"Universal" Doppler formula (one that can be used for any situation):

            Where v is the speed of sound, v_o is the speed of the object, and v_s is the speed of the sound source. When the object is moving toward you, use the upper sign; when the object is moving away from you, use the lower sign.

Doppler Shift Physlet

Doppler Shift-Moving Point Source

An example of the Doppler shift

This is my favorite Doppler applet. I hope it opens for you because on some computers it will not. It has a moving source and a stationary detetector. Simultaneously, it shows the approaching (and receding) wavefront, a picture of the wavelength showing perceived changes, and also plays the sound you would hear. Great Doppler Applet!

Want to see a picture of a real sonic boom? Sonic Boom Picture

Hint: when working Doppler shift problems, associate the word toward with a frequency increase and the words away from or recede with a frequency decrease.

AP Doppler formulas (1-3)

1. Detector moving; source at rest

   

   where v_d is the velocity of the detector, v is the speed of sound, f ' is the detected frequency, and f is the original frequency.

   Here the plus sign indicates that the detector is moving toward the source and the minus sign indicates that the detector is moving away from the source.

   Source moving; detector at rest

   

   where v_s is the velocity of the source.

   Source and detector both moving:

   
   This can be used for all Doppler calculations. If the detector is stationary, v_d=0 and if the source is stationary, v_s=0.

   Doppler problem solving strategy: Establish a coordinant system, decide which direction is positive, and make sure you know the signs of all relevant velocities. A velocity in the direction from the detector and toward the source is positive; a velocity in the opposite direction is negative.
   Please note: because of limitations in how I had to construct the image for the "all-in-one" Doppler formula, there is a slight error in the denominator. The negative sign should be on the top, rather than the positive sign. I was unable to find a corresponding image to use to make a -/+ and had to use the +/- instead.
   The upper signs in the Doppler formula apply is source and/or observer move toward each other; the lower signs apply is they are moving apart. v_d is positive if the detector moves toward the source; if the source moves toward the detector, v_s is positive.

   Hubble Picture from March 9, 2004 of the most distant galaxies found.

   Terms:

   • speed
     The speed of a wave is given by v = l f

     pitch

     frequency

     loudness

     amplitude

     decibels

     unit for measuring sound level

     timbre

     sound quality

     beat

     what a listener hears when two sound waves of slightly different frequency are played

     resonance

     a vibrating object induces a vibration of the same frequency in another object
     The most famous examples of resonance involve bridge collapses. In most physics textbooks, the Tacoma Narrows Bridge collapse is cited as an example of resonance. Some engineers dispute this. Video of Tacoma Narrows Bridge Collapse.

   Sound can be characterized by its frequency, its wavelength, its speed, and its intensity (or loundness). Sound waves carry energy that can do work (example: a sonic boom can break windows).

   Sound intensity (I)

   The intensity of the sound is the power of the wave (or energy/sec) per unit of area (or one square meter. The power of the wave is the amount of energy transported per second. If the sound originates from a point source, you can think of this as a wave front passing through a sphere of area 4p r² Sound intensity depends upon distance; if the distance is doubled, sound intensity is reduced by a factor of 4 (This is only valid for a point source with no reflections.)

   

   Intensity level ( b ) The units of the intensity level of sound are decibel, or dB, in honor of Alexander Graham Bell. Since the intensity level is based on a log scale, every change of 10 dB means that the sound is 10 times louder; a change of 20 dB means that the sound is 10², or 100 times louder. The human ear is sensitve over the range of 0-120 dB. A whisper is 20 dB; a shout is 90 dB. The threshold of pain is 130 dB.

   b = (10 dB) log (I/I_o)

   where I_o is the threshold of sound

   Threshold of soundThe threshold of sound has the value of I_o = 1 x 10^-12 W/m²

   Sources of Sound

   node

   region of zero displacement in a standing wave

   antinode

   region of maximum displacement in a standing wave

   Sources of musical sound: Most instruments involve more than a single vibrating body. For example, in a violin, both the strings and the violin body vibrate.

   • vibrating strings (guitar, piano, violin)
   • vibrating membranes (drums)
   • vibrating air columns (flute, oboe, organ)
   • vibrating steel bars (xylophone)

   1. strings produce transverse waves; sound is produced as string compresses and rarefacts air
      law of strings:
      • frequency is increased as string length is decreased
      • frequency is increased as string diameter is decreased
      • frequency is increased as string tension is increased
      • frequency is increased as string density is decreased

      in a standing wave on a string, each segment is ½ l

      Pipes produce standing waves

      • closed pipes — an antinode is always at an open end and a node is always at a closed end
      • open pipes — an antinode is at each open end

   Instruments produce standing waves. In any instrument, several harmonics are excited at the same time and the resultant sound is the superposition of these components.

   fundamental (1^st harmonic):

   • string, length = ½ l
     
   • closed pipe, length = ¼ l
     
   • open pipe, length = ½ l
     

   2^nd harmonic:

   • string, length = l
     
   • open pipe, length = l
     

   3^rd harmonic:

   • string, length = 3/2 l
     
   • closed pipe, length = ¾ l
     
   • open pipe, length = 3/2 l
     

   Here is a trick to remember: Draw the desired harmonic for the string, open pipe, or closed pipe. Determine how much of a wavelength is represented. Set this equal to the length of the pipe and solve for the wavelength. In the pictures above of the harmonics, if it looks like a "v" it is equal to 1/4 l. If it looks like two "v's" stuck together to form a closed object (a segment), it is equal to 1/2 l.

   Notice: There are no even-numbered harmonics in a closed pipe. A closed pipe only produces odd harmonics. In strings and open pipes,

   n=(n v)/2 l, where n=1, 2, 3, ... In closed pipes,

   n=(n v)/4 l, where n=1, 3, 5, ...

   Where l is the length of the pipe.

   In music, harmonics are called overtones.

   Beats Suppose two sounds with frequencies very close to one another are played simultaneously. We hear an average of the two sounds. The sound is modulated by a slow, wobbling beat note whose frequency is the difference between the two sound frequencies, or beats. For example, when a 552 Hz and a 564 Hz tone are played simultaneously, we hear 564-552, or 12 beats per second. The beat frequency is 12 Hz.

   Beats - you actually HEAR them!

   Sound on the AP exam:

   • Typically on multiple choice questions. There are few free response questions that deal with sound/waves.
   • For a vibrating string, you might be asked to predict how frequency changes if tension is changed.
   • You might be given a drawing that shows a moving source producing a wave train. They may ask you about the relative speed and direction of movement of the source. They also might ask you to predict what relative frequency an observer detects. You might be asked to predict what factors affect the frequency detected by the observer.
   • You might be asked to compare characteristics of sound and light waves.
   • Free response questions - open and closed pipe calculations where you calculate the wavelength, the speed of sound, and predict resonance frequencies.

   Interference of Sound Waves

   Two speakers which emit identical sinusoidal waves of identical frequencies are another example of sound wave interference phenomena. Suppose the speakers are separated by distance d.

   A microphone is placed equidistant from both speakers, on a line perpendicular to the line connecting the speakers as shown below.

   

   Wave crests emitted from the two speakers travel equal distances to arrive at the microphone and thus arrive at the microphone at the same time. According to the principle of superposition, the amplitudes of the two waves add, resulting in constructive interference. If the microphone is moved to another position, destructive interference occurs where the wave from one speaker travels a half-wavelength farther than the wave from the other speaker. According to superposition, the amplitudes of the two waves subtract.

   • You might be asked to calculate the minimum frequency where destructive interference can occur. Remember - destructive interference occurs every 1/2 wavelength. Thus, the minimum frequency would occur when d = 1/2 l. Knowing v=lf, the speed of sound and d can be used to calculate this minimum frequency.
   • You might be asked to graph how intensity varies with horizontal distance. Remember, intensity follows an inverse square relationship.
   • You might be asked to graph how intensity varies with vertical distance. Remember, this looks like double slit diffraction pattern. At the midpoint, the intensity is the greatest. As you move outwards vertically, a minimum next occurs. As you continue to move out vertically, another maximum occurs, but it will not be as intense as the first one. This is followed by another minimum and so forth.

   Sound Sample Problems

   Sound Homework

   Waves & Sound AP Objectives

   AP Sound & Waves Class Problems