[vc_row][vc_column][rev_slider_vc alias=”repository/team”][/vc_column][/vc_row][vc_row][vc_column][vc_empty_space][/vc_column][vc_column][vc_custom_heading source=”post_title” font_container=”tag:h1|text_align:left” use_theme_fonts=”yes”][/vc_column][/vc_row][vc_row][vc_column][vc_column_text][vc_row][vc_column][vc_column_text]Documentation/Explanations regarding the Animations may be found here
1. Reflection = mirror wave (MOV-01.mp4)
The reflection of a (dispersion-free) transversal wave at the fixed termination at one end may be described as superposition of two waves running towards each other. In the picture, the primary wave (shown in blue) is running from left to right while the mirror wave (shown in black) is running from right to left. The superposition of the two waves results in a standing transversal wave (shown in magenta).
Depending on the phase shift between primary wave and mirror wave, any kind of boundary condition is possible. In the picture, the mirror wave meets the primary wave (at the hatched border) in such a way that the displacement always remains zero (fixed termination).
For further details, see chapter 2.2 and A.3 in Physics of the Electric Guitar
2. Propagating transversal wave (MOV-02.mp4)
In a transversal wave (shear wave), the particles of the medium oscillate orthogonally to the propagation direction: they receive a shear deformation but no change in volume. If the propagation direction always remains one and the same, the wave is linearly polarized. Acoustical transversal waves can only exist in solid-state materials but not in a gas.
In the picture, the sections delimited by the blue lines move up and down. The lateral forces appearing at the border surfaces of each section normally are of different strength on both sides of the section. A shear tension that deforms the section results.
The distance between two maxima of the deflection corresponds to the wavelength.
For further details, see chapter 2.1, 1.1, and A.3 in Physics of the Electric Guitar
3. Standing transversal wave (MOV-03.mp4)
In a transversal wave (shear wave), the particles of the medium oscillate orthogonally to the propagation direction: they receive a shear deformation but no change in volume. If the propagation direction always remains one and the same, the wave is linearly polarized. Acoustical transversal waves can only exist in solid-state materials but not in a gas.
In the picture, the sections delimited by the blue lines move up and down. The lateral forces appearing at the border surfaces of each section normally are of different strength on both sides of the section. A shear tension results that deforms the section.
Contrary to the propagating transversal wave, the standing transversal wave includes regions that do not move (vibration nodes) and regions of maximum excursion (anti-nodes). In a node, the shear deformation is at a maximum while in an anti-node it is zero. The distance between two adjacent nodes corresponds to half the wavelength.
For further details, see chapter 2.1, 1.1, and A.3 in Physics of the Electric Guitar
4. Propagating flexural wave (MOV-04.mp4)
In the flexural wave, the particles of the medium oscillate orthogonally to the propagation direction. Contrary to the transversal wave, they are subject to a special shape change: their boundary surfaces do not remain parallel. If the propagation direction always remains one and the same, the wave is linearly polarized.
The centre of the section marked in red in the picture moves orthogonally to the propagation direction and the angles of the boundary surfaces change accordingly. The propagation of the flexural wave is dispersive – the propagation velocity decreases with increasing frequency.
The wavelength corresponds to the distance of two neighboring maxima (or minima) of the transversal displacement.
For further details, see chapter 2.7, 1.3, and A.4 in Physics of the Electric Guitar
5. Standing flexural wave (MOV-05.mp4)
In the flexural wave, the particles of the medium oscillate orthogonally to the propagation direction. Contrary to the transversal wave, they are subject to a special shape change: their boundary surfaces do not remain parallel. If the propagation direction always remains one and the same, the wave is linearly polarized.
The centre of the section marked in red in the picture moves orthogonally to the propagation direction and the angles of the boundary surfaces change accordingly. In contrast to the propagating flexural wave, we find sections in the standing flexural wave that – in their centre – do not move at all. In these areas relatively strong normal stresses appear. Moreover, there are regions that move up and down strongly – here the flexural stress is relatively strong.
For further details, see chapter 2.7, 1.3, and A.4 in Physics of the Electric Guitar
6. Standing flexural wave for different boundary conditions [1] (MOV-06.mp4)
The left end of the vibrating rod is braced (i.e. both deflection and bending moment are zero) while the right end of the rod is clamped down (i.e. both deflection and strain are zero). For the right hand end of the rod a boundary field results – but not for the left hand end.
For further details, see chapter 2.7 and A.4.3 in Physics of the Electric Guitar
7. Standing flexural wave for different boundary conditions [2] (MOV-07.mp4)
The left end of the rod is guided (i.e. shear force and strain are zero) while the right end is disengaged (i.e. shear force and bending moment are zero). For the right hand end of the rod a boundary field results – but not for the left hand end.
For further details, see chapter 2.7 and A.4.3 in Physics of the Electric Guitar
8. Propagating longitudinal wave (MOV-08.mp4)
For the longitudinal wave, the particles of the medium oscillate in the direction of the propagation (or in the opposite direction). In a simple model we imagine small sections of mass connected by helical springs. All these sections oscillate towards the neighboring section or away from it. In a continuum, pure longitudinal waves appear only if the dimensions transverse to the propagation direction are big enough so that transverse contraction may largely be neglected.
Looking at a small volume section that positively maintains its dimensions, we observe a change in the mass of the medium in these sections. Conversely, observing a section of constant mass (shown in green and red, respectively, in the picture) we find that the density of the medium changes – due to the change in volume. In the picture, every section delimited by a dividing line oscillates back and forth sinusoidally; the centre of this oscillation remains fixed, however. A specific state (e.g. maximum compression) nevertheless migrates through the picture from left to right with a propagation velocity c. The distance between neighboring density maxima corresponds to the wavelength.
The angled black lines are there to facilitate recognizing the deflection – they are not part of the oscillating medium.
For further details, see chapter 1.4 and A.2 in Physics of the Electric Guitar
9. Standing longitudinal wave (MOV-09.mp4)
For the longitudinal wave, the particles of the medium oscillate in the direction of the propagation (or in the opposite direction). In a simple model we imagine small sections of mass connected by helical springs. All these sections oscillate towards the neighboring section or away from it. In a continuum, pure longitudinal waves appear only if the dimensions transverse to the propagation direction are big enough so that transverse contraction may largely be neglected.
Looking at small volume section that positively maintains its dimensions, we observe a change in the mass of the medium in these sections. Conversely, observing a section of constant mass (shown in green and red, respectively, in the picture) we find that the density of the medium changes – due to the change in volume. In the picture, every section delimited by a dividing line oscillates back and forth sinusoidally; the centre of this oscillation remains fixed, however. Contrary to the progressing longitudinal wave, we find for the standing longitudinal wave areas where the oscillation velocity equals zero (the so-called oscillation nodes), and between these nodes areas of maximum oscillation velocity (anti-nodes). In the oscillation nodes, the temporal change of density is at its maximum whereas in the anti-nodes the change of the density is zero. The distance between neighboring nodes corresponds to half the wavelength.
The angled black lines are there to facilitate recognizing the deflection – they are not part of the oscillating medium.
For further details, see chapter 1.4 and A.2 in Physics of the Electric Guitar
10. Propagating dilatational wave (MOV-10.mp4)
For the dilatational wave, the particles of the medium oscillate both in parallel and transverse to the direction of propagation. This embodiment of a wave therefore is a mixture of the longitudinal and the transversal wave. Dilatational waves appear in sold state materials with a small transversal dimension, for example in rods.
In the picture, each of the sections delimited by a blue line is both shifted horizontally, and stretched or compressed transversally. Even though the maxima (and the minima) of the deflection are running from left to right through the picture, the centre of the longitudinal movement remains stationary. Every particle on the surface of the rod moves in a circular fashion.
The distance between neighboring maxima of the transversal deflection corresponds to the wavelength.
For further details, see chapter 1.4 and A.2.2 in Physics of the Electric Guitar
11. Standing dilatational wave (MOV-11.mp4)
For the dilatational wave, the particles of the medium oscillate both in parallel and transverse to the direction of propagation. This embodiment of a wave therefore is a mixture of the longitudinal and the transversal wave. Dilatational waves appear in sold state materials with a small transversal dimension, for example in rods.
In the picture, some of the sections delimited by a blue line are predominantly shifted relative to each other shifted. Other sections experience predominantly a change in shape. A more precise specification of the terms “node” and “antinode” is therefore required: the longitudinal movement is smallest at the locations where the transverse movement is largest, and vice versa.
The maximum of the transverse deflection is suited to determine the wavelength: the distance between two neighboring maxima of this kind corresponds to the wavelength.
For further details, see chapter 2.2 and A.2.2 in Physics of the Electric Guitar
12. String vibration: 1st and 2nd harmonic (MOV-12.mp4)
Eigenmodes of an ideal string: 1st and 2nd partial
Every spring-mass-system capable of oscillation has a number (one or more) of natural frequencies (also called eigenmodes). As a response to an excitation, the system oscillates, either at a single eigenmode, or with a superposition of several eigenmodes.
In the top picture, we see the fundamental, i.e. the lowest possible eigenfrequency f, below it the eigenmode at twice the fundamental frequency (2f), and in the bottom picture the sum of the two oscillations.
The fundamental is also termed 1st harmonic and the 2f-oscillation is called 2nd harmonic or 1st overtone.
For further details see chapter A.1.4 in Physics of the Electric Guitar
13. String vibration: 1st and 3nd harmonic (MOV-13.mp4)
Eigenmodes of an ideal string: 1st and 3rd partial
Every spring-mass-system capable of oscillation has a number (one or more) of natural frequencies (also called eigenmodes). As a response to an excitation, the system oscillates, either at a single eigenmode, or with a superposition of several eigenmodes.
In the top picture, we see the fundamental, i.e. the lowest possible eigenfrequency f, under it the eigenmode at three times the fundamental frequency (3f), and in the bottom picture the sum of the two oscillations.
The fundamental is also termed 1st harmonic and the 3f-oscillation is called 3rd harmonic or 2nd overtone.
For further details, see chapter A.1.4 in Physics of the Electric Guitar
14. String vibration: impulse excitation (w/out dispersion), struck string (MOV-14.mp4)
The string is excited by an impulse in the shape of a half-wave sine (in the picture at the location of the black dot). From this point of excitation, transversal waves run in both directions along the string; they are reflected at both supports in phase opposition. As the two progressing waves meet, their displacements superimpose. In this idealization, the propagation velocity is assumed to be independent of the frequency (i.e. without dispersion).
The time function of the displacement of the blue dot (as a point on the string) is shown in the lower picture. The temporal derivative of this deflection (i.e. the vertical velocity) is the input value for the voltage-generating law of induction: the source-voltage of the magnetic pickup is proportional to the speed of the string in the vertical direction.
“Ortsfunktion der Saitenauslenkung” = function in space of the string displacement
“Zeitfunktion der Saitenauslenkung am blauen Punkt” = time function of the string displacement at the blue dot
For further details, see chapter 1.1 and 2.2 in Physics of the Electric Guitar
15. String vibration: step excitation (w/out dispersion), plucked string (MOV-15.mp4)
The string is plucked (triangular deflection) and released. Assuming that the propagation velocity is independent of frequency (i.e. neglecting any dispersion), the string oscillation may be represented as a superposition of two progressing triangular waves (chapter 2).
The lower picture shows the time function of the deflection of the blue dot (as a point on the string). The dot either is at rest or moves with constant velocity (in either direction). The temporal derivative of this displacement (i.e. the vertical velocity) is the input value for the voltage-generating law of induction: the source-voltage of the magnetic pickup is proportional to the speed of the string in the vertical direction.
“Ortsfunktion der Saitenauslenkung” = function in space of the string displacement
“Zeitfunktion der Saitenauslenkung am blauen Punkt” = time function of the string displacement at the blue dot
For further details, see chapter 1.1 and 5.10 in Physics of the Electric Guitar
16. String vibration: impulse excitation (w/dispersion), struck string (MOV-16.mp4)
Due to the unavoidable flexural rigidity of the string, the propagation velocity of the transversal waves is dependent on frequency – signal components of higher frequency propagate faster than those of lower frequency, and a short impulse becomes elongated already a few centimeters further down the string.
In the picture, the string is excited by a short impulse in the shape of a half-wave sine. The frequency dependent group velocity (dispersion) causes high-frequency components (i.e. short wave-lengths) to run faster than low-frequency components, and therefore the impulse is spread out. On a guitar string, the dispersive transversal wave would be reflected after about 65 cm – this is not shown in the picture. The dispersion has its biggest effect on the low E-string (E2): the thicker the string, the more the high- and low-frequency group velocities differ.
„Dispersive Impulswelle (Ortsfunktion)“ = dispersive impulse wave (function in space)
For further details, see chapter1.3 and 2.7 in Physics of the Electric Guitar
17. String vibration: step excitation (w/dispersion), plucked string (MOV-17.mp4)
Due to the unavoidable flexural rigidity of the string, the propagation velocity of the trans- versal waves is dependent on frequency – signal components of higher frequency propagate faster than those of lower frequency, and a short impulse becomes elongated already a few centimeters further down the string.
In the picture, the string is deflected and released (picked). For a non-dispersive and loss-free wave propagation, the string would swing back and forth always in the same way within the dashed rhomboid (see Fig. 1.2). With dispersion, the oscillation is deformed more and more with increasing duration of the string oscillation.
“Sattel” = nut; “Steg” = bridge
For further details, see chapter 1.3 and 2.7 in Physics of the Electric Guitar
18. String bounce: string lifted and released, w/out dispersion (MOV-18.mp4)
In this example, a string is fretted at the 2nd fret and at the same time it is lifted between fretboard and bridge with a plectrum. After the sting has slipped from the plectrum, it oscillates up and down while bouncing repeatedly against various frets (displacement exaggerated).
“2. Bund” = 2nd fret; “Steg” = bridge
For further details, see chapter 7.12.2 in Physics of the Electric Guitar
19. String bounce: string pressed down and released, w/out dispersion (MOV-19.mp4)
In this example, a string is fretted at the 2nd fret and at the same time it is pressed down between fretboard and bridge with a plectrum such that it comes into contact with the last fret (not an uncommon occurrence for thin strings). After the sting has slipped from the plectrum, it oscillates up and down while bouncing repeatedly against various frets (displacement exaggerated).
“2. Bund” = 2nd fret; “Steg” = bridge
For further details, see chapter 7.12.2 in Physics of the Electric Guitar
20. Magnetic flux for a single coil pickup
As the string oscillates, its distance to the permanent magnet varies. Shape and magnetic resistance of the magnetic field in the air gap change correspondingly. With the string approaching the magnet, the magnetic resistance decreases – and consequently the magnetic flux increases. Conversely, increasing the distance results in a reduction of the flux. Within the magnet, these variations are relatively small (about 1%) while they are relatively large within the string – which is fully magnetized into saturation.
The AC-component (i.e. the variation of the magnetic flux over time, dΦ/dt) induced an electrical voltage in the coil (about 50 mV up to more than 1 V).
Total magnetic flux:
AC-component of magnetic flux:
“Magnetische Flussdichte” = magnetic flux density; “Normierter magnetischer Wechselfluss” = normalized magnetic AC-flux.
For further details, see chapter 5.4 in Physics of the Electric Guitar
21. Magnetic flux for a hum bucking pickup
As the string oscillates, its distance to the permanent magnet varies. Shape and magnetic resistance of the magnetic field in the air gap change correspondingly. With the string approaching the magnet, the magnetic resistance decreases – and consequently the magnetic flux increases. Conversely, increasing the distance results in a reduction of the flux. Within the magnet, these variations are relatively small (about 1%) while they are relatively large within the string – which is fully magnetized into saturation.
The ac-component (i.e. the variation of the magnetic flux over time, dΦ/dt) induced an electrical voltage in the coil (about 50 mV up to more than 1 V).
Total magnetic flux:
AC-component of magnetic flux:
“Magnetische Flussdichte” = magnetic flux density; “Normierter magnetischer Wechselfluss” = normalized magnetic AC-flux.
For further details, see chapter 5.4 in Physics of the Electric Guitar
22. Law of induction
A time-variant magnetic field induces an electrical voltage U = dΦ / dt into a conductor loop; this voltage is dependent on the time-derivative of the magnetic flux Φ = BS, with B representing the magnetic flux-density, and S representing the surface area of the conductor loop. Basis for this relation is the 2nd Maxwell equation, which – for time-invariant surface area – may be simplified to U = dΦ /dt.
Whether the right part of the equation includes a plus or a minus sign depends on the directionalities defined for the electrical and magnetic quantities. In the animation, the right- hand connector of the wire winding becomes positive as the magnetic field (oriented along the direction of the arrow) increases with time.
The maximum of the induced voltage is not reached when the magnetic flux is greatest, but when its change is most pronounced. If we defined the variation of the magnetic flux over time as sin(ωt), the induced voltage changes according to d/dt [sin(ωt)] = ω⋅cos(ωt). In a guitar pickup the voltage induced per turn of the winding is very small and only the large number of turns (e.g. 10000) provides an output voltage of about 1 V.
For further details, see chapter 4.10 and 5.4 in Physics of the Electric Guitar
23. Lorentz’s equation, phase shifts in a loudspeaker
A current-carrying conductor placed in a magnetic field (blue arrows) is subjected to a force (black arrow) proportional to the length of the conductor, to the strength of the current (red), and to the strength of the magnetic field.
For a loudspeaker, the conductor is wound resulting in the voice coil. The magnetic field flows radially; the force direction is axial.
Using a simple approach, the loudspeaker membrane may be modeled by a mass, a spring and a friction resistance. The mass and the spring form a resonance system that is, in approximation, spring-inhibited (“Federhemmung”) below the resonance frequency, and mass-inhibited (“Massenhemmung”) above the resonance frequency. At resonance, the load is resistive (“Reibungshemmung”). From this load, which is different depending on the frequency, different phase shifts between current and movement result.
For further details, see chapters 4 and 11 in Physics of the Electric Guitar
24. Spherical wave, impulse and sinusoidal
In an inviscid, infinitely extending fluid (gas or liquid), longitudinal waves propagate from locally limited excitations. The spherical excitation often used to model this scenario results in concentric waves. The direction of the movement of the matter particles is radial i.e. directed away from the center, or towards the centre, respectively (strongly exaggerated in the animation). The waves travel with the velocity of sound c, which in air equals about 344 m/s. It is necessary to distinguish between sound particle velocity v and the speed of sound c: v is the speed each matter particle oscillates with. At usual sound power, v is much smaller than c. In the case of periodic oscillations, the spatial period (wavelength λ) is proportional to the period T in time: λ = cT.
Spherical impulse wave:
Spherical sine wave:
For further details, see chapters 1 and 2 in Physics of the Electric Guitar
25. Partial vibration of a loudspeaker membrane
In the simple model, the membrane of the loudspeaker is taken to be a rigid disk oscillating in the axial direction. However, at higher frequencies the individual points of the membrane do in fact not move with the same amplitude and phase – rather, a complicated movement dependent on the location on the membrane establishes itself.
The simplest oscillation mode is the so-called fundamental mode (01-mode) – it is the shape of the lowest-frequency natural vibration exhibiting no nodal lines. For the 11-mode one single diametrical nodal line appears, with all points on this line remaining at rest; the 21- mode shows two diametrical nodal lines. The actual membrane oscillation results from a superposition of several oscillation shapes.
11 mode:
21 mode:
For further details, see chapter 11.3 in Physics of the Electric Guitar
26. Sub-harmonic vibrations of a loudspeaker membrane
The ideal loudspeaker membrane is rigid. Conversely, the individual points on a real membrane oscillate with various (location-dependent) speed and standing waves with nodal lines (partial oscillations) manifest themselves. The differential equation describing these occurrences includes level-dependent stiffness, i.e. the oscillation system is non-linear. Driving the membrane with sinusoidal excitation at the frequency f may – as a consequence of this non-linearity – result in membrane deformations at a fraction of the excitation frequency (f/2, f/3, …). These distortions are termed sub-harmonics.
For further details, see chapter 11 in Physics of the Electric Guitar
27. Image source: reflection of a spherical wave
As a sound-wave hits a wall, it is reflected to a large extent. There are similarities to the reflection of light: 1) angle of incidence = angle of reflection, 2) incoming and outgoing sound ray as well as the normal of the wall all are in one and the same plane.
Instead of the reflection we may (as a model) also let the incoming wave continue to travel into the wall without reflection, and add another (mirror-) wave exiting the wall (red in the picture).
For further details, see chapter 11 in Physics of the Electric Guitar
28. Diffraction at a baffle
A. The membrane radiates sound only towards the front.
A loudspeaker is mounted in the middle of a finite baffle. The rear of the speaker is hermetically sealed such that it can radiate sound only in one direction. Given an impulse excitation, an impulse wave propagates spherically until it reaches the ends of the baffle. Here, part of the wave is reflected with a phase of 180°, another part is diffracted around the end of the baffle and penetrates the space behind the membrane.
B. The membrane radiates sound towards both the front and the rear.
A loudspeaker is mounted in the middle of a finite baffle. Given an impulse excitation, a positive pressure is generated on one side of the baffle and a negative pressure on the other side. Both impulse waves propagate spherically until they reach the ends of the baffle. Here, they are diffracted around the end of the baffle.
The diffraction results in complex attenuation- and phase-frequency-responses.
For further details, see chapter 11 in Physics of the Electric Guitar
29. Travelling waves inside the cochlea
As sound is received by the ear, a travelling wave propagates on the basilar membrane within the cochlea. Depending on the frequency, local maxima of deflection are created: for high frequencies at the basal end, for low frequencies at the apical end of the basilar membrane. This deflection bends tiny sensory hairs (stereocilia), triggering hair cells associated with the sensory hairs to send nerve impulses to the brain via the auditory nerve.
OW: Oval window; Hel: Helicotrema – respective ends of the cochlea
For further details: Fastl/Zwicker, Psychoacoustics, Springer, 2007.
30. Thermionic emission from oxide-coated cathodes
In conductive metals, the metal ions are placed in regular lattices; in between move the free electrons. The direction of this movement is at random and may carry an electron even a small distance outside of the metal. Electrical field forces appear pulling the electrons back. As temperature rises, the average speed of the electrons also grows and – given appropriate environmental conditions – individual electrons are able to leave the metal. An external electrical field (between cathode and anode) generates additional field forces and draws the electrons away from the cathode. The electrons flow from cathode to anode i.e. with the technical current direction: the current flows from the anode to the cathode.
Cold cathode:
Hot cathode:
Hot cathode w/U+ (voltage at the anode):
For further details, see chapter 10.1 in Physics of the Electric Guitar, as well as K. Küpfmüller: Einführung in die theoretische Elektrotechnik, Springer 1967.
© M. Zollner 2002 – 2014
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