Experiment III-11

Wave Applications


By Aristotle's time, many of the phenomena that we now attribute as wave effects were known, but few were explained by any theory of waves.  While the Greeks understood how to produce sound and music, there was little understanding about what sound and music are.  Only later when there were efforts to understand other phenomena was much thought given to waves.


The wave theory didn't gain much traction as as an explanation for light until 1799 when Thomas Young (b1773, d1829) began writing about optical theory.  Many of the properties of heat and light were already appreciated: polarization, double refraction, and interference.  But the accept theory was that light was composed of corpuscles which radiated from the source.  Remarking in his first paper on the dubious requirement of corpuscles from feeble sparks and the warmest sunlight that their velocities of emission must be identical, Young suggested This difficulty does not exist in the undulatory theory, since all disturbances are known to be transmitted through an elastic fluid with the same velocity.  At the time the wave theory had difficulty explaining diffraction and had no explanation at all for polarization.  A dramatic turn came following a contest to explain diffraction in which Augustin Fresnel (b1788, d1827 ←engraving at left) amplified the wave theory of light.  When one of the judges, Siméon Denis Poisson (b1781, d1840), noticed that Fresnel's theory would require a bright spot in the middle of a shadow, he suggested that should be tested, expecting the lack of such a spot would be a fatal blow to the theory.  But in 1818 the spot was surprisingly found, decisively establish the wave theory of light.

Austrian mathematician Christian Andreas Doppler (b1803, d1853, portrait below left ↵) in 1842 presented to the Royal Bohemian Society a paper Concerning the colored light of double stars and certain other stars of the heavens.  The paper presented the principle which relates the frequency of a source to its velocity relative to an observer.  Doppler derived the principle in a few lines treating both light and sound as longitudinal waves in the ether and matter, respectively.  Doppler regarded (incorrectly) light to be a longitudinal wave.  Fresnel had previously published a theory that light was a transverse wave but Doppler had read Fresnel's work and disagreed.  However Doppler's principle applies to all types of waves.  Doppler tried to illustrate his theory with an application to the colors of double stars.  Although the principle does change the colors of double stars, depending on which star was approaching or receding from the Earth, the effect was too small to verify by observing with the instruments of the time.

Christian DopplerBecause the speed of sound is much slower than the speed of light, the situation with sound was rather different.  By 1845 experiments were conducted with musicians on railway trains as the train approached them and receded from them.  To test his hypothesis, Doppler used two groups of trumpeters:  one group was stationary at a train station and one group rode an open moving train car.  The musicians played the same note and maintained its pitch as the train passed the station.  The frequency of the two notes didn't match, even though the musicians were playing the same note.  In 1846 Doppler published a better version of his principle where he considered both the motion of the source and the motion of the observer.

doppler effectGenerally sound waves, radio waves and light waves propagate uniformly in all directions from a stationary point source.  When a source generating waves from a point location is moving, the wave fronts ahead of its motion are compressed being produced closer together as the source follows each previous wave.  The waves are more spread out behind the source as it moves away from the previous wave front before generating the next wave.  The waves are unaffected in directions perpendicular to the motion of the source.  Christian Doppler was the first to explain this phenomena mathematically.

But over the following years, waves were used to explain many phenomena and to create may useful devices.  Wave phenomena are ubiquitous in science and engineering.  From the obvious and long standing applications in hydrodynamics, through seismics, acoustic noise propagation and radio wave propagation waves underlie many important scientific and engineering areas.  Typically the need to simulate devices based on wave physics requires the solution of an appropriate wave equation such as Maxwell's equations, or the acoustic or elastic wave equation.  In all but the simplest geometry or media this requires the use of numerical techniques.

Hansen's groupIn the early 1930's, William W. Hansen (shown in 1939 far right in back→), proposed using high-frequency waves to accelerate particles to high energy.  Two brothers, Sigurd (standing on left) and Russell (kneeling) Varian, originally interested in generating the very-high-frequency short wavelength signals needed for radar and direction finding, began working with Hansen.  Utilizing Hansen's earlier invention, a cavity resonator, they demonstrated in 1937 the first very-high-frequency source, which they named the klystron.  Hansen pioneered the development of microwave theory and techniques for testing microwave systems during World War II.  After the war, Hansen worked with three graduate students to demonstrate a 4.5 MeV linear accelerator in 1947.  His progress report to the Office of Naval Research which funded the effort contained only four words, We have accelerated electrons.  After his death in 1949, a 1 BeV 220 foot long accelerator was completed which eventually led to the two mile long Stanford Linear Accelerator Center (SLAC) and the first use of the electron storage ring for X-ray spectroscopy at the accompanying Stanford Synchrotron Radiation Laboratory (SSRL).  Smaller klystrons are more commonly used in microwave ovens, radar antennae, transmitting microwave communications, in broadcasting television, and radiation therapy to kill cancers.

In a klystron a narrow electron beam (or sheet) is launched into a drift tube composed of one or more coupled resonant cavities, constrained by a surrounding magnetic field.  Much like the pressure clumping of air inside a musical tube, the resonance of the tuned cavity encourages the electrons to travel in bunches which broadcast a characteristic microwave.  In the synchrotron, electron bunches curving around the ring shaped tube emit more energetic, even shorter wavelength X-radiation as they are accelerated by bending magnets.  Additional radiation can be generated when bunches of electrons are deflected by magnetic wigglers.

Computational techniques for wave equations have been developed since the advent of computers.  In the early days between 1960 and 1970 these were typically low order finite difference methods such as the Yee Finite Difference Time Domain or boundary integral equation methods for time-harmonic scattering. These methods were augmented by celebrated techniques involving approximate or asymptotic representation such as the geometric theory of diffraction and ray tracing.

The desire to solve ever larger and more complex problems resulted in development of more efficient solution techniques.  With a long tradition for emphasis on high-order methods for solving wave problems, linear and nonlinear, recent activities include the development of high order discontinuous Galerkin methods for wave problems in general geometries, methods for nonlinear wave problems with shocks and multiple related issues such as new time integration techniques and methods for dealing with uncertainty in wave scattering and penetration.


Today the Doppler effect is used by police to measure automobile speeds, open super market doors as we approach, map the surface of the earth and other planets, and monitor the weather.  Doppler is used for acoustic echoing (SONAR), radio echoing (RADAR) and light/laser echoing (LIDAR).  Sound travels well through water, ground, metal and other solids.  The speed of sound is slow compared to the speed of light, but it works well at short range and in the ocean depths.  Radar uses electromagnetic waves longer than visible light, so travels at the speed of light.  It can penetrate weather, clouds, and some building materials, but is reflected rather than penetrate metal or ground.  The visible light from lasers cannot penetrate weather; however because of its shorter wavelength, it can be formed into tight beams thousands of times narrower than radar beams, giving higher angular resolution.

Consider the following set of measurements:

  1. Consider moving sound source generating a musical A note (440 Hertz = 440 vibrations/second) and a stationary listener.  If the source travels 30 miles/hr (48 km/hr) when the listener is in front of the approaching source, the listener hears a 457 Hz note (the Doppler frequency change is +17.6 Hz).  As the source begins to pass with the listener 30 degrees off the source's path, 448.8 Hz will be heard.  (A Doppler frequency change of 8.8 Hz = sine 30° times 17.6 Hz).  As the source the listener, at 90 degrees, 440 Hz is heard (no Doppler effect). As the source recedes, the frequency will approach 422.4 Hz (440 Hz -17.6 Hz Doppler effect).  If the source had been moving at 60 miles/hr the Doppler effect would have been twice as great.
  2. Moving listener and stationary sound source:  The same experiment is conducted only now the listener is moving while the 440 Hz note is played by a stationary source.  In this experiment the listener hears the same notes because of identical relative motion to the fixed sound source.  The listener will again hear the higher pitch sound when approaching the source with the lower pitch sound while traveling away from the source.
  3. An echo experiment:  Experiment 2 is repeated but the sound recorded.  Then the motion is repeated with the recording of the higher pitch to lower pitch played from the moving frame of reference.  A stationary listener will here both the recorded Doppler shift plus an identical shift due to the current motion.  That is the Doppler effect will be double because of the effect while the sound was traveling from the stationary source to the moving frame plus the effect while the sound was returning from the moving frame to the stationary listener.
  4. A better echo experiment:  If instead of a moving source, a large flat plate is moving to reflect back the sound from the stationary source.  The listener would hear the same double Doppler shifted notes that was heard in Experiment 3 with out the need for the recording to repeat the note; however, because of the double travel distance of the sound and perhaps less than perfect reflection, the reflected note would be very low in intensity and hard to hear.  The result of experiments 3 and 4 is that echoing or repeating from a moving body produces a doubling of the Doppler shift.  If radio waves are used instead of sound waves, it is called Doppler radar.

The same techniques as Experiment 4 are used by law enforcement with their radar guns giving them a precise measurement of your cars speed.  The National Weather Service uses Doppler weather surveillance radar (NEXRAD) excels in detecting water droplets in the air and any wind caused motion to track storms

Doppler techniques are also used by astronomers to determine the velocity of distant stars and galaxies relative to the earth.  The most distant objects we can detect in the universe are moving away from us at very high velocities  Both light and radio signals from these distant objects are Doppler shifted toward the longer wavelengths, an effect called the RED SHIFT.  Some astronomic objects are rotating producing a difference in Doppler shift from one side from its opposite side.


  1. Obtain any portable device which generates a steady pitch of sound.

  2. Fasten it securely to the end of a strong cord at least several meters long.

  3. Whirl the device in a circle above one's heat.  Caution: make sure not to hit anything and that the sound generating device will not accidentally be released.

  4. Ears in the center of the motion, being perpendicular to the motion, will hear little variation in pitch.  But listeners standing outside the radius of motion should be able to detect a periodic fluctuation in pitch.

  5. Calculate the speed and attempt to compare the change in pitch with that expected from the theory.

Communicating technical information such as observations and findings is a skill used by scientists but useful for most others.  If you need course credit, use your observations in your journal to construct a formal report.



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created 12 February 2007
revised 18 February 2007
by D Trapp
Mac made