Physics for physicians: oscillations and waves

Oscillations are Processes in which a physical quantity changes periodically as a function of time. This includes movements which occur periodically around a rest position.

The most significant form of oscillation is the harmonic oscillation, the temporal change of the physical quantity is sinusoidal here. All other types of oscillation are not harmonic.

Basic concepts of oscillations and waves

A Period is called a complete to-and-fro of the oscillating body.
The Amplitude is the largest deflection from the rest position.
The Period duration T is the time required by a vibrating body for a reciprocating motion, d. h. for a period. It is also called Oscillation period.
The frequency is the quotient of the number of periods and the time needed for this.
At Elongation one understands the instantaneous distance to the rest position.
As Circular frequency is the angular velocity of a circular motion, the projection of which on a straight line results in a harmonic oscillation.
The Phase is determined by the two oscillation quantities elongation and time. It is characteristic of the instantaneous state of the oscillation.
The damped oscillation is an oscillation with an amplitude which mostly decreases continuously due to friction.
The undamped vibration is an oscillation with a constant amplitude. The prerequisite is, however, that the energy supplied to the oscillating system is retained. Due to constant small energy losses an undamped oscillation is only approximately possible in reality. If a truly undamped oscillation is to be produced, the energy losses that occur must be compensated for by a regular supply of energy.

The undamped harmonic oscillation

Harmonic oscillations are usually called sinusoidal oscillations. The reason for this is that the elongation is a sine function of the time.

The harmonic oscillation is an unevenly accelerated motion. Their acceleration is a function of time. You can read more about this in the articles "Mechanics Part I" and "Mechanics Part II".

At any moment of a harmonic oscillation, a force acts in the direction of acceleration, which wants to bring the oscillating body to its center position. This force is called Resetting power. This restoring force is proportional to the elongation.

For mechanical oscillations, the basic equation of dynamics is:

From this principle the Equation of the undamped harmonic oscillation derive:

y ⇒ Elongation
ω ⇒ angular frequency

Forced oscillations

If an oscillatory system is deflected from rest and then oscillates out without additional constraints, it is an free oscillation. The frequency of the free vibration is called natural frequency.

If, however, a periodically varying force acts on the oscillating system, which forces the system to resonate, then one speaks of a forced oscillation. On the vibrating system then three forces act:

Restoring force
Damping force
excitation force

Resonance

If the excitation frequency is equal to the natural frequency of the oscillation, then resonance occurs. With a small damping the amplitude increases very much. In everyday life and in technology, the importance of resonance is very great.

Since most mechanical structures are capable of oscillation and can be excited by external periodic forces, in the case of resonance, the amplitude can be greatly increased and thus the structure can be destroyed.

To be able to prevent such a "resonance catastrophe", you have to avoid periodic forces or keep large differences between the natural frequency and the excitation frequency, create damping possibilities and allow a resonance frequency only for a period of time that is smaller than the settling time. With rotary motion, the resonant frequency is called a critical speed.

Superposition of vibrations

Every oscillating system can perform several oscillations at the same time. The individual oscillations superimpose to form a resultant oscillation. It applies: If vibrating bodies are excited to several vibrations, they overlap without interfering with each other.

The following are to be distinguished oscillations that have the same direction of oscillation and oscillations whose directions of oscillation are at right angles to each other.

If two harmonic oscillations with the same direction and frequency overlap, again a harmonic oscillation of the same frequency is generated, whose amplitude depends on the individual amplitudes. The phases of the initial oscillations also influence the amplitude of the resulting oscillations.

If two oscillations with the same direction but unequal frequency overlap, a non-harmonic oscillation results.

Waves

An oscillatory process in an extended medium is called an oscillating wave. An extended medium consists of a multitude of vibrating particles, which are all coupled with each other. If one of these particles experiences a vibration pulse, it becomes the center of a propagating wave motion.

A Wave is a spatially and temporally periodic process in which energy is transported, but not mass. The propagation direction of a wave is called Wave beam. Perpendicular to the wave beam is the Wave front. A wave front is the geometric location of all particles belonging to the same phase.

The distance between two wave fronts is called Wave length. The distance between two adjacent particles of the same phase of oscillation is the Wave propagation. It is independent of location and time. The wave propagation can be Huygens principle Of the elementary waves can be interpreted.

Huygens’ principle: Each point of a medium captured by a wave motion can itself be regarded as the starting point of a new wave – a so-called elementary wave. Each wave front can be understood as an enveloping one of elementary waves. The hyugens principle is valid for all kinds of waves, even for electromagnetic waves.

Wave types

Longitudinal waves (longitudinal waves): The particles oscillate back and forth in the direction of propagation. If the particles oscillate in the direction of propagation, there is an overpressure (compression). Where they oscillate against the direction of propagation of the wave, there is a negative pressure (dilution). Compaction and thinning alternate.
Transverse waves: The particles oscillate perpendicular to the direction of propagation of the wave. Wave valleys and wave crests alternate.
Linear waves are one-dimensional waves. Their spreading possibility exists only in one direction.
Area waves are two-dimensional waves. Its propagation possibility is areal.
Space waves are three-dimensional waves. Their ability to spread is spatial.

At Plane waves with a point-shaped excitation center the wave fronts are circles. At Space waves with a point excitation center the wave fronts are spherical shells.

The Propagation speed of a wave results from the product of the frequency f at which each particle of the wave oscillates and the wavelength λ:

The following laws are valid for the propagation speed of waves:

Elastic transverse wave in solids:

Elastic longitudinal wave in solids:

Longitudinal wave in liquids:

c ⇒ propagation velocity
F ⇒ tension force
A ⇒ cross-sectional area
ρ ⇒ density of the medium
E ⇒ Young’s modulus
K ⇒ compression module

If two waves overlap that have the same amplitude, frequency and wavelength but travel in opposite directions, the result is a standing wave. In the case of a standing wave, the spatial image does not move any further. Places of maximum deflection (Wave bellies) and places with a deflection equal to zero (Node) remain stationary.

Standing waves can originate in the reflection from a thinner or a denser medium. Most often a standing wave occurs when a linear wave superimposes on itself after reflection.

Reflection

If a wave hits another medium at the boundary layer of its medium, it is completely or partially reflected back. This process is called reflection. The Reflection law is as follows:

At the transition between two media, the wave beam is refracted, the direction of propagation and the speed of propagation change. For the propagation velocities the following applies Law of Refraction:

Diffraction

A further change in the speed of propagation of the wave is found at the boundaries of an obstacle, z.B. of a gap. The obstacle does not cast a sharp shadow. The phenomenon of diffraction can be explained by Huygens’ principle.

$this illustration shows a diffraction at the slit$

The energy of the wave particle diffracted by the opening slit of the wall is distributed among the individual directions after diffraction in such a way that the energy components decrease with increasing diffraction angle. As Angle of diffraction one calls the Angle between the original wave direction and the new wave directions.

Sound waves

Sound waves are mechanical longitudinal waves. They originate in the source of sound, a vibrating body, and propagate in solids, liquids, and gases in the form of pressure waves.

example: To the human ear, frequencies from 16 Hz to 20000 Hz are audible. At frequencies lower than 16 Hz, one speaks of infrasound. At frequencies above 20000 Hz, one speaks of Ultrasound.

In acoustics, a distinction is made between the following Types:

sound: Graphically, the pure tone is a sine wave. The pitch of the tone is determined by the frequency. The height distance of two tones is called interval. In the case of tone intervals, the lower tone is called the fundamental tone. The higher tone is the octave, the fifth, the fourth, etc. The musical interval of two tones that can be detected by hearing is determined by the quotient of their frequencies.
Sound: Several sinusoidal oscillations superimpose to form a non-sinusoidal oscillation. The pitch is determined by the fundamental tone, the other tones convey the timbre.
Noise: a mixture of numerous sounds, of rapidly changing frequencies and strength.
bang: a strong sound event, which starts abruptly and is only effective for a short time.

Sound

Everything that we can perceive with the human ear is called sound. A distinction is made between tones and noises. How we perceive a sound event depends on the loudness, the pitch and the sound color.

Sound propagates from a Sound exciter off – this is a vibrating body.

For sound to reach our ear, this must be transmitted by a sound carrier. For sound propagation, solid, liquid or gaseous bodies are necessary as sound carriers. Sound cannot be transmitted in a vacuum. A Sound source generates longitudinal waves propagating in the sound carrier. Sound perception occurs as soon as these longitudinal waves reach our ear (sound receiver).

Speed of sound

The speed of sound indicates how fast the sound propagates in a given sound carrier. It is a quantity independent of the frequency. In air, the speed of sound is z.B. at 340 m/s, in water at 1440 m/s.

Sound waves, when they encounter another medium, become, reflected at the transition boundary.

Example: In a sonographic measurement, frequencies of a certain magnitude sent by the sound transmitter are reflected at a boundary surface (such as tissue or blood components in the Doppler method) and picked up again at a different frequency by a measuring probe. From the time required, given a known speed of sound propagation (or given known initial frequencies), the distance traveled can be calculated.

Electromagnetic wave

The electromagnetic wave consists of electric and magnetic fields, which are coupled with each other. They are dependent on the frequency.

Electromagnetic waves can propagate both in free space and in vacuum. They do not require a carrier medium. In a vacuum, these waves propagate at the speed of light. According to today’s measurements, the speed of light in vacuum is c = (299792.5 + / – 0.9) km/s.

Infrared light

The white light of an arc lamp or incandescent lamp provides a invisible radiation, which is found in the visible spectrum next to red. It is called infrared radiation and provides the infrared light.

Ultraviolet light

Outside the violet range of the visible color spectrum is the ultraviolet color range. The irritation and tanning of the skin when exposed to the sun is due to the ultraviolet light component. Mercury vapor lamps, which are used as artificial high-altitude suns, also emit ultraviolet light.

The various color impressions are presented to our eye by different frequencies of the visible spectrum conveyed:

this illustration shows the visible color spectrum of the human being

Image: "Electromagnetic Wave Spectrum" by Horst Frank. License: CC BY-SA 3.0

Gamma radiation

Beside the atomic building blocks protons, neurons and electrons we know more than 200 others today Elementary particles. Many of them are the result of the interaction between the Earth’s atmosphere and cosmic rays, or the product of nuclear fragmentation using particle accelerators. The elementary particles are are divided into the following groups:

Leptons: light particles
mesons: medium weight particles
baryons: heavy particles

Most elementary particles exist with an opposite-pole electric charge and an opposite-pole magnetic moment as so-called Antiparticle. If a particle meets with its antiparticle, there is a Dispersion. Their energy is released as gamma radiation.