Waves & Sound (5% of AP Exam)

Traveling waves

Students should understand the description of traveling waves, so they can:

(1)  Sketch or identify graphs that represent traveling waves and determine the amplitude, wavelength, and frequency of a wave from such a graph.

(2)  Apply the relation among wavelength, frequency, and velocity for a wave.

(3)  Understand qualitatively the Doppler effect for sound in order to explain why there is a frequency shift in both the moving-source and moving-observer case.

(4)  Describe reflection of a wave from the fixed or free end of a string. 

(5)  Describe qualitatively what factors determine the speed of waves on a string and the speed of sound.

Wave propagation

Students should understand the difference between transverse and longitudinal waves, and be able to explain qualitatively why transverse waves can exhibit polarization. 

Students should understand the inverse-square law, so they can calculate the intensity of waves at a given distance from a source of specified power and compare the intensities at different distances from the source.

Standing waves

Students should understand the physics of standing waves, so they can:

(1)  Sketch possible standing wave modes for a stretched string that is fixed at both ends, and determine the amplitude, wavelength, and frequency of such standing waves. 

(2)  Describe possible standing sound waves in a pipe that has either open or closed ends, and determine the wavelength and frequency of such standing waves. 


Students should understand the principle of superposition, so they can apply it to traveling waves moving in opposite directions, and describe how a standing wave may be formed by superposition.


Wave Types

For longitudinal waves, the medium is displaced in the direction of travel.

Example #1

A traveling wave on a rope has a frequency of 2.5 Hz.  If the speed of the wave is 1.5 m/s, what are its period and wavelength?

Example #2

The period of a traveling wave is 0.5 s, its amplitude is 10 cm, and its wavelength is 0.4m.  What are its frequency and wave speed?

Waves on a String

Waves traveling on a stretched string will have speeds dependent on the density and the tension of the string.

Example #3

Two ropes of unequal linear mass densities are connected, and a wave is created in the rope on the left, which propagates to the right.  If frequency does not change, how does the speed and wavelength of the wave in the second rope compare to that in the first?

Traveling Wave Demonstration

Wave Reflection

Superposition of Waves

When waves interfere with each other, the amplitude of the resulting wave depends on

 the frequencies

 relative phases (relative positions of the

      crests and troughs)

3)   amplitudes

    of the interfering waves.

Constructive Interference

Occurs at a point where two overlapping or intersecting waves of the same frequency are in phase - that is, where the crests and troughs of the two waves coincide.

The two waves reinforce each other and combine to form a wave that has an amplitude equal to the sum of the individual amplitudes of the original waves.

 Constructive Interference

Destructive Interference

Occurs when two intersecting waves of the same frequency are completely out of phase—that is, when the crest of one wave coincides with the trough of the other.

The two waves cancel each other out.


Destructive Interference


More Complex Interferences

Intersecting or overlapping waves that have different frequencies or that are not entirely in or out of phase with each other have more complex interference patterns.


Interference Between Point Sources

This interference pattern was formed by two rods moving rhythmically up and down in a ripple tank.

If two crests arrive at a point together, they superimpose to form a very high crest; if two troughs arrive together, they superimpose to form a very low trough.

Standing Waves

Standing (stationary) waves are present in the vibrating strings of musical instruments.

A violin string, for instance, when bowed or plucked, vibrates as a whole, with nodes at the ends


Vibration of a String

This is known as the fundamental frequency or first harmonic.


A string also vibrates in halves, with a node at the center, in thirds, with two equally spaced nodes, and in various other fractions, all simultaneously.

The vibration as a whole produces the fundamental tone, and the other vibrations produce the various harmonics.


Vibration of a String

Musical Harmonics

In musical sound the full-length vibration produces the fundamental tone (or first harmonic or first partial), which is usually perceived as the basic pitch of the musical sound.

The subsidiary vibrations produce faint overtones (second and higher harmonics or partials).


C (65.5 Hz)

Here is the harmonic series for low C; black notes show pitches that do not correspond exactly with the Western tuning system.

 Harmonics contribute to the ear's perception of the quality, or timbre, of a sound:


Vibration in an open air column

Vibration in a closed air column


“Beats” occur when two slightly different frequency tones are played simultaneously.

The frequency of the “beats” (beats per second) represents the difference between the two frequencies being played.

Doppler Effect

The change in frequency of wave motion due to the motion of the wave generator or receiver.

Doppler effect conditions

Doppler effect conditions

Homework Problems

Chapter 21

Do Problems 3, 7

Chapter 22

Do Problems 3, 7, 13

Chapter 23

Do Problems 18, 23, 26


Sears, Zemansky, and Young, 1987. College Physics, sixth edition, Addison-Wesley, Reading, Massachusetts.

Halliday, Resnick, and Walker, 2001. Fundamentals of Physics, sixth edition, John Wiley and Sons, New York.

Hewitt, 1985, Conceptual Physics, fifth edition,Little, Brown and Company,Boston, MA.



Oscilloscope was downloaded from http://polly.phys.msu.su/~zeld/oscill.html

Tone Generator was downloaded from http://www.world-voices.com/software/nchtone.html


Internet Simulation Links

Traveling Waves

Constructive and Destructive Interference of Two Pulses

Standing Waves


More Internet Links

The Nature of Waves

from The Physics Classroom Tutorial, at Glenbrook South High School in Glenview, IL.


Originally written for high school physics students by Tom Henderson, science teacher at Glenbrook South High School. A good introduction to wavemotion with checking questions, animations, graphs, etc. to help the student understand wave motion.

Longitudinal and Transverse Wave Motion

by Dan Russell, at Kettering University in Flint, MI .


One of many excellent animations at Russell's Acoustics and Vibration Animations. This page shows animated gifs of mechanical waves propagate through a material medium. Particle motion for Longitudinal, Transverse, Water and Rayleigh Surface Waves are modeled.

Java Applets for High School Physics Education

by S.Kamikawa, high school Physics teacher in Japan.


A variety of simulations for Internet Explorer 3.0. These are simple but show the patterns of waves nicely:
Longitudinal Wave; Reflection of Longitudinal Wave from a Boundary; Transverse wave and Longitudinal wave; Simulation of Transverse wave and Longitudinal wave;
Wave Interference, Animation of Interference Pattern Formed by Two Points Sources.

Physics Java Applets

Fu-Kwun Hwang, National Taiwan University Department of Physics.


A very nice series of wave and oscillation applets: Superposition Principle of Waves and Pulses, a great Fourier Synthesizer,Transmission of Waves (Reflection and Refraction), and Interference Between Two Waves (Two Point Sources).

Spectrum Analysis Java Applet

From JAVA Sound and Hearing Demonstrations by Greg Sandell at the Parmly Hearing Institute, of Loyola University, Chicago to accompany the book Fundamentals of Hearing, An Introduction by William A. Yost.


This Applet allows input of Fourier power spectrum which is synthesized into a waveform played through the computer speaker.

Waves and Sound

from the ExploreScience site of Raman Pfaff at the University of Michigan, Ann Arbor.


In order to use this site you must download and install the Macromedia Shockwave Plug-in on your computer.

These activities -- Sound Pulses, Sonic Doppler Effect and Two Source Doppler Effect all display animated sonic phenomena and allow you to manipulate variables and make onscreen measurements.

Speed of Sound

These sites discuss the speed of sound in a air and other gases.
The speed of sound in a gas depends upon:




Type of Gas:http://hyperphysics.phy-astr

HyperPhysics by C.R. Nave at the University of Gerogia has some other notes on waves as well as the two calculators above.