Table of Contents
Oscillations are processes in which a physical quantity changes periodically for a period of time. These include movements that occur periodically around an idle position.
The most significant vibration form is the harmonic vibration; this periodic change of a physical quantity is sinusoidal. All other vibration modes are not harmonious.
Basic Concepts of Oscillations and Waves
- Complete back and forth movement of the oscillating body is called a period.
- The amplitude is the extent of a wave measured from the highest peak down to the zero position.
- The period duration T is the time needed for an oscillating body to move back and forth, i.e. to fulfill 1 period. It is often referred to as period.
- Frequency is the quotient of the number of periods and the time required for it.
- Elongation is defined as the current distance from the zero position.
- Circular frequency is the angular velocity of a circular movement whose projection on a straight line results in a harmonic oscillation.
- The phase is specified by the 2 oscillation parameters, elongation and time, and is indicative of the current state of oscillation.
- The damped oscillation is an oscillation in which the amplitude of the oscillating quantity decreases with time due to frictional forces.
- Undamped oscillations have constant amplitudes. However, it is necessary that the energy supplied to the vibrating system is maintained. In the case of constant small energy losses, a sustained undamped oscillation, in reality, is only approx. possible. If one wants to create a real undamped oscillation, the energy losses that occur must be compensated by regular energy supply.
Undamped Harmonic Oscillation
Harmonic oscillations are also referred to as sine oscillations. The reason for this is that its elongation is a sinusoidal function of time.
At any given time of harmonic oscillation, a force acts in the direction of acceleration that wants to bring the oscillating body to its center position. This force is known as the restoring force. This restoring force is proportional to the elongation.
The base equation of dynamics for mechanical oscillation is as follows:
restoring force = mass × acceleration of gravity
Fr = m × a
From this principle, the equation of the undamped harmonic oscillation can be derived:
ӱ + y × ω2 = 0
y ⇒ Elongation
ω ⇒ Angular frequency
Whenever an oscillatory system is displaced and released from its zero position, it will begin to oscillate. If no other external forces are applied, it is a free oscillation. The frequency of the free oscillation is called natural frequency, or eigenfrequency.
However, if an external periodic driving force is imposed on the oscillating system to keep it going, then it is known as forced oscillation. Three forces are acting on the oscillating system:
- Conservation force
- Damping force
- Excitation force
Resonance occurs when the excitation frequency matches the natural frequency of the oscillation. When the damping is small, the amplitude decreases greatly. In everyday life—and in technology—resonance plays a vital role.
As most mechanical structures are subject to an oscillating force, they can be stimulated by external periodic forces. In the case of resonance, this would result in an increase of amplitude followed by the destruction of the structure.
To prevent such a ‘resonance catastrophe’, one has to avoid any periodic forces or maintain large margins between the natural frequency and the excitation frequency, provide damping options and allow a resonance frequency only for a short time which is less than the build-up time. For rotary movements, the resonance frequency is referred to as a critical rotational speed.
Superposition of Oscillations
Each oscillating system can execute several oscillations simultaneously. The individual oscillations overlap and form a resultant oscillation. In conclusion: If the oscillating bodies execute several oscillations, they then superpose themselves without interfering with each other.
A distinction is made between oscillations that possess the same oscillating directions and oscillations whose directions of oscillation are perpendicular to each other.
If 2 harmonic vibrations with the same direction and same frequency are superposed, it results in a harmonic oscillation of the same frequency, whose amplitude depends on the individual amplitudes. The phases of the initial oscillations also affect the amplitude of the resulting oscillations.
When 2 oscillations in the same direction, but not the same frequency are superposed, the result is a non-harmonic oscillation.
Oscillation process in an extended medium is called an oscillating wave. Extended medium consists of a vast amount of oscillating particles, all of which are interconnected. If one of these particles receives an oscillation impulse, it then becomes the center of a propagating wave movement.
Wave is a spatial and temporary periodic process, where energy—but no matter—is transported. The direction of energy propagation is called the wave ray. Perpendicular to the wave ray runs the wavefront. Wavefront is the ge'[ometric location of all the particles that belong to the same phase.
The distance between 2 successive wavefronts is called the wavelength. The distance between 2 neighboring particles of the same oscillation phase is the wave propagation. It is independent of location and time. The wave propagation can be analyzed according to Huygens’ principle of wavelets.
- Longitudinal waves: The particles oscillate back and forth in the direction of propagation. If the particles oscillate in the direction of travel of the wave, an overpressure (compression) occurs. If they oscillate opposite to the direction of wave travel, a vacuum (rarefaction) occurs. Compressions and rarefactions alternate among each other.
- Transverse waves: The particles oscillate perpendicular to the direction of travel of the wave. Transverse waves have troughs and crests which alternate.
- Plane waves are one-dimensional waves. They travel only in 1 direction.
- Surface waves are two-dimensional waves. They propagate in an expanding surface.
- Sky waves are three-dimensional waves. They propagate in the form of an expanding sphere.
When surface waves have a punctiform excitation center, the wavefronts are circles. When sky waves have a punctiform excitation center, the wavefronts are spherical shells.
The velocity of wave propagation c is equal to the product of its frequency f with which each particle of the wave oscillates, and its wavelength λ:
c = f λ
The velocity of wave propagation is given by the following equations:
Elastic transverse wave in solids:
If 2 waves superpose and coincide in amplitude, frequency, and wavelength, but run in the opposite direction, a so-called standing wave occurs. In a standing wave, the spatial image does not travel any further. Locations at which the amplitude is maximum (antinodes) and locations at which amplitude is minimum or equal to 0 (nodes) remain in a constant position.
Standing waves can occur with the reflection on a thinner or a denser medium. A standing wave occurs most frequently when a plane wave superposes with itself after a reflection.
When a wave strikes the boundary of its medium onto another medium, it is entirely or partially reflected. This process is called reflection. The law of reflection is as follows:
angle of incidence = angle of reflection
Further change in the propagation velocity of the wave can be found at the edge of an obstacle, such as a slit. The obstacle does not cast sharp shadows. The Huygens’ principle can explain the phenomenon of diffraction.
The energy of the wave-particle diffracted through the opening slit of the wall is distributed in individual directions after diffraction so that the energy parts decrease with increasing diffraction angle. The diffraction angle is the angle between the original wave direction and the new wave directions.
The electromagnetic wave is caused by the oscillation of electric and magnetic fields coupled together. They are dependent on the frequency.
Electromagnetic waves can propagate both in free space and in a vacuum. They do not need a carrier medium. In the vacuum, these waves propagate with the speed of light. The speed of light in a vacuum is according to current measurements c = (299,792.5 +/- 0.9) km/s.
The white light of an arc lamp or light bulb provides invisible radiation that can be found in the visible spectrum next to the red. It is referred to as infrared radiation and results in the infrared light.
The ultraviolet color range lies outside the violet range of the visible color spectrum. The irritation and tanning of the skin under the sun are due to the ultraviolet light component. Mercury vapor lamps that are used as artificial tanning lamps also send out ultraviolet light.
The different impressions of color are projected to our eye by different frequencies of the visible spectrum:
Apart from the nuclear components—protons, neutrons, and electrons—today, more than 200 other elementary particles are known. A lot of them are the result of the interactions between the terrestrial atmosphere and cosmic radiation or the product of nuclear disintegrations supported by particle accelerators. The elementary particles are classified into the following groups:
- Leptons: light particles
- Mesons: medium-weight particles
- Baryons: heavy particles
Most elementary particles exist with an antipole electric charge and a reversed magnetic momentum—so-called antiparticles. If a particle meets with its antiparticle, the result is annihilation. Its energy is released as gamma radiation.
Sound waves are mechanical longitudinal waves. They have their origin in the sound source (a vibrating body) and propagate in solids, liquids, and gases in the form of compression waves.
In acoustics, a distinction is made between the following types:
- Tone: The pure tone is graphically displayed as a sinusoidal wave. The frequency specifies the pitch of the tone. The ratio between the 2 tones is called the interval. In musical intervals, the lowest note is called the root—the fundamental tone. The higher notes are the octave, the perfect 5th, the perfect 4th, etc. The musical interval of 2 tones, which can be detected by the sense of hearing, is defined by the quotient of their frequencies.
- Sound: Several sinusoidal oscillations superpose to a non-sinusoidal vibration. The pitch is determined by the root; the other sounds convey the timbre.
- Noise: Variety of sounds with rapidly changing frequencies and strength.
- Bang: A sudden loud noise that lasts for a very brief moment.
Everything perceived with the human ear is called sound. We distinguish between tones and noises. The way we perceive a sound event depends on its volume, pitch, and timbre.
The sound comes from a sound generator—which is an oscillatory body.
In order for sound to reach our ears, it must be transmitted by a sound carrier. The propagation of sound requires solid, liquid, or gaseous bodies as a sound carrier. Sound cannot be transmitted in a vacuum. Sound source generates progressive longitudinal waves in the sound carrier. The perception of sound occurs when these longitudinal waves reach our ears (sound receivers).
Speed of Sound
The speed of sound specifies how fast the sound propagates in a particular sound carrier. It is independent of frequency. In the air, for example, the speed of sound is 340 m/s, and in water, it is 1440 m/s.
When sound waves hit another medium, they are reflected at the transition boundary.