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Acoustics Part 2




Initial phase of the signal


This is the position of the body from which it starts its oscillation.
It is measured in degrees.
If the body has begun fluctuation from equilibrium position, its initial phase is 0.
If we deviate it in extremely positive attitude and let him go, the phase is 90.
If two strings, membranes, pendulums or air columns begin their movement with a temporary delay to each other, we see a shift in phase between them.


 If it is a quarter period, the shift is 90 degrees, if a half – 180, if in 3 / 4-270 if it is entire period – 360 degrees.

Signal polarity


     We should not confuse the terms phase and polarity. Polarity defined whether the signal is positive or negative, ie whether it is above or below the x-line on the graph with sinusoidal or other periodic signal.
     In the context of the sound waves we can say that the compressions are with right polarity and the rarefactions – with reverse polarity, because they transform into a positive or negative voltage.


     Here we see a reversal of signal polarity. This is a very simple procedure from mathematical or electrical standpoint.
     Phase change on the other hand is a time function, not a function of polarity. The fact that in this situation, the polarity is changed frequently, it is a reason for the replacement of one with another term, which is also common.

    Here is a phase change of the signal. As we know, it is measured with degrees of deviation, but we can also describe it with millisecond delay of one signal to the other:


    We see the main signal together with 90 degrees, 180 degrees and 270 degrees phase shifts copies thereof.




     At different frequencies, however, certain phase shift in milliseconds of delay leads to a different degree deviation. This is the cause of many specificities in combining phases in the propagation of sound.
    For example, so-called. standing waves in the rooms are frequency dependent and are therefore a manifestation of the nonlinearity of the rooms. At the same time deviation resultant level of the transmitted and reflected sound is different depending on its frequency.
      The second peculiarity is that almost all waves in nature are complex. In this case the reversal of the phase (in degrees) with a level of 180 degrees does not automatically lead to a reversing of the polarity upside down, as in the sine wave or in another one that is absolutely symmetric about the front of ascent towards the front of the drop:


Polarity inversion











    As we see in complex waveforms phase shift can not be called a “reversal” of phase because it does not lead to reverse polarity.
   For this reason “phase inverse” button in the mixers and pre-amp units is actually a button which is used for electrical reversing the polarity of the input signal.

Free and forced vibrations. RESONANCE



    When we deflect agile body from its equilibrium position and release him, it starts to execute oscillations with a frequency, witch is a function of its mass and agile force.
These are called free vibrations of the body.
   The frequency of the oscillation is called its natural frequency or the resonance frequency and is measured by the formula:

sobstvena chestota

where k is the hardness and m – the mass of the body.
    A body can have more than one natural frequency
     The natural frequency is not a function of the amplitude, it changes only insofar as for example the tension of the string changes slightly with changing amplitude and thus the agile force. But in general, we can conclude that its own frequency is intrinsic to the agile body.
     When this body affects other elastic body, it gives its vibrations in the form of mechanical energy. The vibrations of the second body are called forced oscillations.


A typical example of this process is when we put a tuning fork on the table after we have produced vibration with it:
In this case we say that the table performs forced vibrations. 


     How to combine this with the natural frequency of the second body, which carries out the forced vibrations?
     At first it tries to vibrate with its own frequency. But in the process of forcing him from the first body it begins to oscillate with its frequency until such time as we have force, causing it to carry out these oscillations.
    Once this force stops its operation, the vibrations fade by passing back through a time segment with a natural frequency.
    This is one of the reasons why the transient processes in the begin an in the end of the sound differ from the stationary timbre of a musical instrument or other vibrant body.
    What happens when the natural frequency of the first body is close or equal to that of the second?

Then we observe the phenomenon of resonance.

     When we have a vibrating system that performs forced vibrations under the influence of an external force, we observe a sharp increase in the amplitude of the oscillation frequency in and around the area of ??the natural frequency of this system.
This increase in amplitude is called resonance.
     The principle of the swing is an example of combining the two frequencies:


    When we try to push the swing forcibly at a frequency other than its own, we need to apply much effort. But if we use the natural frequency of the pendulum, we can limit ourselves with a force equal to the loss from friction and air resistance. The rest of the energy is stored in the system.
    The decay rate in the system determines how large increase in volume will result from resonance.Characteristic of this resonance is that it exists mainly in the point of overlap between the two frequencies – the feed rate on the one hand, and 
that of the body carrying out the forced oscillations of the other.
    However, we have also a partial resonance at frequencies close to each other. The closer they are, the resonance is more pronounced:
     If a system has zero damping, the increase caused by the resonance would increase logarithmically to infinity, scoring the system, if it is a sound one, in a feedback.
      In practice there is always a damping. 
      In systems with a small damping the resonance curve is higher and more narrow, i.e. it has a lower bandwidth.
     In contrast, in systems with a high damping, they are less susceptible to resonance, but have a wider curve of  influence from frequencies that are adjacent to its own:Screen Shot 2014-03-16 at 5.47.45 PM











Different q-factor in resonant systems with different attenuation.


     Resonance phenomenon has great application in the music practice. On it are based many cases, when you enhance and enrich the sound of a musical instrument. In some string instruments there are used some resonator strings, witch are enriching and reinforcing the fundamental.
     Corpuses of the musical instruments are also resonator boxes. In such cases, we seek a low selectivity, allowing more uniform resonance at larger frequency range.
     The human vocal apparatus is also a complex resonator system. The human body is full of microcavities concentrated in the chest and in the head, which help to enhance and enrich the voice emitted by the vocal apparatus. They thus establish the uniqueness of the the timbre of each individual.
      The resonance has downsides also. For example, in the field of architecture applying a relatively small force, but periodic and in the fundamental frequency of a building structure may result in many problems including it’s destruction. For this reason, for example it is forbidden to pass military infantry marched on bridges because accidentally coincidence with their own frequency can lead to unpredictable consequences.
    Resonance is avoided in the design of the speakers because it increases dramatically their frequency nonlinearity. For this reason, the goal in their design is to look for a frequency of
the loudspeaker, witch is lower than the lowest frequency it can reproduce. This is achieved by increasing the weight (and the mass) of the speaker.
    However, this is viable only for relatively large and heavy speakers. In other cases the aim is to their higher internal damping to completely reduce the tendency of the acoustic body to work as a resonator.


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