Standing Waves

The Physics Hypertextbook™
© 1998-2008 by Glenn Elert -- A Work in Progress
All Rights Reserved -- Fair Use Encouraged

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Discussion

introduction

Maybe you've noticed or maybe you haven't. Sometimes when you vibrate a string, or cord, or chain, or cable it's possible to get it to vibrate in a manner such that you're generating a wave, but the wave doesn't propagate. It just sits there vibrating up and down in place. Such a wave is called a standing wave and must be seen to be appreciated.


A Traveling Wave in Action


A Standing Wave in Action

I first discovered standing waves (or I first remember seeing them) while playing around with a phone cord. If you shake the phone cord in just the right manner it's possible to make a wave that appears to stand still. If you shake the phone cord in any other way you'll get a wave that behaves like all the other waves described in this chapter; waves that propagate -- traveling waves. Traveling waves have high points called crests and low points called troughs (in the transverse case) or compressed points called compressions and stretched points called rarefactions (in the longitudinal case) that travel through the medium. Standing waves don't go anywhere, but they do have regions where the disturbance of the wave is quite small, almost zero. These locations are called nodes. There are also regions where the disturbance is quite intense, greater than anywhere else in the medium, called antinodes.


The Parts of a Standing Wave

Standing waves can form under a variety of conditions, but they are easily demonstrated in a medium which is finite or bounded. A phone cord begins at the base and ends at the handset. (Or is it the other way around?) Other simple examples of finite media are a guitar string (it runs from fret to bridge), a drum head (it's bounded by the rim), the air in a room (it's bounded by the walls), the water in Lake Michigan (it's bounded by the shores), or the surface of the earth (although not bounded, the surface of the earth is finite). In general, standing waves can be produced by any two identical waves traveling in opposite directions that have the right wavelength. In a bounded medium, standing waves occur when a wave with the correct wavelength meets its reflection. The interference of these two waves produces a resultant wave that does not appear to move.

Standing waves don't form under just any circumstances. They require that energy be fed into a system at an appropriate frequency. That is, when the driving frequency applied to a system equals its natural frequency. This condition is known as resonance. Standing waves are always associated with resonance. Resonance can be identified by a dramatic increase in amplitude of the resultant vibrations. Compared to traveling waves with the same amplitude, producing standing waves is relatively effortless. In the case of the telephone cord, small motions in the hand result will result in much larger motions of the telephone cord.

Any system in which standing waves can form has numerous natural frequencies. The set of all possible standing waves are known as the harmonics of a system. The simplest of the harmonics is called the fundamental or first harmonic. Subsequent standing waves are called the second harmonic, third harmonic, etc. The harmonics above the fundamental, especially in music theory, are sometimes also called overtones. What wavelengths will form standing waves in a simple, one-dimensional system? There are three simple cases.

one dimension: two fixed ends

If a medium is bounded such that its opposite ends can be considered fixed, nodes will then be found at the ends. The simplest standing wave that can form under these circumstances has one antinode in the middle. This is half a wavelength. To make the next possible standing wave, place a node in the center. We now have one whole wavelength. To make the third possible standing wave, divide the length into thirds by adding another node. This gives us one and a half wavelengths. It should become obvious that to continue all that is needed is to keep adding nodes, dividing the medium into fourths, then fifths, sixths, etc. There are an infinite number of harmonics for this system, but no matter how many times we divide the medium up, we always get a whole number of half wavelengths (1∕2, 2∕2, 3∕2, … , n∕2).

There are important relations among the harmonics themselves in this sequence. The wavelengths of the harmonics are simple fractions of the fundamental wavelength. If the fundamental wavelength were 1 m the wavelength of the second harmonic would be 1∕2 m, the third harmonic would be 1∕3 m, the fourth 1∕4 m, and so on. Since frequency is inversely proportional to wavelength, the frequencies are also related. The frequencies of the harmonics are whole-number multiples of the fundamental frequency. If the fundamental frequency were 1 Hz the frequency of the second harmonic would be 2 Hz, the third harmonic would be 3 Hz, the fourth 4 Hz, and so on.
 

one dimension: two free ends

If a medium is bounded such that its opposite ends can be considered free, antinodes will then be found at the ends. The simplest standing wave that can form under these circumstances has one node in the middle. This is half a wavelength. To make the next possible standing wave, place another antinode in the center. We now have one whole wavelength. To make the third possible standing wave, divide the length into thirds by adding another antinode. This gives us one and a half wavelengths. It should become obvious that we will get the same relationships for the standing waves formed between two free ends that we have for two fixed ends. The only difference is that the nodes have been replaced with antinodes and vice versa. Thus when standing waves form in a linear medium that has two free ends a whole number of half wavelengths fit inside the medium and the overtones are whole number multiples of the fundamental frequency.
 

one dimension: one fixed end -- one free end

When the medium has one fixed end and one free end the situation changes in an interesting way. A node will always form at the fixed end while an antinode will always form at the free end. The simplest standing wave that can form under these circumstances is one-quarter wavelength long. To make the next possible standing wave add both a node and an antinode, dividing the drawing up into thirds. We now have three-quarters of a wavelength. Repeating this procedure we get five-quarters of a wavelength, then seven-quarters, etc. In this arrangement, there are always an odd number of quarter wavelengths present. Thus the wavelengths of the harmonics are always fractional multiples of the fundamental wavelength with an odd number in the denominator. Likewise, the frequencies of the harmonics are always odd multiples of the fundamental frequency.

The three cases above show that, although not all frequencies will result in standing waves, a simple, one-dimensional system possesses an infinite number of natural frequencies that will. It also shows that these frequencies are simple multiples of some fundamental frequency. For any real-world system, however, the higher frequency standing waves are difficult if not impossible to produce. Tuning forks, for example, vibrate strongly at the fundamental frequency, very little at the second harmonic, and effectively not at all at the higher harmonics.

filtering

The best part of a standing wave is not that it appears to stand still, but that the amplitude of a standing wave is much larger that the amplitude of the disturbance driving it. It seems like getting something for nothing. Put a little bit of energy in at the right rate and watch it accumulate into something with a lot of energy. This ability to amplify a wave of one particular frequency over those of any other frequency has numerous applications.

two dimensions

The type of reasoning used in the discussion so far can also be applied to two-dimensional and three-dimensional systems. As you would expect, the descriptions are a bit more complex. Standing waves in two dimensions have numerous applications in music. A circular drum head is a reasonably simple system on which standing waves can be studied. Instead of having nodes at opposite ends, as was the case for guitar and piano strings, the entire rim of the drum is a node. Other nodes are straight lines and circles. The harmonic frequencies are not simple multiples of the fundamental frequency.

The diagram above shows six simple modes of vibration in a circular drum head. The plus and minus signs show the phase of the antinodes at a particular instant. The numbers follow the (D, C) naming scheme, where D is the number of nodal diameters and C is the number of nodal circumferences.

Standing waves in two dimensions have been applied extensively to the study of violin bodies. Violins manufactured by the Italian violin maker Antonius Stradivarius (1644-1737) are renowned for their clarity of tone over a wide dynamic range. Acoustic physicists have been working on reproducing violins equal in quality to those produced by Stradivarius for quite some time. One technique developed by the German physicist Ernst Chladni (1756-1794) involves spreading grains of fine sand on a plate from a dismantled violin that is then clamped and set vibrating with a bow. The sand grains bounce away from the lively antinodes and accumulate at the relatively quiet nodes. The resulting Chladni patterns from different violins could then be compared. Presumably, the patterns from better sounding violins would be similar in some way. Through trial and error, a violin designer should be able to produce components whose behavior mimicked those of the legendary master. This is, of course, just one factor in the design of a violin.

Chladni pattern Chladni pattern Chladni pattern Chladni pattern

Chladni patterns on the plates of a typical violin in order of increasing frequency.
Source: Joe Wolf, University of New South Wales

three dimensions

The Probability Density of a Ground State Electron in a Hydrogen Atom
| 1, 0, 0 >

In the one-dimensional case the nodes were points (zero-dimensional). In the two-dimensional case the nodes were curves (one-dimensional). The dimension of the nodes is always one less than the dimension of the system. Thus, in a three-dimensional system the nodes would be two-dimensional surfaces. The most important example of standing waves in three dimensions are the orbitals of an electron in an atom. On the atomic scale, it is usually more appropriate to describe the electron as a wave than as a particle. The square of an electron's wave equation gives the probability function for locating the electron in any particular region. The orbitals used by chemists describe the shape of the region where there is a high probability of finding a particular electron. Electrons are confined to the space surrounding a nucleus in much the same manner that the waves in a guitar string are constrained within the string. The constraint of a string in a guitar forces the string to vibrate with specific frequencies. Likewise, an electron can only vibrate with specific frequencies. In the case of an electron, these frequencies are called eigenfrequencies and the states associated with these frequencies are called eigenstates or eigenfunctions. The set of all eigenfunctions for an electron form a mathematical set called the spherical harmonics. There are an infinite number of these spherical harmonics, but they are specific and discrete. That is, there are no in-between states. Thus an atomic electron can only absorb and emit energy in specific in small packets called quanta. It does this by making a quantum leap from one eigenstate to another. This term has been perverted in popular culture to mean any sudden, large change. In physics, quite the opposite is true. A quantum leap is the smallest possible change of system, not the largest.

| 2, 0, 0 > | 2, 1, 0 > | 2, 1, 1 >
| 3, 0, 0 > | 3, 1, 0 > | 3, 1, 1 >
| 3, 2, 0 > | 3, 2, 1 > | 3, 2, 2 >

Some Probability Densities for Excited Electrons in a Hydrogen Atom

mathematics

In mathematics, the infinite sequence of fractions 1∕1, 1∕2, 1∕3, 1∕4, … is called the harmonic sequence. Surprisingly, there are exactly the same number of harmonics described by the harmonic sequence as there are harmonics described by the "odds only" sequence: 1∕1, 1∕3, 1∕5, 1∕7, …. "What? Obviously there are more numbers in the harmonic sequence than there are in the 'odds only' sequence." Nope. There are exactly the same number. Here's the proof. I can set up a one-to-one correspondence between the whole numbers and the odd numbers. Observe. (I will have to play with the format of the numbers to get them to line up correctly on a computer screen, however.)

01, 02, 03, 04, 05, 06, 07, 08, 09, …
01, 03, 05, 07, 09, 11, 13, 15, 17, …

This can go on forever. Which means there are exactly the same number of odd numbers as there are whole numbers. Both the whole numbers and the odd numbers are examples of countable infinite sets.

There are an infinite number of possible wavelengths that can form standing waves under all of the circumstances described above, but there are an even greater number of wavelengths that can't form standing waves. "What? How can you have more than an infinite amount of something?" Well I don't want to prove that right now so you'll have to trust me, but there are more real numbers between 0 and 1 than there are whole numbers between zero and infinity. Not only do we have all the rational numbers less than one (1∕2, 3∕5, 733∕2741, etc.) we also have all the possible algebraic numbers (square root 2, 7 minus square root 13, etc.) and the whole host of bizarre transcendental numbers (π, e, eπ, Feigenbaum's number, etc.). All of these numbers together form an uncountable infinite set called the real numbers. The number of whole numbers is an infinity called aleph null (א0) the number of real numbers is an infinity called c (for continuum). The study of infinitely large numbers is known as transfinite mathematics. In this field, it is possible to prove that aleph null is less than c. There is no one-to-one correspondence between the real numbers and the whole numbers. Thus, there are more frequencies that won't form standing waves than there are frequencies that will form standing waves.

Summary

Problems

practice

  1. Write something.
    • Answer it.
  2. Write something else.
    • Answer it.
  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 electrical 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 Otto Schumann (1888-1974, Germany), the scientist who predicted their existence in 1952.
    1. Complete the following table …
      Schumann Resonances
      harmonic λ (km) ƒpredicted (Hz) ƒobserved (Hz) Δƒ/ƒobserved
      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.

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

    [magnify — see all the harmonics]
    1. For the wavelength, divide the circumference of the earth by the number of the harmonic.
             
      λ =  C  =  2πr
      n n
             
      Use the wave speed equation to get the frequency.
             
      c = ƒλ ⇒ ƒ =  c
      λ
             
      Compute the relative uncertainty with the given equation.
             
      relative uncertainty =  Δƒ  =  |ƒpredicted − ƒobserved|
      ƒobserved ƒobserved
             
      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 = 8 014.95684 km
      		 
      the speed of light / (8 014.95684 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) ƒpredicted (Hz) ƒobserved (Hz) Δƒ/ƒobserved
      first 40,100 7.48 7.8 0.041
      second 20,000 15.0 14 0.069
      third 13,400 22.4 20 0.12
      fourth 10,000 29.9 26 0.15
      fifth 8,010 37.4 33 0.13
      sixth 6,680 44.9 39 0.15
      seventh 5,720 52.4 45 0.16
       
    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).
  4. Write something completely different.
    • Answer it.

numerical

  1. A common form of hearing loss is associated with resonance in the ear canal. When this happens, there is reduced sensitivity to sounds around 4000 Hz (since this frequency is consistently louder than all the others).
    1. What fraction of a wavelength fits in the ear canal while it is resonating at its fundamental frequency? (Recall that the ear canal starts at the opening in the outer ear and ends at the eardrum.)
    2. From this information determine the length of the ear canal in the average human. (Assume that the speed of sound in the air in the ear canal is about 348 m/s.)
  2. The water level in a vertical tube 1.00 m long can be adjusted to any position in the tube. A tuning fork of vibrating at 660 Hz is held just over the open top end of the tube. At what positions of the water level will resonance occur? (Hey You! Remember the tube is only 1.00 m long.)

statistical

  1. resonance-tube.txt
    A tuning fork was held over a half closed tube, the length of which was adjusted until the sound from the tuning fork was at its loudest. Use this data to determine the speed of sound in air at room temperature.
    The columns in this data set are as follows:
    1. Note of tuning fork (scientific scale except where indicated)
    2. Frequency in hertz
    3. Length of tube in meters
  2. vibrating-string.txt
    A one meter piece of ordinary string was connected to a variable oscillator that was fixed at both ends. The oscillator was dialed through different frequencies of vibration until transverse standing waves formed in the string. A photogate was then used to time the period of vibration since the oscillator was not calibrated in any way. Use this data to determine the speed of transverse waves in the string.
    The columns in this data set are as follows:
    1. Number of antinodes (or the number of the harmonic)
    2. Period of oscillation in seconds
  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 electrical 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 which then travel around the earth. Those with just the right wavelength will wrap around the earth and interfere constructively with themselves. The resulting amplification would result in peaks in the electromagnetic spectrum at the fundamental frequency of the waveguide and its harmonics. These peaks are known as the Schumann resonances in honor of Winfried Otto Schumann (1888-1974, Germany), the scientist who predicted their existence in 1952.
     
    Schumann Resonance -- 5th Harmonic [magnify]
     
    1. Determine the number of times a radio wave traveling at the speed of light in a vacuum (c = 299,792 km/s) can circle the earth (r = 6371 km) in the gap between the earth's surface and its ionosphere (h = 110 km) in one second.
    2. Complete the following table …

      Schumann Resonances
      harmonic λ (km) ƒpredicted (Hz) ƒobserved (Hz) Δƒ/ƒobserved
      first 7.8
      second 14
      third 20
      fourth 26
      fifth 33
      sixth 39
      seventh 45

    3. Do the predicted Schumann resonances agree with the observed values to a reasonable degree? Account for any significant discrepancies.

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