Why does a wave on a string invert after reflection?

Question: Why does a wave pulse on a string invert after it is reflected from a rigid boundary?

Below is a conventional explanation on this question: We may assume an end of a string is rigidly clamped to a wall and a wave pulse on a string moves from left to right towards this end which is fixed. As the wave pulse approaches the fixed end, the string exerts an upward force on the wall or rigid boundary. According to Newton's third law of motion, the wall exerts an equal downward (or restoring) force on the end of the string. This restoring force generates an inverted wave pulse that propagates from right to left, having the same wave speed and amplitude as the incident wave. In addition, the displacement of the wave remains zero at the rigid boundary and the there is a phase change of 180^o. (To have a more complete understanding of the phenomenon, some textbook authors include explanations on a string that has a “free end.”) 

In general, we can derive a wave equation of a vibrating string that requires a small segment of the string obeying Newton’s second law of motion. However, we need more information to specify the complete movement of the vibrating string. That is, it is necessary to know initial conditions (position and velocity) of the string at a particular time and boundary conditions at a particular location on the string. The boundary condition may refer to the displacement of the string that is zero at the rigid boundary or the slope of the string that is zero if the end of the string is attached to a frictionless boundary. 

Alternatively, Pierce (2006) explains the inverted wave pulse by using a voltage wave. Imagine a voltage wave is traveling down a transmission line. If the transmission line is open at one end, the reflected voltage wave will be the negative of the incident voltage wave. If the transmission line is shorted at the point of reflection, the voltage is zero at that point instead. In other words, during the process of reflection, the incident voltage wave plus the reflected voltage wave must always be zero. Hence, we may infer that the reflected voltage wave must be the negative of the incident voltage wave during the process of reflection. 

How would Feynman answer?: 

Feynman did not provide an intuitive explanation by using Newton’s Third law of motion. In The Feynman Lectures on Physics, he has provided a mathematical explanation:  “Suppose that the string is held at one end, for example by fastening it to an “infinitely solid” wall. This can be expressed mathematically by saying that the displacement y of the string at the position x = 0 must be zero because the end does not move… we know that the general solution for the motion is the sum of two functions, F(x − ct) and G(x + ct)… (Feynman et al., 1963, section 49–1 The reflection of waves).” Note that the mathematical expression F(x − ct) represents a wave traveling in the string to the right at the speed c and G(x + ct) represents another wave traveling to the left at the speed c. Thus, the displacement y of the string can be expressed as F(x − ct) + G(x + ct) by using the principle of superposition. This physical principle is applicable to the vibrating string because it is a linear system.

We should not assume that Feynman has the attitude of “shut up and calculate” or he simply believes in the unreasonable effectiveness of mathematics. More important, Feynman would have a concern on the different interpretations of mathematics. According to Feynman, “[t]he next great awakening of human intellect may well produce a method of understanding the qualitative content of equations. Today we cannot. Today, we cannot see whether Schrödinger’s equation contains frogs, musical composers, or morality - or whether it does not. We cannot say whether something beyond it like God is needed, or not. And so we can all hold strong opinions either way (Feynman et al., 1964, p. 41-12).” It is possible to have different philosophical perspectives for a mathematical equation that is applied in the physical world. The “shut up and calculate” attitude could be attributed to Mermin (2004) instead of Feynman.

Interestingly, Feynman elaborates that we can imagine a hypothetical wave traveling in the opposite direction that is behind the wall. In his own words, “We say hypothetical because, of course, there is no string to vibrate on that side of the origin. The total motion of the string is to be regarded as the sum of these two waves in the region of positive x. As they reach the origin, they will always cancel at x = 0, and finally, the second (reflected) wave will be the only one to exist for positive x and it will, of course, be traveling in the opposite direction… (Feynman et al., 1963, section 49–1 The reflection of waves).” That is, we can understand the reflected wave by imagining an inverted wave that comes out from behind the wall. In short, we may assume that the string is connected to an infinitely massive string at x = 0. This explains the boundary condition in which the displacement of the string at x = 0 must always be zero.

Note: 

Feynman might mention problems of defining waves. During a BBC interview, Feynman (1994) explains that “[i]f I'm sitting next to a swimming pool, and somebody dives in, and she's not too pretty, then I can think about something else. I like to think about the waves that are formed in the water, and when lots of people have dived into the pool, there's a very great choppiness of all these waves all over the surface. Now to think that it's possible, maybe, that in those waves there's a clue as to what's happening in the pool: that an insect of sufficient cleverness could sit in the corner of the pool, and just by being disturbed by the waves and by the nature of the irregularities, the insect could figure out who jumped in where, and when, and what's happening all over the pool. It seems incredible, but that’s what we're doing when we looking at something… (p. 130).” 

References: 
1. Feynman, R. P. (1994). No Ordinary Genius: The Illustrated Richard Feynman. New York: W. W. Norton & Company. 

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley. 

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley. 

4. Mermin, N. D. (2004). Could Feynman have said this. Physics Today, 57(5), 10.

5. Pierce, J. R. (2006). Almost all about waves. New York: Dover.