Wave Mechanics: Harmonic Motion, Euler's Formula, and Complex Exponentials

Wave Mechanics: Harmonic Motion, Euler’s Formula, and Complex Exponentials

WM01 introduced oscillation at one point without assuming a particular mathematical shape. WM02 then introduced the sinusoid, angular frequency, and phase. This supplemental derivation asks a deeper question:

(1)

The answer links mechanics, differential equations, geometry, and wave analysis. A linear restoring force produces a second-order differential equation whose real solutions are sine and cosine. The same equation also admits complex exponential solutions. Euler’s formula then shows that these are not different physical models. They are different mathematical representations of the same harmonic motion [1, 2, 3, 4].

This connection is central to later wave mechanics, circuits, RF engineering, antennas, communications, and GNSS because differentiation, phase shifts, propagation, and superposition become much easier to manipulate in complex-exponential form.

1 Learning objectives

After working through this entry, the reader should be able to:

1. derive the simple harmonic oscillator equation from a linear restoring force;
2. verify why sine and cosine solve that equation;
3. convert between the forms C cos ωt + D sin ωt and A cos(ωt + ϕ);
4. derive Euler’s formula from the power series of the exponential;
5. derive sine and cosine from the two exponentials e^i𝜃 and e^−i𝜃;
6. interpret e^i𝜃 geometrically as rotation on the complex plane;
7. explain why differentiation of a complex sinusoid becomes multiplication by iω;
8. connect one-point harmonic motion to the traveling-wave phase kx − ωt + ϕ₀.

2 From restoring force to the harmonic differential equation

The simplest physical route to sinusoidal motion is a mass attached to an ideal spring. Let u(t) denote displacement from equilibrium. For sufficiently small deformation, Hooke’s law gives

(2)

where κ > 0 is the spring stiffness. The minus sign is the essential physical feature: when u > 0 the force points in the negative direction, and when u < 0 the force points in the positive direction. The force therefore tends to restore the mass toward u = 0.

Newton’s second law gives

(3)

Using Equation (2),

(4)

or

(5)

Define the natural angular frequency by

(6)

Then

(7)

This is the ideal simple harmonic oscillator equation.

Equation (7) contains the key mathematical requirement:

(8)

The second derivative must reproduce the original function with a negative scale factor.

3 Why sine and cosine solve the equation

Differentiate a cosine twice:

cos(ω0t)	= −ω0 sin(ω0t),	(9)
cos(ω0t)	= −ω02 cos(ω 0t).	(10)

Therefore

(11)

Likewise,

sin(ω0t)	= ω0 cos(ω0t),	(12)
sin(ω0t)	= −ω02 sin(ω 0t).	(13)

Thus sine also satisfies Equation (7).

Because the differential equation is linear, any linear combination of two solutions is also a solution. Hence

(14)

The two constants are not accidental. Equation (7) is second order, so two independent initial data are needed. Usually they are

(15)

From Equation (14),

(16)

and

(17)

so

(18)

Therefore

(19)

This is already a complete real solution of the harmonic oscillator.

4 From two coefficients to amplitude and phase

The same motion can be written as a single shifted cosine:

(20)

Use the angle-difference identity

(21)

Then

u(t)

= A cos(ω0t) cos ϕ + A sin(ω0t) sin ϕ.

(22)

Compare this with Equation (14). The coefficients must satisfy

(23)

Squaring and adding gives

C2 + D2	= A2 cos 2ϕ + A2 sin 2ϕ	(24)
	= A2.	(25)

Hence

(26)

The phase must retain quadrant information, so the robust expression is

(27)

Thus

(28)

The sine form is equally valid because

(29)

There is no physical preference for sine or cosine. Choosing one rather than the other only changes the phase convention.

5 Reading the general sinusoid

Write a general one-point harmonic signal as

(30)

Its instantaneous phase is

(31)

The important quantities are

The period and frequency satisfy

(32)

while one complete cycle is 2π radians, so

(33)

The equation is easiest to interpret as uniform phase accumulation. Since

(34)

we have

(35)

Angular frequency is therefore the rate at which phase advances.

6 The unit circle: the geometry behind sine and cosine

Consider a point moving around a circle of radius A. Let its angle be 𝜃. Its Cartesian coordinates are

(36)

If the angle increases uniformly,

(37)

then the horizontal projection is a cosine in time and the vertical projection is a sine in time.

This geometric picture explains why sine and cosine occur as a natural pair. They are not unrelated functions that happen to solve the same equation. They are the two orthogonal coordinates of rotation.

7 A second route: the exponential trial solution

For constant-coefficient linear differential equations, exponentials are especially useful because differentiating an exponential reproduces the same exponential.

Try

(38)

Then

(39)

Substitute into Equation (7):

(40)

Since e^rt is never zero,

(41)

Thus

(42)

where

(43)

The two exponential solutions are therefore

(44)

In electrical engineering the symbol j is often used instead of i so that i remains available for electric current. The mathematics is identical.

At first these complex solutions may seem to have introduced something unphysical. The next section shows that they are simply a compact representation of sine and cosine.

8 Deriving Euler’s formula from power series

The exponential function has the Maclaurin expansion

(45)

Set

(46)

Then

ei𝜃

= 1 + i𝜃 + + + + .

(47)

The powers of i repeat in a four-step cycle:

(48)

Therefore

ei𝜃 =

1 + i𝜃 − − i + + i −.

(49)

Group the real terms and the imaginary terms:

ei𝜃 =		(50)
	+ i.	(51)

But the Maclaurin series of cosine and sine are

cos 𝜃	= 1 − + −,	(52)
sin 𝜃	= 𝜃 − + −.	(53)

Hence

(54)

This is Euler’s formula.

The result is algebraic, but its geometry is equally important. Equation (54) places the point e^i𝜃 at coordinates

(55)

on the unit circle. Its magnitude is

(56)

Changing 𝜃 rotates the complex number without changing its magnitude.

9 Recovering cosine and sine from exponentials

Euler’s formula gives

ei𝜃	= cos 𝜃 + i sin 𝜃,	(57)
e−i𝜃	= cos 𝜃 − i sin 𝜃.	(58)

Add the two equations:

(59)

Therefore

(60)

Subtract the equations:

(61)

so

(62)

This is an important conceptual result. A real sinusoid contains a conjugate pair of complex exponentials. Later, Fourier analysis will interpret those exponentials as positive- and negative-frequency components.

10 Why the complex exponential solves the oscillator equation

Consider

(63)

Differentiate once:

(64)

Differentiate again:

(65)

Therefore

(66)

Taking the real part gives

(67)

and taking the imaginary part gives

(68)

Thus the physical real-valued sinusoid is contained inside the complex solution.

The complex number is a mathematical bookkeeping device. It simultaneously stores amplitude and phase:

(69)

has magnitude A and argument ϕ₀.

11 The general real solution as a conjugate pair

The two characteristic roots in Equation (44) give the general complex solution

(70)

For the result to be real for every t, choose conjugate coefficients:

(71)

Let

(72)

Then

z(t)	=	(73)
	= A cos(ω0t + ϕ0).	(74)

The sine/cosine basis and the complex-exponential basis therefore span the same two-dimensional solution space.

12 Why complex exponentials simplify differentiation

This is one of the most important practical reasons for using the complex representation. For

(75)

where Z is independent of time,

(76)

Thus

(77)

for a single-frequency complex sinusoid.

Similarly,

(78)

A differential operation has become algebraic multiplication. This is why complex exponentials are so effective in oscillation theory, AC circuits, linear systems, Electromagnetism, and signal processing [5, 4].

12.1 Multiplication by i is a quarter-cycle rotation

From Euler’s formula,

(79)

If

(80)

then

(81)

Therefore multiplying by i rotates a complex phasor by 90^∘ counterclockwise without changing its magnitude. Differentiation of e^iωt both scales the magnitude by ω and advances the phase by π∕2.

13 Phasor notation

For a single fixed angular frequency ω, write

(82)

Define the complex amplitude, or phasor,

(83)

Then

(84)

The common time factor e^iωt can often be suppressed while performing algebra. The constant complex number U carries the amplitude and phase.

This notation is powerful only when all quantities under comparison share the same angular frequency. A phasor is not a replacement for an arbitrary time-dependent signal.

14 From one-point oscillation to a traveling wave

The next conceptual step is to allow phase to depend on position as well as time. A one-dimensional traveling sinusoid can be written

(85)

The total phase is

(86)

The spatial phase rate is the Wavenumber

(87)

and the temporal phase rate is the angular frequency

(88)

At fixed time, moving by Δx changes the phase by

(89)

At fixed position, waiting for a time Δt changes the phase by

(90)

for the sign convention in Equation (85).

The complex representation is

(91)

This is the direct bridge from elementary harmonic motion to wave mechanics and RF signal representation.

15 Phase velocity from constant phase

A recognizable crest or trough corresponds to a fixed value of phase. Set

(92)

Differentiate with respect to time:

(93)

Hence the phase velocity is

(94)

Using ω = 2πf and k = 2π∕λ,

(95)

The familiar wave-speed relation is therefore also a statement about the ratio of temporal phase accumulation to spatial phase accumulation.

16 Sign conventions

Two common complex traveling-wave conventions are

(96)

and

(97)

They are complex conjugates. Either convention is valid if used consistently. The observable real cosine is unchanged by a consistent switch of convention, but signs in derivatives, propagation factors, Fourier transforms, and phasor formulas may change.

Before comparing formulas from two sources, identify the assumed time convention.

17 Worked example: initial displacement and velocity

Consider

(98)

with

(99)

The natural angular frequency is

(100)

Write

(101)

From u(0) = 3,

(102)

Differentiate:

(103)

At t = 0,

(104)

so

(105)

Thus

(106)

The amplitude is

(107)

and

(108)

Therefore

(109)

The two expressions are exactly the same motion.

18 Worked example: from a traveling wave to its parameters

Suppose

(110)

Comparison with Equation (85) gives

(111)

Then

(112)

and

(113)

The phase velocity is

(114)

A displacement of

(115)

corresponds to

(116)

This example links the one-point harmonic language of WM02 directly to the spatial phase language used in later wave and RF articles.

19 Common conceptual errors

19.1 The imaginary part is not an imaginary physical displacement

Complex notation is a mathematical representation. If the physical quantity is real, the measurable signal is usually obtained by taking the real part after the complex algebra has been completed.

19.2 Amplitude and phase are different pieces of information

The complex number

(117)

contains both. Its magnitude is A; its angle is ϕ.

19.3 Sine and cosine are not physically different kinds of motion

They differ by a phase shift of π∕2. Choosing sine or cosine as the reference form is a convention.

19.4 Angular frequency is not ordinary frequency

The two are related by

(118)

The factor 2π converts cycles to radians.

19.5 A phasor is not an arbitrary time signal

The compact replacement d∕dt ↔ iω assumes a single harmonic time dependence. General non-sinusoidal signals require superposition or Fourier methods.

20 Why this matters for RF and GNSS

An RF carrier is commonly represented physically as

(119)

Its complex representation is

(120)

The real waveform is recovered through

(121)

If propagation changes the path length by Δr, the carrier phase changes by

(122)

This is why Euler’s representation, phase, wavenumber, path length, I/Q processing, antenna arrays, and GNSS carrier tracking all become tightly connected. The complex exponential is not an RF-specific trick. It is the natural algebraic language of harmonic waves.

21 Summary

The central chain is

F = −κu	mü + κu = 0	(123)
	ü + ω02u = 0	(124)
	u = C cos ω0t + D sin ω0t	(125)
	⇔u = A cos(ω0t + ϕ0)	(126)
	⇔u = ℜ.	(127)

Euler’s formula provides the bridge:

(128)

For a traveling wave, the phase acquires spatial dependence:

(129)

The temporal phase rate is ω, the spatial phase rate is k, and their ratio gives the phase velocity:

(130)

WM03 can now treat phase difference as a physical comparison between two oscillations or two points on a wave, rather than as an isolated trigonometric parameter.

References

References