The overlap of two wave pulses that are moving in opposite directions along the same line in a linear, nondissipative and nondispersive medium is used to discuss the compatibility of wave superposition and of energy–momentum conservation. What happens to the energy and to the momentum when the pulses overlap in such a situation is examined. The treatment is applicable to electromagnetic waves or to small-amplitude linear mechanical waves on an ideal string. It is argued that introductory-to-intermediate textbooks and the pedagogical literature have neglected these questions, particularly the one relating to the conservation of linear momentum.

1.
L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Pergamon, New York, 1975), 4th revised ed., Chap. 6, p. 127, Eqs. 52.11 and 52.12.
2.
D. C. Giancoli, Physics for Scientists and Engineers (Prentice–Hall, Saddle Creek, NJ, 2000), 3rd ed.
3.
H. C. Ohanian, Principles of Physics (Norton, New York, 1994).
4.
R. A. Serway and J. W. Jewett, Jr., Principles of Physics, A Calculus-Based Text (Harcourt, New York, 2002), 3rd ed.
5.
A. P. French, Vibrations and Waves (Norton, New York, 1971).
6.
See Ref. 5, Chap. 7, pp. 228–230.
7.
H. J. Pain, The Physics of Vibrations and Waves (Wiley, New York, 1999), 5th ed.
8.
R. C.
Levine
, “
False paradoxes of superposition in electric and acoustic waves
,”
Am. J. Phys.
48
,
28
31
(
1980
).
9.
F. A. Jenkins and H. E. White, Fundamentals of Physical Optics (McGraw–Hill, New York, 1976), 4th ed.
10.
J. Strong, Concepts of Classical Optics (Freeman, San Francisco, 1958).
11.
W. N.
Mathews
, Jr.
, “
Superposition and energy conservation for small amplitude mechanical waves
,”
Am. J. Phys.
54
,
233
238
(
1986
).
12.
W. C. Elmore and M. A. Heald, Physics of Waves (Dover, Mineola, NY, 1969).
13.
See Ref. 5, Chap. 7, pp. 233–234.
14.
See Ref. 12, pp. 45–48.
15.
R.
Benumof
, “
Momentum propagation by traveling waves on a string
,”
Am. J. Phys.
50
,
20
25
(
1982
).
16.
I. Bloch, The Physics of Oscillations and Waves: With Applications in Electricity and Mechanics (Plenum, New York, 1997).
17.
K. U. Ingard, Fundamentals of Waves and Oscillations (Cambridge U.P., New York, 1998).
18.
T. D. Rossing and N. H. Fletcher, Principles of Vibration and Sound (Springer, New York, 1995).
19.
P. Lorrain, D. R. Corson, and D. R. Lorrain, Electromagnetic Fields and Waves (Freeman, New York, 1988), 3rd ed.
20.
W. N.
Mathews
, Jr.
, “
Energy in a one-dimensional small amplitude mechanical wave
,”
Am. J. Phys.
53
,
974
978
(
1985
).
This content is only available via PDF.
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.