Abstract
In the December 2018 issue, Perkins and Ruiz 1 point out that if a 1D rightward traveling wave is written as A sin(kx -t), then a leftward traveling wave should be written as A sin(-kx -t) and not as A sin(kx + t). Use of the correct expression is crucial in deriving the Fresnel equations for the reflection and transmission coefficients of a wave incident on an interface, such as when a wave encounters a knot joining a string of one linear mass density to another. 2 Thus it is important to convince students of the correct leftward expression. Perkins and Ruiz explain it by saying the sign of the wave vector k should be switched. That is a good argument for students who have been exposed to the modern physics idea that the momentum vector of a 1D wave is k and its direction reverses upon reflection. (In contrast, the angular frequency of a wave is determined by its source and is unchanged for waves traveling in any direction in any medium that the waves may transmit into or reflect off.) However, for introductory students I suggest the following argument instead. Draw a sinusoidal wave on a piece of paper and then fold it in half as illustrated in Fig. 1 . Let the fold define the position of the origin x = 0. The folding operation will map positive x-values onto negative ones, and thus the sign of x in the argument of the wave function switches as the wave propagates past the fold. If one instead reversed the sign of t, a point on the wave that is just past a crest say would incorrectly start returning in phase back toward the crest rather than continuing on toward a trough! 1. J. Perkins and M.J. Ruiz, "A reliable wave convention for oppositely traveling waves, " Phys. Teach. 56, 622-623 (Dec. 2018).

In the December 2018 issue, Perkins and Ruiz 1 point out that if a 1D rightward traveling wave is written as A sin(kx -t), then a leftward traveling wave should be written as A sin(-kx -t) and not as A sin(kx + t). Use of the correct expression is crucial in deriving the Fresnel equations for the reflection and transmission coefficients of a wave incident on an interface, such as when a wave encounters a knot joining a string of one linear mass density to another. 2 Thus it is important to convince students of the correct leftward expression. Perkins and Ruiz explain it by saying the sign of the wave vector k should be switched. That is a good argument for students who have been exposed to the modern physics idea that the momentum vector of a 1D wave is k and its direction reverses upon reflection. (In contrast, the angular frequency of a wave is determined by its source and is unchanged for waves traveling in any direction in any medium that the waves may transmit into or reflect off.) However, for introductory students I suggest the following argument instead. Draw a sinusoidal wave on a piece of paper and then fold it in half as illustrated in Fig. 1 . Let the fold define the position of the origin x = 0. The folding operation will map positive x-values onto negative ones, and thus the sign of x in the argument of the wave function switches as the wave propagates past the fold. If one instead reversed the sign of t, a point on the wave that is just past a crest say would incorrectly start returning in phase back toward the crest rather than continuing on toward a trough! Fig. 1 . After folding the paper in half backward, the wave on the right half of the page that was originally traveling rightward in the +x region becomes a reflected wave traveling leftward in the -x region (on the back side of the page).
Online Appendix for "Sign Convention for Backward Traveling Waves"
Suppose that two ideal strings of linear mass densities  1 and  2 are tied together (at a point defined to be 0 x  ) and stretched out horizontally (along a line defining the x axis with the positive direction pointing from string 1 toward 2) with the same tension T in both of them, set by hanging a weight around a pulley from the free end of string 2. The free end of string 1 is connected to a simple harmonic oscillator that can shake the end up and down at angular frequency  in the y direction. The phase speed and wavenumber of the resulting harmonic waves will respectively be k   on string 2. Now turn on the oscillator to generate a wave incident in string 1 toward the knot connecting the strings. Consider the primary reflected and transmitted waves from that knot, before they hit the pulley and oscillator at the far ends of the strings and generate secondary waves reflecting back toward the knot. If the incident wave has amplitude A i then it is described by the wavefunction
Online Appendix for "Sign Convention for Backward Traveling Waves"
where the zero of time is chosen so that there is no phase constant. The reflected wave has amplitude A r so that its wavefunction is
Online Appendix for "Sign Convention for Backward Traveling Waves"
and the transmitted wave of amplitude A t has wavefunction
Online Appendix for "Sign Convention for Backward Traveling Waves"
The net displacement y net of string 1 is ir ( , ) ( , ) y x t y x t  , while that of string 2 is simply t ( , ) y x t . Assuming the string does not break, this net displacement must be continuous across the knot, so that the first boundary condition is
Online Appendix for "Sign Convention for Backward Traveling Waves"
after substituting Eqs. (1) to (3) (1) and (3). The second boundary condition is that the slope net / yx  must be continuous across the knot. Otherwise, a free-body diagram shows that the tension forces T on the two sides of the knot would not point in opposite directions, which would result in a nonzero force on the massless knot and give it an infinite acceleration! Taking the partial derivatives of Eqs. (1) to (3) and substituting 0 x  leads to
Online Appendix for "Sign Convention for Backward Traveling Waves"
after canceling the common factor of cos t  . Simultaneous solution of Eqs. (4) and (5) gives the reflection coefficient
Online Appendix for "Sign Convention for Backward Traveling Waves"
The final result is valid not just for string waves but also for water, sound, and electromagnetic waves that strike an interface perpendicularly. One sees that a reflected wave will be inverted (so that r is negative) if and only if the wave travels slower in the transmitted than in the incident medium. (For a string wave, that would mean string 2 is denser than string 1. For an optical wave, it would imply medium 2 has a larger refractive index than medium 1, which is the origin of the familiar phase change upon reflection.) Likewise the transmission coefficient is
Online Appendix for "Sign Convention for Backward Traveling Waves"
Unlike r,  can never be negative and so the transmitted wave will always be upright (relative to the incident wave orientation). On the other hand, whereas the absolute value of r can never exceed unity, that of  will whenever 21   . But the latter fact does not violate energy conservation because the power carried by a harmonic string wave is given by 2 2 2 / 2 / AA    whose ratios in reflectance and transmittance correctly add up to unity. Equations (6) and (7) are the Fresnel coefficients for the special case of normal incidence. C are must be taken representing waves traveling in opposite directions. Erroneous results can occur when combining waves with the popular convention where a wave traveling to the right is R (x,t) = A sin (kx -t) and a left-traveling wave is L (x,t) = A sin (kx + t). An easily overlooked pitfall is pointed out and a way to avoid it with a secure convention is presented. As an exercise in critical thinking, students can be presented with the pitfall and asked to find a way out.
Background
A common way to introduce a traveling disturbance is to first shift a function f(x) along the positive x-axis a distance d by replacing f(x) with g(x) = f(x -d). Students can easily appreciate this effect if you sketch a peaked function f(x) at the origin so that f(0)= peak. Refer to Fig. 1(a) . Then, from Fig. 1(b) where the pulse at the origin (left) now moves to the right with speed v and the shifted pulse (at right) moves to the left with the same speed v.
indicating that the peak has shifted a distance d along the positive x-axis.

Finally, setting d = vt gives the right-traveling disturbance f(x -vt). Switching the sign in front of the speed leads to the left-traveling wave f(x + vt). See
Introductory textbooks 1-8 often present students with sine waves using the convention described above. Sine waves traveling to the right and left, respectively, are given by
Finally, setting d = vt gives the right-traveling disturbance f(x -vt). Switching the sign in front of the speed leads to the left-traveling wave f(x + vt). See
where kx -t = k(x -vt). The familiar parameters k and are the wave number and angular velocity, respectively. The common description 2A sin(kx) cos( t) related to standing waves can be obtained by adding Eqs. (1a) and (1b). However, the example in the following section demonstrates an application leading to an unsatisfactory result.
A pitfall
To illustrate the treacherous nature of the convention described above for waves traveling to the right and left, first see Fig. 2 for an arrangement where two speakers are separated by a distance 2d. The speakers emit sine waves in phase, which are picked up by the microphone at the middle position between the two speakers. 9 Let the wave leaving the left speaker be y R (x,t) = A sin (kx -t), where the subscript R designates that the wave travels to the right. For the wave emitted from the right speaker, we arrive at the function by shifting a sine wave a distance 2d to the right and changing -t to + t so that the wave moves to the left: y L (x,t) = A sin[k(x -2d) + t]. The superposition of the waves at the microphone, i.e., at x = d, is then
A pitfall
which disagrees with the experimental observation. The waves at the microphone undergo constructive interference since the sources are emitting in phase and the waves are traveling the same distance in order to reach the center. Equation (2) indicates an incorrect result since it predicts destructive interference at the center point. An inspection of Figs. 3(a) and (b) reveals that the convention does not work for the sine wave, which is an odd function. A disguised phase shift has been introduced, as illustrated in Fig. 3(b) , where a leading trough travels from left to right, while a leading crest travels from right to left. However, the correct result is obtained for Fig. 3(a) , where the cosine function (an even function) is used. In the next section we discuss Figs. 3(c) and (d) .
A correct approach
Jonathan A. Jones at the University of Oxford points out to his students that there are several wave conventions: the good, the bad, and the ugly. 10 We have employed in the previous section a convention that Jones lists as bad. 10 He also reminds us that the rules presented in Eqs. The analysis of wave conventions can sharpen a student's critical thinking, forcing the student to ponder deeply in order to understand the physics behind each wave scheme. Finally, students will appreciate the wave-vector convention, which is the more reliable approach for analyzing problems with waves traveling in opposite directions.
A correct approach
The good convention given in Ref. 10 is the one used in advanced physics where the sign change for the left-traveling wave is made by reversing the sign of k rather than the sign of . We will refer to reversing the sign of k as the wave-vector convention since in three dimensions one finds  k . r -t.
A correct approach
The good convention is readily found in current texts for upper-level courses in optics, 12 electromagnetic theory, [13] [14] and quantum mechanics. [15] [16] With the wave-vector convention, Eqs. (1a) and (1b) become
A correct approach
Equation (2) with x = d then becomes
A correct approach
representing constructive interference, in agreement with the experiment. Refer to Figs. 3(c) and (d) for an illustration of the correct result for both cosine and sine waves.
Conclusion
Conventions that switch the sign of t to represent waves traveling to the left introduce an inadvertent hidden phase shift, which can lead to misconceptions. We have shown that the common wave convention for traveling waves found frequently in introductory texts can result in the erroneous conclusion given by Eq. (2). A challenge can be posed to students to properly describe the physics with the common convention by introducing a phase shift. When the waves leave in phase, the waves interfere constructively at the microphone, which is located at the midpoint between the two speakers. However, the wave convention used with the odd sine functions to soniasoto@example.com. (2) introduces an accidental phase shift of 180 o for one of the waves.
Conclusion
Using the wave-vector convention, one can proceed in a straightforward fashion to shift the source of the wave and reverse the traveling direction to arrive at the correct result. 

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