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Opus in profectus

Standing Waves

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Practice

practice problem 1

Tuning forks
note pitch (Hz)
C4 256
D4 288
E4 320
F4 341
G4 384
A4 427
B4 480
C5 512
D5 576
E5 640
F5 682
G5 768
A 1.00 m vertical tube is partially submerged in water. The height of the tube above the water can be adjusted to any value from 0 m (the tube is completely submerged) to 1 m (the bottom of the tube is just touching the top of the water). A vibrating tuning fork is held just over the open top end of the tube. Pick one of the notes from the table on the right and answer the following questions.
  1. Determine the wavelength of the sound emitted from the tuning fork in air at room temperature (vsound = 343 m/s).
  2. How long is ¼, ½, ¾, 1¼, 1½, 1¾ of the wavelength you calculated in part a.?
  3. At what heights will resonance occur? (Just highlight the answers from part b. that satisfy this condition.)

To see a similar experiment using a shorter tube and, more importantly, to hear what resonance sounds like watch the video below.

youtu.be/YpCA9CVsVos

solution

The entire set of solutions is summarized in the table below the numbered list. The problem said to pick one of the notes.

  1. Wavelengths are calculated using the formula below. Replace v with 343 m/s (the speed of sound in air at room temperature). Compute with whatever value of f you chose.

    λ =  v  ⇐  v = fλ
    f
  2. Multiply the results of part a. by ¼, ½, ¾, 1¼, 1½, 1¾. A boring, but simple task.

  3. A tube submerged in water is a half open, half closed system. Resonance will only occur when an odd number of quarter wavelengths fit inside the tube. Since the tube is only one meter long, only heights that are less than one meter will fit at all. The answers that satisfy both of these conditions are highlighted in the table below.

Tube lengths that will resonate for various tuning forks

note
pitch
(Hz)
¼ λ
(m)
½ λ
(m)
¾ λ
(m)
1 λ
(m)
1¼ λ
(m)
1½ λ
(m)
1¾ λ
(m)
C4 256 0.335 0.670 1.005 1.340 1.675 2.010 2.345
D4 288 0.298 0.595 0.893 1.191 1.489 1.786 2.084
E4 320 0.268 0.536 0.804 1.072 1.340 1.608 1.876
F4 341 0.251 0.503 0.754 1.006 1.257 1.509 1.760
G4 384 0.223 0.447 0.670 0.893 1.117 1.340 1.563
A4 427 0.201 0.402 0.602 0.803 1.004 1.205 1.406
B4 480 0.179 0.357 0.536 0.715 0.893 1.072 1.251
C5 512 0.167 0.335 0.502 0.670 0.837 1.005 1.172
D5 576 0.149 0.298 0.447 0.595 0.744 0.893 1.042
E5 640 0.134 0.268 0.402 0.536 0.670 0.804 0.938
F5 682 0.126 0.251 0.377 0.503 0.629 0.754 0.880
G5 768 0.112 0.223 0.335 0.447 0.558 0.670 0.782

practice problem 2

Write something else.

solution

Answer it.

practice problem 3

Schumann Resonances
The ionosphere is a layer in the Earth's upper atmosphere where a large portion of the atoms and molecules have been ionized by exposure to the ultraviolet radiation of the Sun. With so many charged particles free to roam around, the ionosphere is a reasonably good conductor of electricity. The surface of the Earth is also a reasonably good conductor. This should be somewhat obvious since 70% of the Earth's surface is covered in saltwater, which will short out electrical equipment as everyone knows, and the remaining 30% is exposed rock or soil, the stuff that electric circuits are grounded to. The layer of atmosphere in between these two conductors is ordinary, non ionized air, which is transparent to radio waves. For extremely low frequency (ELF) radiation, the gap between the Earth and its ionosphere acts as a spherical wave guide — a kind of racetrack for radio waves. Lightning and other natural phenomena generate ELF waves at all sorts of different frequencies. Those frequencies that are just right will travel around the Earth, meet themselves in phase, and form standing waves. The set of frequencies that will do this are known as the Schumann resonances in honor of Winfried Schumann (1888–1974), the scientist who predicted their existence in 1952.
  1. Complete the following table…
    Schumann resonances
    harmonic λ (km) fpredicted (Hz) fobserved (Hz) f/fobserved (%)
    first     7.8  
    second     14  
    third     20  
    fourth     26  
    fifth     33  
    sixth     39  
    seventh     45  
  2. Do the predicted Schumann resonances agree with the observed values to a reasonable degree? Account for any significant discrepancies.

solution

Start with a picture. Here's what the 5th harmonic looks like as an example…

Magnify

  1. For the wavelength, divide the circumference of the Earth by the number of the harmonic.

    λ =  C  =  r
    n n

    Use the wave speed equation to get the frequency.

    f =  c  ⇐  c = fλ
    λ

    Compute the relative uncertainty with the given equation.

    relative uncertainty  =  f
    fobserved
    relative uncertainty  =  |fpredicted − fobserved|
    fobserved

    Repeat the procedure over and over again. I suggest letting a computer do the work for you. Here's a suggested method using the 5th harmonic as an example …

    (2*pi*radius of earth)/5 = 8014.95684 km
    speed of light/(8014.95684 km) = 37.4041263 Hz
    (37.4041263Hz-33 Hz)/(33 Hz) = 0.133458373

    Compile your results. (Please report only a reasonable number of significant digits.) You should get something like this …

    Schumann resonances
    harmonic λ (km) fpredicted (Hz) fobserved (Hz) f/fobserved (%)
    first 40,100 7.48 7.8 4.1
    second 20,000 15.0 14 6.9
    third 13,400 22.4 20 12
    fourth 10,000 29.9 26 15
    fifth 08,010 37.4 33 13
    sixth 06,680 44.9 39 15
    seventh 05,720 52.4 45 16
  2. The disagreement between theory and observation is significant (more than 10% in most cases). The flaw lies in our use of a one-dimensional wave model. The earth is a sphere, an object whose surface is a two-dimensional object. A proper solution to this problem is beyond the scope of this book (and the skills of the author).

practice problem 4

Write something completely different.

solution

Answer it.