$ represents a different rotation than 0 or $-2\mathord{\mbox{\boldmath $\pi$}}$ or $4\mathord{\mbox{\boldmath $\pi$}}$--suppose for example that we are measuring rotation of a wheel that winds up a spring.

The other sorts of angles are directional angles, which measure the direction that a vector is pointing with reference to some conventional 0 direction (for complex numbers, 0 is the directional angle of the vector represented by $1$). Directional angles must be in the half-open interval $[0,2\mathord{\mbox{\boldmath $\pi$}})$. Many books and articles prefer to describe directional angles in the interval $(-\mathord{\mbox{\boldmath $\pi$}},\mathord{\mbox{\boldmath $\pi$}}]$ (so, for example, the angle $3\mathord{\mbox{\boldmath $\pi$}}/2$ in our notation becomes $-\mathord{\mbox{\boldmath $\pi$}}/2$). It makes no essential difference which interval is used, since all arithmetic on directional angles is done on a circle of circumference $2\mathord{\mbox{\boldmath $\pi$}}$, rather than the usual real line. We may convert rotational angles to directional angles with the function $\bmod 2\mathord{\mbox{\boldmath $\pi$}}$.

$$x\bmod 2\mathord{\mbox{\boldmath$\pi$}} = x-2\mathord{\mbox{\boldmath$\pi$}}\lfloor x/(2\mathord{\mbox{\boldmath$\pi$}})\rfloor$$ (1.32)

$$0\leq x\bmod 2\mathord{\mbox{\boldmath$\pi$}}<2\mathord{\mbox{\boldmath$\pi$}}$$ (1.33)

When $x\bmod z=y\bmod z$ we often write $x=y\pmod{z}$ instead. This form suggests an alternate view of modular arithmetic: $x=y\pmod{z}$ means that $x$ and $y$ are two names for the same thing in the $\pmod{z}$ universe, even though they may be different numbers in the usual real number universe. If we apply a rotational angle $x$ to rotate a vector from a starting position with directional angle 0, we get a new vector with directional angle $x\bmod 2\mathord{\mbox{\boldmath $\pi$}}$. Notice that all rotational angles $x+2k\mathord{\mbox{\boldmath $\pi$}}$ for integers $k$ correspond to the same directional angle. Given a directional angle $x$ resulting from a rotation, there is no way to tell which of the infinitely many possible rotational angles generated $x$. To avoid becoming confused by the ambiguity in the word ``angle,'' visualize each angle as either an amount of rotation or a static direction, instead of a pure abstract real number.

The angle of a complex number is a directional angle, so it is restricted to the interval $[0,2\mathord{\mbox{\boldmath $\pi$}})$. We denote the length (also called magnitude) of a complex number $\alpha$ by $\vert\alpha\vert$, and its angle (also called argument) by $\arg(\alpha)$.

$$\vert x+\mathord{\mbox{\boldmath$i$}}y\vert = \sqrt{x^2+y^2}$$ (1.34)

$$\arg(x+\mathord{\mbox{\boldmath$i$}}y) = \left\{\begin{array}{ll} \arctan(y/x) \bmod 2\mathord{\mbox{\bold... ...rd{\mbox{\boldmath$\pi$}} & \text{if }x=0\text{ and }y\neq 0 \end{array}\right.$$ (1.35)

$$\vert\alpha\vert \geq$$ 0 (1.36)

$$\arg(\alpha) \geq$$ 0 (1.37)

$$\arg(\alpha) < 2\mathord{\mbox{\boldmath$\pi$}}$$ (1.38)

$$\alpha=\beta \left\{ \begin{array}{lcl} \multicolumn{2}{l}{\vert\alpha\vert=\v... ...(\alpha)=\arg(\beta)\pmod{2\mathord{\mbox{\boldmath$\pi$}}} \end{array} \right.$$ (1.39)

$\arg(0)$ is undefined, since it makes no sense to take the angle of a vector with magnitude 0. $\arctan(x)$ means an angle $\theta$ such that $\tan(\theta)=x$. Figure 1.9 shows the relation between $\Re(\alpha)$, $\Im(\alpha)$, $\vert\alpha\vert$, and $\arg(\alpha)$ when $\alpha$ is drawn as a vector in a two-dimensional space. Even though the angle of a complex number is a directional angle, rotational angles arise naturally in complex analysis. The derivative of the angle of a complex number is a rotational angle. The imaginary component of the logarithm of a complex number represents a rotational angle, and the rules for logarithms are complicated by the problem of choosing the appropriate rotational version of a given directional angle.

Figure 1.9: Complex number: Cartesian and polar coordinates

$\vert\alpha\vert$ and $\arg(\alpha)$ are called the polar coordinates of the complex number $\alpha$. Addition of complex numbers is easiest to do by manipulating the real and imaginary parts, but multiplication and division may be defined very nicely on the magnitude and angle.

$$\vert\alpha\beta\vert = \vert\alpha\vert\vert\beta\vert$$ (1.40)

$$\arg(\alpha\beta) = \arg(\alpha)+\arg(\beta)\bmod 2\mathord{\mbox{\boldmath$\pi$}}$$ (1.41)

$$\vert\alpha/\beta\vert = \vert\alpha\vert/\vert\beta\vert$$ (1.42)

$$\arg(\alpha/\beta) = \arg(\alpha)-\arg(\beta)\bmod 2\mathord{\mbox{\boldmath$\pi$}}$$ (1.43)

$$\vert-\alpha\vert = \vert\alpha\vert$$ (1.44)

$$\arg(-\alpha) = \arg(\alpha)+\mathord{\mbox{\boldmath$\pi$}}\bmod 2\mathord{\mbox{\boldmath$\pi$}}$$ (1.45)

$$\vert x\alpha\vert=\vert-x\alpha\vert = x\vert\alpha\vert x\geq 0$$ (1.46)

$$\arg(x\alpha) = \arg(\alpha) x>0$$ (1.47)

$$\vert\overline{\alpha}\vert = \vert\alpha\vert$$ (1.48)

$$\arg(\overline{\alpha}) = -\arg(\alpha)\bmod 2\mathord{\mbox{\boldmath$\pi$}}$$ (1.49)

$$\alpha = \vert\alpha\vert(\cos(\arg(\alpha))+\mathord{\mbox{\boldmath$i$}}\sin(\arg(\alpha)))$$ (1.50)

Using Equations 1.40 and 1.41, we see that multiplication by a complex number $\alpha$ of magnitude $1$ acts as a rotator: the multiplication $\alpha\beta$ rotates $\beta$ by the angle $\arg(\alpha)$. In particular, multiplication by $\mathord{\mbox{\boldmath $i$}}$ rotates a vector counterclockwise by $\mathord{\mbox{\boldmath $\pi$}}/2$ (right angle). So, letting the single complex number $\rho=x+\mathord{\mbox{\boldmath $i$}}y$ represent the rotor state $\langle x,y\rangle$, we may express Equations 1.5 and 1.6 as a single equation.

$$\rho' = \mathord{\mbox{\boldmath$i$}}A\rho$$ (1.51)

The elliptical rotor system of Equations 1.3 and 1.4 may also be expressed as a single equation using the conjugate operation.

$$\rho' = \mathord{\mbox{\boldmath$i$}}((A+B)\rho+(A-B)\overline{\rho})/2$$ (1.52)

1.2.2.1.3 Complex number to real power.

Equation 1.41 is also the key to understanding exponentiation of complex numbers. Notice that the angle behaves logarithmically with respect to addition and multiplication (think of the analogous equation $\ln(xy)=\ln(x)+\ln(y)$). Since exponentiation is essentially iterated multiplication, and complex multiplication is additive on angles, complex exponentiation has a multiplicative effect on angles. Consider first a complex number $\alpha$ raised to the power of a real number $x$.

$$\vert\alpha^x\vert = \vert\alpha\vert^x$$ (1.53)

$$\arg(\alpha^x) = x\arg(\alpha)$$ (1.54)

$$\alpha^{x+y} = \alpha^x\alpha^y$$ (1.55)

$$\alpha^{xy} = {(\alpha^x)}^y$$ (1.56)

$${(\alpha\beta)}^x = \alpha^x\beta^x$$ (1.57)

$$\overline{\alpha}^x = \overline{\alpha^x}$$ (1.58)

$$\alpha^1 = \alpha$$ (1.59)

$$\alpha^0 = 1$$ (1.60)

Equation 1.54 begins to reveal the power of complex numbers for analyzing vibration. Think of a rotor with amplitude $R$, frequency $F$, and phase 0 (starting value $R+\mathord{\mbox{\boldmath $i$}}0$ at time $t=0$). The state of the rotor at any time $t$ is $\rho=R\mathord{\mbox{\boldmath $i$}}^{4Ft}$ (we multiply $F$ by $4$ because $\arg(\mathord{\mbox{\boldmath $i$}})=\mathord{\mbox{\boldmath $\pi$}}/2$ is $1/4$ of $2\mathord{\mbox{\boldmath $\pi$}}$ radians, which is a full rotation). For an elliptical rotor with starting value $R_1$ that crosses the imaginary axis at $\mathord{\mbox{\boldmath $i$}}R_2$, the state at time $t$ is $(R_1+R_2)\mathord{\mbox{\boldmath $i$}}^{4Ft}/2+(R_1-R_2)\overline{\mathord{\mbox{\boldmath $i$}}^{4Ft}}/2$. Equivalent expressions include $(R_1+R_2)\mathord{\mbox{\boldmath $i$}}^{4Ft}/2+(R_1-R_2)(-\mathord{\mbox{\boldmath $i$}})^{4Ft}/2$ and $(R_1+R_2)\mathord{\mbox{\boldmath $i$}}^{4Ft}/2+(R_1-R_2)\mathord{\mbox{\boldmath $i$}}^{-4Ft}/2$. The last of these is the most popular, and leads to the notion of a $-F$ frequency component in an elliptical rotor.

1.2.2.1.4 Complex exponents.

The most important fact about complex numbers for the study of vibration is the rule for raising the real number $e=2.71828\ldots$ to a complex power.

$$e^{\mathord{\mbox{\boldmath\scriptsize$i$}}y} = \cos(y)+\mathord{\mbox{\boldmath$i$}}\sin(y)$$ (1.61)

$$e^{x+\mathord{\mbox{\boldmath\scriptsize$i$}}y} = e^x(\cos(y)+\mathord{\mbox{\boldmath$i$}}\sin(y))$$ (1.62)

$$\vert e^{\alpha}\vert = e^{\Re(\alpha)}$$ (1.63)

$$\arg(e^{\alpha}) = \Im(\alpha)\bmod 2\mathord{\mbox{\boldmath$\pi$}}$$ (1.64)

$$e^{\overline{\alpha}} = \overline{e^\alpha}$$ (1.65)

$$\alpha = \vert\alpha\vert e^{\mathord{\mbox{\boldmath\scriptsize$i$}}\arg(\alpha)}$$ (1.66)

$$\alpha = e^{\ln(\vert\alpha\vert)+\mathord{\mbox{\boldmath\scriptsize$i$}}\arg(\alpha)}\text{ for }\alpha\neq 0$$ (1.67)

Equation 1.61, known as Euler's formula in honor of the famous mathematician who discovered it, is the most important single equation for the study of vibration. It allows us to reason about trigonometric functions by using the relatively easy-to-remember properties of exponentiation. Computer algebra systems typically convert trigonometric formulae into exponential form in order to simplify them more efficiently. The conventional real-valued trigonometric functions $\sin$ and $\cos$ are defined in terms of complex exponentials:

$$\sin(x) = \Im(e^{\mathord{\mbox{\boldmath\scriptsize$i$}}x})=\frac{e^{\math... ...e^{-\mathord{\mbox{\boldmath\scriptsize$i$}}x}}{2\mathord{\mbox{\boldmath$i$}}}$$ (1.68)

$$\cos(x) = \Re(e^{\mathord{\mbox{\boldmath\scriptsize$i$}}x})=\frac{e^{\math... ...{\boldmath\scriptsize$i$}}x}+e^{-\mathord{\mbox{\boldmath\scriptsize$i$}}x}}{2}$$ (1.69)

The rightmost forms of these definitions generalize nicely to define trigonometric functions of complex inputs, and also the hyperbolic functions $\sinh$ and $\cosh$.

For our purposes, the derivation of Euler's formula is not as important as the formula itself. To see why the formula is sensible, consider the ordinary differential equation defining the exponential function for real numbers:

$$x' = Ax$$ (1.70)

The most interesting solution to equation 1.70 is the one with initial condition $x(0)=1$, and this leads to

$$x(t) = e^{At}$$ (1.71)

That is, the (scaled) exponential function $e^{At}$ is characterized by its initial value and the fact that its slope at each time $t$ is $A$ times its value at time $t$--the larger it gets, the faster it grows. Notice that equation 1.51 has the same form as equation 1.70, but it describes a complex-valued function, and the multiplier is an imaginary number $\mathord{\mbox{\boldmath $i$}}A$ rather than a real number $A$. So, it is sensible to regard the natural solution to equation 1.51, which is a rotor, as the function $e^{\mathord{\mbox{\boldmath $i$}}At}$.

Euler's formula also gives us another way to represent each complex number $\alpha$--instead of the usual form $\Re(\alpha)+\mathord{\mbox{\boldmath $i$}}\Im(\alpha)$ we may write $\vert\alpha\vert e^{\mathord{\mbox{\boldmath\scriptsize $i$}}\arg(\alpha)}$. For $\alpha\neq 0$ we may also write $e^{\ln(\vert\alpha\vert)+\mathord{\mbox{\boldmath\scriptsize $i$}}\arg(\alpha)}$. Unlike the additive Cartesian form $x+\mathord{\mbox{\boldmath $i$}} y$, the exponential polar $re^{\mathord{\mbox{\boldmath\scriptsize $i$}}w}$ form for a complex number is not unique, since $re^{\mathord{\mbox{\boldmath\scriptsize $i$}}w}=re^{\mathord{\mbox{\boldmath\scriptsize $i$}}(w+2k\mathord{\mbox{\boldmath\scriptsize $\pi$}})}$ and $0e^{\mathord{\mbox{\boldmath\scriptsize $i$}}w_1}=0e^{\mathord{\mbox{\boldmath\scriptsize $i$}}w_2}$.

$$r_1e^{\mathord{\mbox{\boldmath\scriptsize$i$}}w_1}=r_2e^{\mathord... ...ox{\boldmath$\pi$}})\pmod{2\mathord{\mbox{\boldmath$\pi$}}} \end{array} \right.$$ (1.72)

We may convert polar to Cartesian form using Euler's formula (Equation 1.61):

$$\Re(re^{\mathord{\mbox{\boldmath\scriptsize$i$}}w}) = r\cos(w)$$ (1.73)

$$\Im(re^{\mathord{\mbox{\boldmath\scriptsize$i$}}w}) = r\sin(w)$$ (1.74)

$$re^{\mathord{\mbox{\boldmath\scriptsize$i$}}w} = r\cos(w)+\mathord{\mbox{\boldmath$i$}}r\sin(w)$$ (1.75)

In essence, exponentiation is a kind of conversion between Cartesian and polar coordinates: the polar coordinates of $e^\alpha$ are $\vert e^{\alpha}\vert=e^{\Re(\alpha)}$ and $\arg(e^{\alpha})=(\Im(\alpha)\bmod 2\mathord{\mbox{\boldmath $\pi$}})$. So, the Cartesian coordinates of $\alpha$ turn into the polar coordinates of $e^{\alpha}$. Notice that $\Im(\alpha)$ is naturally understood as a rotational angle, while $\arg(e^{\alpha})$ is a directional angle.

Euler's formula (Equation 1.61) allows an even nicer way to analyze a rotor with amplitude $R$, frequency $F$, and phase 0 (starting value $R+\mathord{\mbox{\boldmath $i$}}0=Re^0$ at time $t=0$): the state at any time $t$ is just $\rho=Re^{\mathord{\mbox{\boldmath\scriptsize $i$}}2\mathord{\mbox{\boldmath\scriptsize $\pi$}} Ft}$ (such exponential expressions are sometimes called phasors in the engineering literature). (For the elliptical rotor starting at $R_1$ and crossing the imaginary axis at $\mathord{\mbox{\boldmath $i$}}R_2$, the state at time $t$ is $((R_1+R_2)e^{\mathord{\mbox{\boldmath\scriptsize $i$}}2\mathord{\mbox{\boldmat... ...{\boldmath\scriptsize $i$}}2\mathord{\mbox{\boldmath\scriptsize $\pi$}} Ft}})/2$, or $((R_1+R_2)e^{\mathord{\mbox{\boldmath\scriptsize $i$}}2\mathord{\mbox{\boldmat... ...x{\boldmath\scriptsize $i$}}2\mathord{\mbox{\boldmath\scriptsize $\pi$}} Ft})/2$.) And, it is particularly easy to construct a complex number with magnitude $1$ to rotate other numbers by a given angle $w$: use $e^{\mathord{\mbox{\boldmath\scriptsize $i$}}w}$. Look back at Figures 1.3 and 1.6 again, and interpret them in terms of complex numbers.

Now, the way to understand exponentiation $\beta^{\alpha}$ with an arbitrary complex base $\beta$ is to first write $\beta=\vert\beta\vert e^{\mathord{\mbox{\boldmath\scriptsize $i$}}\arg(\beta)}$, and then use the rules for exponentiation with base $e$.

$$(re^{\mathord{\mbox{\boldmath\scriptsize$i$}}w})^{x+\mathord{\mbox{\boldmath\scriptsize$i$}}y} = r^xe^{-wy}e^{\mathord{\mbox{\boldmath\scriptsize$i$}}(wx+y\ln(r))}$$ (1.76)

$$\vert\beta^{\alpha}\vert = \vert\beta\vert^{\Re(\alpha)}e^{-\arg(\beta)\Im(\alpha)}$$ (1.77)

$$\arg(\beta^{\alpha}) = \arg(\beta)\Re(\alpha)+\ln(\vert\beta\vert)\Im(\alpha)$$ (1.78)

These equations are rather complicated, and fortunately we will not be using them. Work them through for exercise with complex numbers, and convince yourself that they follow from the earlier rules. Notice how exponentiation mixes together the Cartesian coordinates of the exponent with the polar coordinates of the base.

1.2.2.1.5 Complex logarithms.

Euler's formula makes it easy to define the natural (base $e$) logarithm of a complex number.

$$\ln(\alpha) = \ln\mathopen{\vert}\alpha\mathclose{\vert}+\mathord{\mbox{\boldmath$i$}}\arg(\alpha)\text{ if }\alpha\neq0$$ (1.79)

$$e^{\ln(\alpha)} = \alpha \alpha\neq0$$ (1.80)

$$\Re(\ln(\alpha)) = \ln(\mathopen{\vert}\alpha\mathclose{\vert})$$ (1.81)

$$\Im(\ln(\alpha)) = \arg(\alpha)$$ (1.82)

$\ln(0)$ is undefined, since there is no power $\alpha$ with $e^\alpha=0$. Notice that, for positive real numbers $x>0$ there is a unique real number $y$ such that $e^y=x$. But, even for positive real numbers $x$ there are infinitely many complex numbers $\beta$ such that $e^{\beta}=x$. That is, while Equation 1.80 defines the natural logarithm uniquely as a real value, it has infinitely many complex solutions, since $e^{\alpha}=e^{\alpha+\mathord{\mbox{\boldmath\scriptsize $i$}}2k\mathord{\mbox{\boldmath\scriptsize $\pi$}}}$ for all integers $k$. The particular choice above for the imaginary part of $\ln(\alpha)$ is arbitrary, just as the particular interval $[0,2\mathord{\mbox{\boldmath $\pi$}})$ for directional angles is arbitrary. Notice that this choice restricts all complex logarithms $\ln(\alpha)$ to the horizontal stripe in the complex plane where $0\leq\Im(\ln(\alpha))<2\mathord{\mbox{\boldmath $\pi$}}$.

$$\log_{\beta}(\alpha) = \ln(\alpha)/\ln(\beta) \alpha,\beta\neq 0,\beta\neq 1$$ (1.83)

In many cases, it is useful to think of complex logarithms as inhabiting a cylindrical universe $\bmod\mathord{\mbox{\boldmath $i$}}2\mathord{\mbox{\boldmath $\pi$}}$ in which the real axis is a line, and the imaginary axis is a circle. Visualize this cylinder in your mind, and think about how it represents polar co-ordinates for all of the complex numbers except 0.

1.2.2.1.6 Trick for dividing complex numbers in Cartesian form.

Dividing two complex numbers in Cartesian form appears to require three steps:

1. convert to polar form by Equations 1.34 and 1.35;
2. divide the polar forms by Equations 1.42 and 1.43;
3. convert back to Cartesian form by Equations 1.73 and 1.74.

These steps are computationally rather expensive compared to addition, subtraction, and multiplication of complex numbers in Cartesian form--the $\arctan$ operation in Equation 1.35 is probably the worst, but even the square root in Equation 1.34 and the $\sin$ and $\cos$ in Equations 1.73 and 1.74 require nontrivial programs. Fortunately, there's a simple trick to speed up complex division, since multiplication of a complex number with its conjugate always yields a real number:

$$\alpha/\beta = \alpha\overline{\beta}/(\beta\overline{\beta}) = \alpha\overline{\beta}/\mathopen{\vert}\beta\mathclose{\vert}^2$$ (1.84)

$$\frac{x_1+\mathord{\mbox{\boldmath$i$}}y_1}{x_2+\mathord{\mbox{\boldmath$i$}}y_2} = \frac{x_1x_2+y_1y_2}{x_2^2+y_2^2}+\mathord{\mbox{\boldmath$i$}}\frac{-x_1y_2+x_2y_1}{x_2^2+y_2^2}$$ (1.85)

Equation 1.84 shows how every complex division can be done by a conjugation, two complex multiplications, and division of a complex by a real. Equation 1.85 works out the details for complex numbers in Cartesian form, and avoids computing the imaginary part of the denominator times its conjugate, since that is known in advance to be 0.

1.2.3 Sound Signals in the Time Domain

In general, the sounds that we would like to create and analyze are much more complicated than the sounds produced by simple rotors. But, we will continue to model sounds by complex-valued functions $\sigma$ depending on a real number parameter $t$ standing for time. Such functions are called sound signals in the time domain. Later, in Chapters 4 and 6, we will see other mathematical representations of sound, but signals in the time domain are the easiest models to relate intuitively to the physical signals that enter the ear. Widely used digital input and output devices for sound are also most easily understood in terms of signals in the time domain. Most books and papers on sound consider real-valued time signals, and most electronic devices, both digital and analog, for analyzing or creating sound deal only with real values. Many analysis and synthesis techniques, however, are best understood in terms of a complex signal $\sigma$. We may always project the complex signal $\sigma$ to a real signal by taking $\Re(\sigma)$. Just as many systems for manipulating graphic images deal with three dimensional models, and project them to two dimensions at the last stage before displaying them on video screens, we will think of sound signals as two dimensional, and project to one dimension at the last stage before rendering them through loudspeakers.

Given a sound signal $\sigma$ in the time domain, and a particular time $t$, the instantaneous amplitude of $\sigma$ at time $t$ is $\mathopen{\vert}\sigma(t)\mathclose{\vert}$, the instantaneous phase is $\arg(\sigma(t))$, and the instantaneous frequency is $(d\arg(\sigma)/dt)(t)$. These are interesting quantities to discuss, and may be useful in analyzing sound, but they do not necessarily have the perceptual impact of the corresponding constant quantities associated with a simple rotor.

Not every complex-valued function $\sigma$ of $t$ makes sense as a sound signal in the time domain. Some reasonable relation must hold between the real and complex components of $\sigma$. But, it is not clear precisely what relation to require in general. Particular physical interpretations of $\sigma$ impose certain constraints--for example if $\Re(\sigma)$ is the velocity of a physical object, and $\Im(\sigma)$ is the displacement of the same object, then $\Re(\sigma)=d\Im(\sigma)/dt$. When $\Im(\sigma)=R\sin(Ft)$, $d\Im(\sigma)/dt=FR\cos(Ft)$, so this derivative constraint forces rotors to be elliptical, with aspect ratio proportional to the frequency. Circular rotors are much more convenient mathematically. Roughly speaking, we would like to restrict sound signals $\sigma$ so that $\Im(\sigma)$ is essentially the same as $\Re(\sigma)$ with a phase difference of $\mathord{\mbox{\boldmath $\pi$}}$ ( $90^{\circ}$)--such signals are said to be in quadrature, since the angle $\mathord{\mbox{\boldmath $\pi$}}$ is one quarter of the full circle. The problem is that many different frequencies may be present in $\sigma$. In Chapter 4 we see a precise definition of this quadrature constraint, using the Hilbert transform.

Figures 1.10, 1.11, and 1.12 show examples of sound signals in the time domain that are slightly more complicated than the basic helix. In each case, part (a) shows a three-dimensional plot of the complex-valued function, and part (b) shows a two-dimensional plot of the real and imaginary components.

Figure: The signal $e^{-t/8+\mathord{\mbox{\boldmath\scriptsize $i$}}2\mathord{\mbox{\boldmath\scriptsize $\pi$}} t/4}$: decaying helix

Figure: The signal $e^{\mathord{\mbox{\boldmath\scriptsize $i$}}2^{t/8}2\mathord{\mbox{\boldmath\scriptsize $\pi$}} t/4}$: increasing frequency

Figure: The signal $e^{\mathord{\mbox{\boldmath\scriptsize $i$}}2\mathord{\mbox{\boldmath\scriptsi... ...{\boldmath\scriptsize $i$}}16\mathord{\mbox{\boldmath\scriptsize $\pi$}} t/4}/8$: sum of 4 helixes

1.3 Exercises

1. Take the spring system of Figure 1.1, rotate the track to a vertical orientation, and let a constant gravitational force act on the mass. The stable position about which the mass oscillates is no longer at the point where the spring attaches to the track, but some distance below that point where the force exerted by the spring exactly cancels gravity. Does the frequency of the vibrating spring increase or decrease as a result of the influence of gravity? Explain briefly.
2. Consider a vibrating spring system in which the motion of the mass is opposed by a certain amount of friction. In order to analyze such a system, do we change Equation 1.1, Equation 1.2, both, or neither? Explain briefly.
3. Consider a vibrating system with a mass that is attracted to the center of vibration by a gravitational force instead of a spring. In such a system, the period of vibration depends on the amplitude. As the amplitude of the vibration increases, does the period increase or decrease? Explain briefly.
4. Notice how Equation 1.5 has a positive multiplier $A$, while Equation 1.6 has a negative multiplier $-A$. There are three other possibilities: (a) both multipliers negative, (b) both multipliers positive, (c) the first multiplier negative and the second positive. Describe briefly and qualitatively the behavior of a system described by each of the variants (a-c). Draw pictures analogous to Figure 1.3(b) to help explain.
5. When $A>B$ in Equations 1.3 and 1.4, the path of the state vector $\langle x,y\rangle$ is an ellipse. Which axis of the ellipse is longer, the $x$ axis or the $y$ axis? Explain briefly, using precise mathematical information derived from Equations 1.3 and 1.4. Hint: Derive slightly different equations relating $C_yy'$ to $C_xx$ and $C_xx'$ to $C_yy$ for cleverly chosen constant multipliers $C_x$ and $C_y$.
6. Derive simple formulae representing the frequency and period of the vibrating system of Equations 1.3 and 1.4 in terms of the constants $A$ and $B$. Hint: Look at Equations 1.7 and 1.8. Differentiate both sides of both equations. Solve the special case where $A=B$. Then, apply the scaling of $x$ and $y$ by constants $C_x$ and $C_y$ that you used in Exercise 5.
7. In an elliptical rotor system obeying Equations 1.3 and 1.4 the speed with which the state point travels around the ellipse is not constant.
   1. Where is this speed the least, and where is it greatest? Explain briefly.
   2. Answer the same question for the angular speed of the state vector--the speed at which its angle with the $x$ axis changes.
8. For each of the following operations on complex numbers $\alpha$ and $\beta$, state whether it is more convenient to represent each number in Cartesian or polar coordinates, or whether both are equally convenient. Sometimes the answer is different for $\beta$ than for $\alpha$.
   1. $\alpha+\beta$
   2. $\alpha-\beta$
   3. $\overline{\alpha}$
   4. $\alpha\beta$
   5. $\alpha/\beta$
   6. $\beta^{\alpha}$
   7. $\log_{\beta}(\alpha)$
9. Derive formulae for the Cartesian coordinates $\Re(\beta^\alpha)$ and $\Im(\beta^\alpha)$ of $\beta^\alpha$ in terms of Cartesian and/or polar coordinates of $\alpha$ and $\beta$.
10. Derive the following trigonometric identities, using Euler's formula (Equation 1.61) and easy algebraic manipulations of additions, subtractions, multiplications, and divisions of complex numbers. Note that $\cos^2(x)$ and $\sin^2(x)$ are conventional ways of writing $(\cos(x))^2$ and $(\sin(x))^2$, respectively.
    $$\begin{array}{lrcl} \text{(a)} & \cos(2x) & = & \cos^2(x)-\s... ...)} & \sin(x+y) & = & \sin(x)\cos(y)+\cos(x)\sin(y) \end{array}$$

11. We saw how to express the state of an elliptical rotor at time $t$ in the form $ae^{\mathord{\mbox{\boldmath\scriptsize $i$}}2\mathord{\mbox{\boldmath\scripts... ...box{\boldmath\scriptsize $i$}}2\mathord{\mbox{\boldmath\scriptsize $\pi$}} ft}}$, where $a$ is the average of the real and imaginary intercepts of the ellipse, and $b$ is half their difference. Derive a nice formula for the state of an elliptical rotor whose major and minor axes are different from the real and imaginary axes.