When one end of a taut horizontal elastic string is shaken repeatedly up and down, a transverse wave (assume sine waveform) will be produced and travel along it.1 College students know this type of wave motion well. They know when the wave passes by, each element of the string will perform an oscillating up‐down motion, which in mechanics is termed simple harmonic2. They also know elements of the string at the highest and the lowest positions—the crests and the troughs—are momentarily at rest, while those at the centerline (zero displacement) have the greatest speed, as shown in Fig. 1. Irrespective of this, they are less familiar with the energy associated with the wave. They may fail to answer a question such as, “In a traveling string wave, which elements have respectively the greatest kinetic energy (KE) and the greatest potential energy (PE)?” The answer to the former is not difficult; elements at zero position have the fastest speed and hence their KE, being proportional to the square of speed, is the greatest. To the PE, what immediately comes to their mind may be the simple harmonic motion (SHM), in which the PE is the greatest and the KE is zero at the two turning points. It may thus lead them to think elements at crests or troughs have the greatest PE. Unfortunately, this association is wrong. Thinking that the crests or troughs have the greatest PE is a misconception.3

Topics
Wave mechanics, Wave propagation, Education
1.
Any realistic string must be extensible (elastic) although the extension may be extremely small. This requirement is essential in our model because we are going to show the string is extended by different amounts at different locations.
2.
The small amplitude approximation is adopted. Under that model, the motion of each element is purely vertical. Mathematically, the horizontal component of tension is Tx = T0 cos θ, where T0 is the undisturbed tension (see Ref. 5) and θ is the angle between the horizontal and the string. Using the identities cosθ = 1/1 + tan2 θ and tan θ = ∂y / ∂x, we obtain Tx = T0 − (T0/2)(∂y / ∂x)2 +….. Obviously, Tx is not constant along the string unless the second and the higher order terms are dropped (the small amplitude approximation).
3.
Seems common. See, for example, http://hyperphysics.phy‐astr.gsu.edu/hbase/waves/powstr.html and http://www.physics.upenn.edu/courses/gladney/phys151/lectures/lecture_mar_31_2003.shtml.
4.
D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics, 7th ed. (Wiley, 2004), p. 423. Few popular textbooks on general physics have and depict correctly the PE variation in a string wave like this.
5.
Let T = T0 + T1, where T0 is the tension in the string before the propagation of the wave (called “undisturbed tension”) and T1 is the part contributed from the uneven extension of the string. Because T1 itself is a term proportional to the square of the slope of the waveform (see Ref. 6), so each element only moves vertically if the small amplitude approximation is still assumed.
6.
See, for examples, http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/AnalyzingWaves.htm; H. Benson, University Physics, rev. ed. (Wiley, 2006), p. 337; and
W. N.
Mathews
Jr., “
Energy in a one‐dimensional small amplitude mechanical wave
,”
Am. J. Phys.
53
,
974
–
978
(Oct.
1985
).
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7.
As defined in Ref. 5, the tension is split into two parts, T0 and T1. Each element of the string always possesses the same amount of PE corresponding to T0, but the wave does not transfer this kind of PE. What this paper focuses on is the PE corresponding to T1, which is found to be position‐dependent. Hence, at the crests or troughs, the PE is minimum but not zero.
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2010
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