Harmonic Oscillations and Specific Vibrations

Introduction to Harmonic Oscillations

Humans are constantly surrounded by objects that vibrate or oscillate around them. Sound perceived by the ears is fundamentally a vibration of the surrounding air. In physics and mathematics, these vibrations are modeled as harmonic oscillations. The simplest physical model to demonstrate this is a mass-spring system. When a mass is attached to a spring and allowed to hang in rest, it sits at the equilibrium point. If the mass is displaced from this equilibrium point and released, it performs an oscillatory motion around that point. This motion is quantified by an x-axis where the position $x$ represents the displacement from equilibrium, measured in meters ($m$). The function $x(t)$ is referred to as the vibration equation or oscillation equation.

A system performs a harmonic oscillation if its vibration equation can be expressed as a general sine function in the form:

$x(t) = a \cdot \sin(bt + c) + d$

Mathematical Parameters of Harmonic Oscillations

The general sine function contains four parameters ($a, b, c, d$) that describe the physical characteristics of the oscillation. If the origin of the x-axis is chosen such that it coincides with the equilibrium point, the parameter $d$ becomes zero, simplifying the equation to:

$x(t) = a \cdot \sin(bt + c)$

The parameter $a$ represents the maximum displacement relative to the equilibrium point. In physics, this is defined as the amplitude and is typically denoted as $A$.

The parameter $b$ determines the period $T$ of the oscillation, which is the time required for one complete cycle. It is also known as the pulsation or angular frequency, denoted by the Greek letter $\omega$, and is expressed in radians per second ($rad/s$). The relationship between these variables is given by:

$\omega = \frac{2 \cdot \pi}{T} = 2 \cdot \pi \cdot f$

Here, $f$ represents the frequency, which is the number of oscillations occurring per second, measured in Hertz ($Hz$) or inverse seconds ($s^{-1}$). The frequency is the reciprocal of the period:

$f = \frac{1}{T}$

The parameter $c$ determines the specific position of the system at time $t = 0\,s$. This is known as the initial phase of the oscillation, denoted as $\phi_0$.

Phase, Initial Phase, and Initial Displacement

A harmonic oscillation is fully characterized by its amplitude $A$, pulsation $\omega$, and initial phase $\phi_0$. When the equilibrium point is at the origin, the equation is:

$x(t) = A \cdot \sin(\omega \cdot t + \phi_0)$

In this context, $\phi(t) = \omega \cdot t + \phi_0$ is defined as the phase of the harmonic oscillation. While a spring is a common demonstration tool, it is not a perfect example because real-world springs experience damping, causing the oscillation to stop over time. Theoretically, a pure harmonic oscillation maintains a constant amplitude indefinitely.

It is important to distinguish between the initial phase ($\phi_0$) and the initial displacement ($x_0$). The initial phase is an angle, whereas the initial displacement is the coordinate on the x-axis at $t = 0\,s$. The following relationships apply for different starting positions within the interval of $0$ to $2 \cdot \pi$:

If the starting displacement is at the maximum positive amplitude ($x_0 = A$), then $A \cdot \sin(\phi_0) = A$, implying $\sin(\phi_0) = 1$. Thus, $\phi_0 = \frac{\pi}{2}, \frac{5 \cdot \pi}{2}, \dots$

If the starting displacement is at the equilibrium point ($x_0 = 0$), then $A \cdot \sin(\phi_0) = 0$, implying $\sin(\phi_0) = 0$. Thus, $\phi_0 = 0, \pi, 2 \cdot \pi, \dots$

If the starting displacement is at the maximum negative amplitude ($x_0 = -A$), then $A \cdot \sin(\phi_0) = -A$, implying $\sin(\phi_0) = -1$. Thus, $\phi_0 = \frac{3 \cdot \pi}{2}, \frac{7 \cdot \pi}{2}, \dots$

The Link Between Harmonic Motion and Circular Motion (Phasors)

Harmonic oscillations are intrinsically linked to Uniform Circular Motion (UCM). If one views the projection of a point moving in UCM onto a vertical axis, that projected point follows a harmonic oscillation. This allows for the visual representation of the initial phase using a "phasor" or "rotor," which is a rotating vector. The angle between the horizontal axis and the phasor at $t = 0\,s$ is equal to the initial phase $\phi_0$.

As the point oscillates, it describes a projection of a UCM with an angular velocity equal to the pulsation:

$\omega = \frac{\Delta\phi}{\Delta t} = \frac{2 \cdot \pi}{T}$

The phase $\phi(t)$ represents the angle between the phasor and the horizontal axis at any time $t$, calculated as:

$\phi(t) = \omega \cdot t + \phi_0$

Examples of Harmonic Motion Calculations

Example 1: The Hanging Lamp Consider a heavy hanging lamp performing a harmonic oscillation with the equation:

$x(t) = 4,0\,cm \cdot \sin(\frac{\pi}{1,5\,s} \cdot t + \frac{\pi}{4})$

To find the frequency ($f$), we identify $\omega = \frac{\pi}{1,5\,s^{-1}}$. Since $\omega = 2 \cdot \pi \cdot f$, we calculate:

$f = \frac{\pi}{2 \cdot \pi \cdot 1,5}\,Hz = \frac{1}{3,0}\,Hz = 0,33\,Hz$

To find when the lamp first passes the equilibrium position ($x = 0$), we set the sine term to zero. This occurs when the phase $\phi(t) = k \cdot \pi$. Setting the equation:

$\frac{\pi}{1,5\,s} \cdot t + \frac{\pi}{4} = k \cdot \pi$

Solving for $t$:

$t = -0,38\,s + 1,5\,s \cdot k$

For $k=1$, the result is $t = 0,63\,s$. Therefore, the lamp passes the equilibrium point for the first time after $0,63\,s$.

Example 2: Marine Buoy A buoy floating at sea provides a practical application for determining amplitude, frequency, and phase directly from observed time-series data of its displacement.

Phase Differences Between Oscillations

When two systems perform harmonic oscillations with the same pulsation $\omega$, their respective displacement equations are:

$x_1(t) = A_1 \cdot \sin(\omega \cdot t + \phi_{01})$

$x_2(t) = A_2 \cdot \sin(\omega \cdot t + \phi_{02})$

The phase difference $\Delta\phi$ is constant over time and is defined as:

$\Delta\phi = \phi_2(t) - \phi_1(t) = \phi_{02} - \phi_{01}$

There are two critical cases for phase differences (where $k$ is an integer):

Case 1: In Phase When $\Delta\phi = k \cdot 2 \cdot \pi$, the systems are "in phase." They pass the equilibrium point at the same moment moving in the same direction and reach their positive maxima simultaneously. Their graphs are shifted relative to each other by $k \cdot T$.

Case 2: Anti-Phase When $\Delta\phi = (2 \cdot k + 1) \cdot \pi$, the systems are in "anti-phase." They pass equilibrium at the same time but move in opposite directions. When one reaches its positive maximum, the other is at its negative maximum. Their graphs are shifted by $\frac{(2 \cdot k + 1) \cdot T}{2}$.

Kinematics of Harmonic Oscillations: Velocity and Acceleration

The velocity and acceleration of a point performing harmonic motion are themselves harmonic functions with the same period $T$ as the displacement.

From the displacement $x(t) = A \cdot \sin(\omega \cdot t + \phi_0)$, velocity ($v_x$) is the first derivative with respect to time:

$v_x(t) = \frac{dx(t)}{dt} = A \cdot \omega \cdot \cos(\omega \cdot t + \phi_0) = A \cdot \omega \cdot \sin(\omega \cdot t + \phi_0 + \frac{\pi}{2})$

The amplitude of the velocity is $A \cdot \omega$, and it is phase-shifted by $\frac{\pi}{2}$ relative to displacement. When $x=0$, the magnitude of velocity $|v_x|$ is at its maximum.

Acceleration ($a_x$) is the derivative of velocity:

$a_x(t) = \frac{dv_x(t)}{dt} = -A \cdot \omega^2 \cdot \sin(\omega \cdot t + \phi_0) = A \cdot \omega^2 \cdot \sin(\omega \cdot t + \phi_0 + \pi)$

The amplitude of acceleration is $A \cdot \omega^2$, and it is phase-shifted by $\pi$ relative to displacement. In terms of timing, the acceleration graph is shifted by $\frac{T}{2}$ and the velocity graph by $\frac{T}{4}$ relative to the displacement graph. When $|x|$ is maximum, $|a_x|$ is also maximum, while $v_x = 0$.

Dynamics: Force, Strength Constants, and Eigenfrequency

In a system like a person on a swing, acceleration and velocity change constantly due to a variable force. According to Newton's Second Law ($F = m \cdot a$), the force acting on a system of mass $m$ performing harmonic oscillation is:

$F_x = m \cdot a_x = -m \cdot \omega^2 \cdot A \cdot \sin(\omega \cdot t + \phi_0) = -m \cdot \omega^2 \cdot x(t)$

We define the force constant $k$ as $k = m \cdot \omega^2$, leading to the formula for the restoring force:

$F_x = -k \cdot x(t)$

This force is always directed toward the equilibrium position and is proportional to the displacement. The frequency at which a system naturally oscillates is called its eigenfrequency ($f$), determined by the mass and the force constant:

$f = \frac{1}{2 \cdot \pi} \cdot \sqrt{\frac{k}{m}}$

Physical implications include:

• A higher force constant $k$ (e.g., a tighter guitar string) results in a higher frequency/pitch.
• A larger mass $m$ (e.g., a double bass vs. a violin) results in a lower frequency/pitch.

Energy in Harmonic Systems

The total mechanical energy ($E_{mech}$) of a harmonic oscillator is constant and proportional to the square of both the amplitude and the frequency:

$E_{mech}(t) = \frac{1}{2} \cdot k \cdot A^2$

Substituting frequency into the equation:

$E_{mech}(t) = 2 \cdot \pi^2 \cdot m \cdot f^2 \cdot A^2$

In a horizontal mass-spring system, where the normal force and gravity cancel out and do not perform work, mechanical energy is conserved through the exchange of kinetic energy ($E_k$) and potential energy ($E_p$):

$E_k(t) = \frac{1}{2} \cdot k \cdot A^2 \cdot \cos^2(\omega \cdot t + \phi_0)$

$E_p(t) = \frac{1}{2} \cdot k \cdot A^2 \cdot \sin^2(\omega \cdot t + \phi_0)$

$E_{mech}(t) = E_k(t) + E_p(t) = \frac{1}{2} \cdot k \cdot A^2 \cdot [\cos^2(\dots) + \sin^2(\dots)] = \frac{1}{2} \cdot k \cdot A^2$

Specific Oscillating Systems: Springs and Pendulums

Vertical Mass-Spring System When a mass is hung vertically, it stretches the spring by $\Delta l$. At rest, gravity and spring force are equal: $k_{spring} \cdot \Delta l = m \cdot g$. If disturbed, it oscillates with:

$T = 2 \cdot \pi \cdot \sqrt{\frac{m}{k_{spring}}}$

Horizontal Mass-Spring System Similar to the vertical case, the restoring force is defined by Hooke's Law as $F_v = -k_{spring} \cdot x$. The period remains the same as the vertical system.

Pendulum (Slinger) A mass $m$ at the end of a cord of length $L$ performing small oscillations (where $\sin\theta \approx \theta$) follows harmonic motion. The restoring force along the arc $s$ is $F_s = -m \cdot g \cdot \frac{s}{L}$. Thus, the force constant for a pendulum is $k = \frac{m \cdot g}{L}$. The eigenfrequency and period are:

$f = \frac{1}{2 \cdot \pi} \cdot \sqrt{\frac{g}{L}}$

$T = 2 \cdot \pi \cdot \sqrt{\frac{L}{g}}$

Crucially, for small angles, the period of a pendulum is independent of the mass and the amplitude; it depends only on the length $L$ and gravitational acceleration $g$.

Types of Oscillations

Vreie Trillingen (Free Oscillations) A system oscillating only under an elastic force moves at its eigenfrequency with a constant amplitude.

Gedempte Trillingen (Damped Oscillations) In reality, friction and resistance cause mechanical energy to convert into heat. The amplitude decreases exponentially over time:

$x(t) = A_0 \cdot e^{-\frac{\lambda \cdot t}{2 \cdot m}} \cdot \sin(\omega \cdot t + \phi_0)$

Here, $\lambda$ represents the damping coefficient; a larger $\lambda$ means faster decay.

Gedwongen Trillingen (Forced Oscillations) and Resonance Energy loss in a damped system can be compensated by an external periodic force with frequency $f_u$. After an initial chaotic phase, the system will oscillate harmonically at the frequency of the external force. If $f_u$ matches the system's eigenfrequency, the amplitude becomes exceptionally large. This phenomenon is called resonance, representing maximal energy transfer between the external driver and the system.

Oscillations in Music and Sound

Sound is perceived when air vibrations reach the eardrum, are converted into electrical pulses in the inner ear, and interpreted by the brain.

Tones and Tone Quality

• Tones: Sound with a periodic signal.
• Simple Tone: A sound with a pure sine wave signal (e.g., a tuning fork).
• Intensity (Toonsterkte): Determined by the amplitude. A larger amplitude results in a louder sound.
• Pitch (Toonhoogte): Determined by frequency. Higher frequency results in a higher pitch.

Hoorbaarheidsgebied (Hearing Range) The human ear typically detects frequencies between approximately $20\,Hz$ and $20\,kHz$. This range narrows with age or noise exposure. Frequencies below $16\,Hz$ are infrasound (detected by elephants and whales). Frequencies above $20000\,Hz$ are ultrasound (used by bats and dogs).

Timbre (Toonklank) Instruments can play the same note (same frequency and loudness) but sound different due to their timbre or klankkleur. Jean-Baptiste Joseph Fourier demonstrated that any periodic signal can be represented as a sum of sine functions with varying amplitudes and frequencies. This composition of a fundamental frequency and overtones (boventonen) defines the sound. For a guitar, a fundamental frequency of $321\,Hz$ may have overtones at $642\,Hz, 963\,Hz, \dots, 2247\,Hz$. If high-frequency tones dominate the spectrum, the sound is bright or sharp; if low frequencies dominate, it is dull.