A short introduction to the sound spectrum

• The sound spectrum comprises the part of the oscillations with audible frequencies. Which means, the frequencies between 20 Hz and 20.000 Hz.
• The sinus is the most simple tone. All other tones can be decomposed into a sum of different sinusses.
• We can use this web-app to generate a sine wave , with the possibility to select the desired frequency.

Vibrating objects

• The properties of a materials, the geometric shape and the contact with the surrounding materials gives the object specific acoustic properties.
• The geometric properties of a membrane (or a thin plate) result in particular properties. The tense, flat and thin properties allow good sound propagation
• [Demonstrate the membrane, pinching it to show that it produces a sound.]

[ Russel, D ]

Forced vibrations

• An object that is subjected to an acoustic signal, will vibrate along with that signal, using the same spectral content, but the sinusoids can be modified in timing or in amplitude because the object is imposing its acoustic character onto it.
• Objects with big surfaces, such as membranes or plates have a great contact surface area where acoustic energy can be transferred, so they are easily influenced by sound waves in the air.

Traveling waves

• Because a typical membrane or plate is constant in thickness and material composition, the vibrations that go through it are constant in speed.
• The length of an entire up-and-down cycle of a vibration on the membrane is called the wavelength:

[ Vinay Prajapati - Feb 16, 2020 ]

• Because higher frequencies go faster up-and-down and the speed in the membrane is constant, these vibrations produce shorter wavelengths.

Reflecting waves

• Traveling waves that reach the extremities of the membrane are bouced back, because the energy must go somewhere!
• This means that a new traveling wave is produced, running in the opposite direction.
  • [See (part of) video "Similarities of wave behaviour"]

Standing waves

• There can be several traveling waves at once traveling through the same membrane. However, each section of the membrane can only be in one place!
  • The membrane will therefore take the shape of the sum of all traveling waves going through it.
    • [See  animation explaining how two traveling waves combined become superposed ]
  • What happens with two traveling waves going in opposite direction?
    • See animation showing combined traveling waves that create a standing wave:

 
[ Wolfgang Christian and Francisco Esquembre CC BY-SA 4.0, via Wikimedia Commons ]

(copy: https://youtu.be/zGDsZ737PE0 )

• When looking at the sum of a traveling wave with its reflection, this gives…
  • [See animation demonstrating how this transforms the waves into a standing wave

 [ Dan Russel, 2011 CC BY-NC-ND ]

(copy: https://youtu.be/ivrA7X1nxPE )]

• Certain locations on the membrane (the vibration antinodes) are moving maximally up-and-down, while other locations (the vibration nodes) are (in practice: almost) stationary.
• (In Dutch we say "knots" and "bellies" for "nodes" and "antinodes" - which is a nice way to imagine it!)

[ Los Senderos Studio, LLC ]

• On a membrane, a similar thing happens, but in two dimensions. This results in a two-dimensional pattern.

Vibration modes

• The wave reflections are occurring at each extremity of the membrane, so that the membrane will be full of waves!

[ (Creating musical sounds). An OpenLearn chunk used/reworked by permission of The Open University copyright © (2020) ]

• When the wavelength of the vibration in the membrane, or a multiple of it, is in a particular mathematical relationship with a characteristic dimension of the membrane (such as its diameter)
  • all the traveling waves will have a "helping" effect on each other, they will be synchronised.
  • This is called a "vibration mode" of the membrane.
  • The mode occurs at a particular "resonance frequency".
  • For other frequencies, there may not be a vibration mode, so the membrane will be less responsive.
• [See  video of the vibration modes in a rubber band: ]
• Something similar happens in a membrane or a plate, but in 2D.
  • [Check the  "Vibrational Modes of a Circular Membrane" website ]
  • For sound frequencies, which are much faster, the up-and-down movement of the membrane is very small, so it is not really visible like this!

Visualisations

• Chladni patterns
  • There is the well-known 'Chladni patterns' experiment to visualise the vibration modes.
    • It can be done by putting salt on a horizontally installed membrane.
    • The salt will escape the strongly up-and-down moving locations (the antinodes) and it will settle at the nodes of the vibration, which therefore makes the nodes visible!
    • [Show Chladni patterns]
      • Show video of membrane with sand, excited by voice and whistle
      • Demonstrate on the balloon membrane with salt, using the tone generator app (the sound needs to be strong enough).
• Vibration-reflected laser
  • Using a directional light source such as a laser, pointed onto a mirror fixed on the membrane, is another way to visualise the vibrations in the membrane.
    • Contrary to the Chladni experiment, this method will visualise what is happening at a single point on the membrane.
      • (However, we will know more of the details about what's happening at that point compared to the Chladni experiment.)
    • Here, it is important to realise what is going to be visualised.
      • The tiny up and down movements of the mirror are very small, so they will only move the laser point a tiny little bit, which will not really produce a visible effect.
        
      • It is rather the little mirror rotations that will deflect the laser beam and produce the patterns that we see. Indeed, a tiny rotation can produce a big effect when the distance between the mirror and the projection surface is great.
        
        • [Invite people to press the membrane to produce a translation, and see the effect on the wall. Compare to what happens when producing mirror rotations.]
      • The rotation over the horizontal axis will create a vertical line on the wall, and vice versa.
        
        
    • So, when we think again about those vibration modes
      • The strongest twist motion happen
        • on one hand at the resonance frequencies
        • and on the other hand at the node location in the membrane
      • Depending on where the mirror is located and which mode is being excited, the 1st and 2nd twist movement will be different in amplitude, and phase.
        • [Show with my hand (mimicing the mirror) in which ways a mirror can be moving, depending on where it is located on a vibration mode.]

Lissajous patterns

• For a complex sound it is very difficult to find out how and why a piece of the membrane will react, because many vibration modes are being excited together.
  • (With the Chladni experiment, such sounds would also not produce any particular geometric pattern, as the salt grains won't find any place where they can rest.)
• A simple sound like a sinusoid is more easy to study.
  • When a sine wave is sent to the membrane, only the most nearby vibration mode of the membrane will be excited.
  • The two twist movements of the mirror have the same frequency, the frequency of the sinus, but as we've seen earlier, they can be different in amplitude and phase.
    • This can result in a range of oval shapes, from a circle to a diagonal oval that theoretically could go into a line.
    • [Do the experiment]
      • and/or invite them to do this, using the sine generator website.
    • Even with the same sound, the result is mostly different, because the vibration modes and mirror position are easily very different.
    • Non-oval shapes occur due to
      • The wall not being perpendicular to the laser beam projected onto it
      • The laser could be pointed on the border of the mirror
      • The vibrations can be too strong, so that some vibrating structures are saturating (which is called 'nonlinear' behaviour)

Pattern amplitudes

• The sound source is a primary factor that influences the amplitude of the projected pattern.
  • Loud sounds clearly produce a greater pattern than soft sounds.
• Secondly, the question is how much of that sound energy is being transferred to the membrane.
  • In the case of the low-res version, the distance between the sound source and the mebrane plays a role.
  • In the case of the high-res version, this is constant: the speaker is mounted onto the plate.
• Another parameter is how reactive the membrane is, which we have explained earlier : the optimal properties of a membrane make it very resonant.
  • If we for instance damp the membrane with a cloth, this takes away much of the energy in the membrane, decreasing its vibration amplitude.
• Next, the question is: how strongly is the mirror swinging?
  • As we have seen, standing waves in a membrane have vibration nodes and antinodes. This means that certain locations on the membrane are vibrating stronger than others. So, even when the membrane is vibraing strongly, it could be that the mirror is located on or near an antinode, where the membrane is not rotating much.
  • Also, higher frequencies require more energy to get the same amplitudes (because of the mass of the membrane). So this also explains our observation that the patterns become smaller and smaller when choosing higher frequencies.
• Finally, the pattern of a beam that is reflected on a fluctuating rotation will become bigger the greater the distance between the mirror and the wall.

Pattern saturation

• When the amplitude of the source goes from soft to loud, with the same sound, the shape should be the same while its amplitude simply increases.
• When this is not anymore the case, there are 'nonlinear' physical processes occurring, as we've mentioned earlier.

Downloads

• Presenter notes (rtf)
• Audience-shared notes (rtf)
• Presentation slides (pdf)