Periodic phenomena

RECALL OF MOVEMENT SINUSOIDAL

                 s = a * sin (w*t + j)

s Þ elongation

a Þ amplitude

w*t + j Þ phase

j Þ dephasing or phase in the beginning

        w Þ  pulsation

w = 2p*f   Þ f   Þ  fréquence  (in hertz)

T = 1 / f  Þ period (in secondes)

s = a*sin(2p* t / T + j)

The periodical movement is the projection on an axis of a uniform circular movement, to angular speed w .

PERIODIC MOVEMENT UNSPECIFIED

A periodic movement is not necessarily sinusoidal. So that a movement be periodic, it suffices that the movable one rediscovers itself to the same place with the same speed to instants separated by a constant time interval T called the period.

If s = F(t)      s = F(t + T)        alors         F(t) = F(t + T)  

Harmonic FOURIER theorem

This therem establishes that a periodic function of T period is the sum of periodic functions of T periods, T/2, T/3 ....

It returns as equal to say that if the sinusoidal function to a f frequency, one can bring back it to the sum of periodic functions of f ... frequencies, 2F,, kf (entire k). In acoustics it is preferable to speak of frequencies that give the height of the sound. The harmonic analysis allows to calculate the different sinusoidal functions that compose a periodic function. A device called analyseur of frequencies, allows this experimental decomposition. The sinusodal function of same f frequency that the periodic function is said basic, the others of frequencies ...2F, 3F are said harmonic.

EBRANLEMENT AND TRANSVERSE VIBRATION

Experience: attach an elastic wire to a fixed point. The wire is not eeither soft or rigid completely. With a stick give a brief blow on the wire in a point A (placed in a known distance of the bridge sets up) in order to provoke a deformation. This part of the wire is secured to the near parts of an elastic manner; the near parts will be located inducing they also towards the bottom, but with a certain delay: the initial deformation is propagated itself alongside the wire. It imports to notice that it did not have there matter transportation alongside the wire: each not at all equipment such that m vibrated on the spot; only the deformation propagated itself in the direction of the wire.

The vibration is transverse for the matter vibrates perpendicular to the propagation direction.

PROPAGATION SPEED

If during the t time the deformation propagated itself x's distance one will call V propagation speed:

V = x / t

This speed is constant all alongside the wire: it does not depend on form or of the magnitude of the initial shock, it depends only of the nature of the environment.

The propagation movement of a vibratoire movement is uniform. One notes that: - speed grows with the tension of the rope

- speed diminishes if the mass of the rope by length unity is bigger

If F is the tension of the rope (force tension), m his linear mass, V is given by:

V = Ö ( F / m)

Exercise: a rubber rope stretched by a weight of 10 N to a length of 2M and a mass of 50G. Calculate V.

V =Ö ( F / m) = Ö (10*2 / 50exp(-3)) = Ö 2000/5 = Ö 400 = 20m/s

CASES OF A WAVE TRAIN

If one replaces the stick by a vibratoire source (diapason, vibreur ...) one produces a deformation succession. One observes then a wave train that propagates itself alongside the rope with V speed. A point of the rope begins vibrating during the passage of the wave train then returns to the rest.

MAINTAINED VIBRATION; WAVE LENGTH

Let us suppose that the vibratoire source is animated of a periodical movement of  T period, and of AB amplitude = a . All the preceding results apply again. One observes to the t times, t + T/4, t + T/2, t + 3T/4, t + T. During this T interval times, the vibration propagated itself of a certain length l , called wave length:

l = V * T

or      l = V / f

The wave length is a characteristic magnitude of the movement that depends on source and environment.

Two m and m'  points , distant of a wave length, have at each instant, same elongation and same V speed: they vibrate in phase. The wave length characterizes thus the periodicity of the phenomenon in the space.

All that we have just signaled applies to a unspecified periodic movement.

Exercise: An extremity of the rope in rubber of the preceding exercise is animated of a periodic movement of frequency 20HZ. Which is the wave length?

V = 20 m/s                      l = V / f = 20/20 = 1m.

SHOCK AND LONGITUDINAL VIBRATION

Experience: Stretch by hand a long hung ressort to the wall by his other extremity. Whith the other hand let us compress a number of spires and let us loosen them abruptly. They stretch resumes it balance position while relaxing; but in this movement they compress the nearest spires; the compression zone is itself therefore propagated.

This new compression zone plays the same role that the first screw-to-live following spires and so on of near one in near.

It imports to notice that it some would be of even if one initially had separated the spires instead of to compress them: a depression zone would have propagated itself. At the same time the spires budged to the neighborhood of their balance positions. There was propagation of a deformation of the ressort in the direction of the propagation.

A vibration is longitudinal when the matter vibrates in the direction of the propagation.

Propagation speed.

If during the t time the deformation propagated itself x length one will call V propagation speed:

V = x / t

This speed is constant alongside the ressort; it does not depend on magnitude of the initial deformation but only depend on the nature of the environment.

CASES OF A FLUID

The preceding experience gives the picture of what would happen in an elastic fluid, for example a contained air column in a pipe: each air slice plays the same role that a whorl of a spring

In the case of a gas, V speed is given by:

V = Ö (g* p / r)

g is a coefficient depending on nature of the gas

p is the pressure

r is the gaz density

The product gp measures the elasticity of the gaseous environment.

Exercise: Calculate the speed of waves in the air to 0°C and under the normal pressure p=1atm; g = 1,4.

T=273K R=8 , 3J/K M=29

PV = RT for a mole. M is the Molaire mass.

V = Ö (g* p / r) = Ö (g* RT / M)= 330 m/s

Wave trains and maintained waves

All what precedes applies in the case of wave train or in the case of maintained waves.

If the vibratoire source vibrates longitudinally to the frequency f = 1 / T and of amplitude a = AB, the fluid (or the ressort) will be traversed by an alternated continuation of condensed half-waves and of dilated half-waves.

Wave surfaces

In an air tube or of gasses admitted us that we could envisage slicees glide air in vibration. One says that the vibratoire movement propagates itself by waves glide.

One calls surface of waves the body of points attained by the vibraton to the same instant.

Seems free and either in a tube the vibratoire movement propagates itself by centered spherical waves on the surface vibrating (sonorous source for example).

VIBRATORY STATE OF AN UNSPECIFIED POINT

That is to say the sinusoidal vibration being propagated by plane waves in the Sx direction starting from the source S, whose vibration is given by:

                         V ®

---|-----------------------------------|------------------------®  x

    S                                                   A

s = asin 2pt/T

That is to say A a point located at a distance X of S when A and S are at rest. The vibration produced in S spends a time

q = x / V To traverse SA. The L'élongation in A at the t time is therefore the even that the l'élongation of the source to the time

t -q = t - x / V    

Sa = asin 2pt/T(t - x / V) = asin 2p (t / T - x / l)

because        l= VT

The point A has therefore the same movement that the source but with a certain delay translates by a déphasage

2p*x/l .

Thus the points that vibrate in phase with the source are such that:

2p*x/l = 2kp          x = kl    k natural

They are therefore situated to multiples distances of the wave length. The points that vibrate in phase opposition with the source are such that:

2p*x/l = p + 2kp        x = (2k+1)*l/2

x is odd multiple of the half-length wave.

If one took an instantaneous photograph of a rope vibrating to the t time, the photograph would give the form of the rope Sa= F(x) to this instant. The formula Sa = asin (2pt/T-2px / l) shows that for fixed t the rope has the form of a sinusoïde. The wave length l play live-to-live x's abscissa in the space the same role that the T period lives-to-live t time.

When the time flows itself, this sinusoïde seems to advance in pad with V speed: the amplitude propagates itself; one says that V is a phase speed.

PRESSURE VARIATION

Consider two voisines sections A and B of a gaseous column to the rest of x abscissas and x + AB in comparison with the source. When the gas vibrates, A and in A' to the t time, and B in B'.

Sa=AA'=asin (2pt/T-2px / l)

Sa=BB'=asin2p(t/T-(x+AB) / l)

The AB thickness of the gasses slice became A'B' :

A'B' = AB - AA' + BB'

Be A'B' = AB - asin (2pt/T-2px / l) + asin2p(t/T-(x+AB) / l)

or A'B' = AB - 2asin (pAB / l)*cos2p(t /T-(x+AB/2) / l)

Suppose that the AB thickness be very small in front of l one then can confuse the sinus with the angle and neglect AB/2 in front of x :

A'B' = AB - 2apAB/l * cos2p(t /T - x / l)

If one designates by V the volume of the AB slice and by V' the one of A'B' one has:

(V'-V) / V = (A'B' - AB) / AB = -2ap/l * cos2p(t /T - x / l)

But according to thermodynamics, to a dimintion -dV of the volume corresponds a dp increase pressure of the gas such that:

dp/p = -g dV/V

dp/p = g * 2ap/l * cos2p(t /T - x / l)

It is seen that the variation of pressure is a sinusoidal function of time, but in squaring with the elongation. The pressure is maximum when the elongation is null, and minimum when the elongation is maximum.

WAVE REFLEXION

Transverse waves

Reflexion on a fixed obstacle indeformable

Let us take again the experiment on the propagation of a transverse deformation along a tended rubber wire. Rubber being fixed at the wall one notes that the deformation return towards the obsevator: it is reflected by the wall where is a fixed point of the cord. But the deformation with the return is in contrary direction of the deformation to the outward journey. In addition, the propagation velocity is always V =Ö ( F / m).

In a reflexion on an indeformable fixed obstacle, there is change of sign of the elongations: speed, except for the sign, is preserved. Reflexion on a deformable medium Let us let hang a rubber cord vertically and cause a deformation there: this one is propagated to the bottom at the speed V; arrival with the bottom of the cord it makes some draw aside the end: all occurs then as if at this place one had produced a deformation which goes back with same speed as to the incidence without change of sign.

Logitudinal waves

Reflection on a stationary obstacle indeformable

Reflexion on an indeformable fixed obstacle Let us consider an oar of coaches A, B, C, D in front of a stop and suppose that coach A receives an impulse towards the line. The springs of the buffers between A and B are compressed, the coach B is pushed back towards the line however which A takes again its place; in its turn it pushes back the coach C and so on until D is thorough on the stop. But the stop is fixed; under the action of the springs thus compressed D is pushed back on the left, it pushes back C which pushes back B and so on. It is seen that with the outward journey displacements of the coaches are done from left to right, and with the return of right-hand side on the left. But they are always compressions of the springs which intervene and not of the tensions. In a reflexion on an indeformable fixed obstacle, there is change of the sign of the elongations; speed, except for the sign, is preserved. If with the incidence a compression is propagated, it is still a compression which is propagated after reflexion (in the same way for a dilation). Reflexion on a deformable medium Let us take again the whole of an oar of coaches but without stop. If coach A receives an impulse, it pushes back B of left on the right, which pushes back C, etc always by compressions of the springs. D moves of left on the right but does not meet a stop; its movement is not constrained. Launched on the line it draws from left with drote C which draws B and so on. As well with the incidence as after the reflexion all the coaches moved in the same direction; but with the incidence there was compression of the springs, with the reflexion extension of the springs. In a reflexion on a deformable medium, there is no change of the sign of the elongations; speed, except for the sign, is consevée. If, with the incidence, a compression is propagated, it is a dilation which is propagated after reflexion, and conversely. This result applies to the whorls of a spring whose end is free, or to the sections of air contained in an open pipe.

SYSTEMS OF STANDING

Interferences of incidental and reflected vibrations

Be a maintained vibrations source of which the movement equation is:

s = asin2p t / T

The waves which are propagated starting from this source in a given direction meet an obstacle, or mirror R, at a distance D of the source. In a point P of the medium, at a distance X of R, will superimpose the incidental vibrations and the considered vibrations. Let us study these phenomena of interferences. 1st case: R is an indeformable fixed obstacle. In a point P whose distance to the source is D - X the incidental movement is:

 

si = asin2p( t / T - (d - x)/l)

-----------½-----------------½--------------½

                 S                          P                     R

                  <- - - - - - - - - - d - - - - - - - - > 

                                              <- - - -x- - - ->

In R the vibration is reflected while changing sign and traversed 2x before returning out of P where it has as a value:

sr = -asin2p( t / T - (d - x )/l - 2x / l) = -asin2p( t / T - (d+x)/ l)

The vibration out of P resulting from the interference of incidental and considered is:

s = si + sr = 2asin2px/l *cos 2p( t / T - d/ l)

s = Acos 2p( t / T - d/ l)

It is there the equation of a sinusoidal movement of amplitude A = 2a*sin 2px/ l variable with the position of the P point.

It is seen that this amplitude is constantly null for the points P such as:

sin2px/l = 0     t   2px/l = kp

   x = kl/2   entire k

These points are called nodes of vibration. Two consecutive nodes are distant of a half-length of wave. It will be noted that there is a node of vibration on the fixed obstacle in X = 0. The amplitude will be extreme and have as a value  2A for:

sin (2px's/l) = ± 1

sin(2px/l) = ± 1

     2px/l = p/2 + kp     or     x = (2k + 1)l/4

These half-points are called bellies of vibration. Two belly consecutive are distant of a half-length of wave. Let us see the aspect of the medium in the case of a vibrating cord attached to a fixed point R. One observes a succession of spindles, separated by nodes N; the broadest parts are the bellies V One should not confuse this phenomenon with that which corresponds to the propagation of a single wave. The resulting wave that we have just studied does not propagate; it remains on the spot. One has a phenomenon of standing waves characterized by the following points: 1° All the points located between two nodes vibrate in phase. 2° the points located on both sides of a node vibrate in opposition of phase. One sees these property on:

s = 2asin2px/l *cos 2p( t / T - d/ l)

In the vibratory term cos 2p( t / T - d/ l)  position x of the point P does not appear: the phase remains fixed except when sin2px/l change of sign what occurs with each node. 3° the amplitude varies from a point to another of 0 with the nodes with 2x with the bellies, from where the sinusoidal aspect taken by the cord at a given moment.

2 case: R is a déformable environment

Raisonnemant is the same one but there is no change of sign in R at the time of the reflexion. There will be this time:

s = si + sr = 2asin2p(t/T - (d-x)/ l) + asin2p(t/T - (d+x)/ l)

 s = 2acos(2px/l) * sin 2p( t / T - d/ l)

The node's position is defined by:

cos(2px/l) = 0          Þ    2px/l = kp + p/2

or x = (2k + 1)l/4

The one of the bellies by:

cos(2px/l) = ± 1      Þ    2px/l = kp

x = kl / 2

The results thus are reversed exactly compared to the preceding case. There is a belly of vibration at the place of the reflexion.

Conditions to the limits

That is to say a cord fixed at its two ends (piano, violin...); if one puts it in vibration it will produce a system of standing waves if each end corresponds to a node i.e. if the length of the cord contains an integer of spindles. If F is the tension of the cord V =Ö ( F / m)      l = V * T  =  T*Ö ( F / m)

l = Kl/2 = KT/2 * Ö ( F / m)

T is given,  as such as m . F tension must satisfy the condition on  l .

Exercise

A cord fixed at its two ends vibrates by giving two spindles. How is it necessary to modify F to have three spindles? (the period T is the same one).

l = 2 l / 2 = l

be l' the new wave length such that l = 3 l' / 2

l' / l =( T Ö ( F' / m) ) / ( T Ö ( F / m)) = Ö (F' / F)             F' / F = 4/9

F' = 4/9 F  < F

FAMILIAR EXAMPLES

A bell which oscillates is a resonator if the bell ringer requests it periodically, the period being the clean period of the bell which behaves like a made up pendulum. If the bell ringer draws on the cord with one period much smaller the leaf strikes the bell à each period. It still oscillates but with a low amplitude. It is said that it undergoes a forced oscillation. Other familiar example: the swing. A resonator is put in vibration if the exiting system has even period that the clean period of the resonator

DAMPING

If one places far from resonance, one notes that the system diminishes. If damping is weak, the amplitude of the oscillations presents an acute maximum for the frequency of resonance. If damping is significant, the maximum widens and becomes badly defined. Resonance is all the more selective since damping is weaker.

RRESONANCE IN ACOUSTICS

Let us make vibrate a tuning fork by pressing its stem on a table: the sound is reinforced. To reinforce the sound of a tuning fork one fixes this one on a case of resonance which is not different than a sound pipe closed at an end; of calculated size so that the condition of resonance is satisfied. The violin is a case of resonance on which are tended the cords; it is not necessary that the case reinforces a note much more than the others. One carries out what is called a close connection by a wood stem, called the heart, tightened between the two walls of the case under the rest. It is still in the same way that résone a sounding board of a piano. These phenomena of resonance have a gravitational role: a slightly désacordé violin will play just in the presence of the other violins thanks to a phenomenon of resonance. The loudspeakers, the tympanum are very deadened resonators what enables them to vibrate regularly in a wide range of frequencies (resonance fuzzy).

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