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Energy in a String Wave
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 T
x = 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 and tan θ = ∂y / ∂x, we obtain T x = T0 − (T0/2)(∂y / ∂x)2 +….. Obviously, T x 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).
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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