Section 37–3 An experiment with waves

 (Idealized conditions / Measure wave intensity / Interference pattern)

 

In this section, Feynman discusses the idealized experimental conditions, measurement of wave intensity, and interference pattern of double slit experiment involving water waves.

 

1. Idealized conditions:

“A small object labeled the ‘wave source’ is jiggled up and down by a motor and makes circular waves. To the right of the source we have again a wall with two holes, and beyond that is a second wall, which, to keep things simple, is an ‘absorber,’ so that there is no reflection of the waves that arrive there. This can be done by building a gradual sand ‘beach’ (Feynman et al., 1963, p. 37–3).”

 

The double slit experiment involving water waves can be described in terms of idealized experimental conditions. Based on Feynman’s description of the experiment, there are three possible idealizations: 1. Ideal Water Wave Generator: The wave source produces circular water waves with a constant wavelength. 2. Ideal wall: The first wall (with two holes) allows coherent water waves with the same amplitude to pass through, fulfilling the conditions for stable interference pattern. 3. Perfect absorber (second wall): The wall absorbs the water waves completely without any reflection; the interference pattern is formed due to the primary wave interactions without reflected waves. In summary, it is important to have coherent waves (or constant phase difference) such that the interference pattern is stable and visible.

 

It is possible that Feynman realized some shortcomings of his Caltech lecture and included two improvements in his Cornell lecture. Firstly, he introduces a single hole before the double-slit setup: the single hole acts as a source of coherent waves, ensuring the waves coming through the first barrier are of equal amplitude. The use of a single hole also ensures the waves through the double-slit are coherent (or same frequency), allowing the formation of clear interference pattern. Secondly, he excludes the absorber (gradual sand beach) possibly because it is not an ideal place for making measurements. The interference pattern near the absorber is less visible due to the lower wave intensity. However, Feynman’s placing of a cork in water only gives a basic visualization of the wave motion, but it is not an accurate method to measure wave intensity.

In his Cornell lecture, Feynman (1965) explains that, “[t]he source is now a big mass of stuff which is being shaken up and down in the water. The armour plate becomes a long line of barges or jetties with a gap in the water in between. Perhaps it would be better to do it with ripples than with big ocean waves; it sounds more sensible. I wiggle my finger up and down to make waves, and I have a little piece of wood as a barrier with a hole for the ripples to come through. Then I have a second barrier with two holes, and finally a detector. What do I do with the detector? What the detector detects is how much the water is jiggling. For instance, I put a cork in the water and measure how it moves up and down, and what I am going to measure in fact is the energy of the agitation of the cork, which is exactly proportional to the energy carried by the waves. One other thing: the jiggling is made very regular and perfect so that the waves are all the same space from one another (pp. 133-134).”

 

2. Measure wave intensity:

“In front of the beach we place a detector which can be moved back and forth in the x-direction, as before. The detector is now a device which measures the “intensity” of the wave motion... If we cover one hole at a time and measure the intensity distribution at the absorber we find the rather simple intensity curves shown in part (b) of the figure (Feynman et al., 1963, p. 37–3).”

 

It does not seem appropriate to measure the intensity of water waves at the absorber in the double slit experiment. By the time the waves reach the absorber (sand beach), much of their energy is absorbed, and therefore, the intensity measured at the absorber would be significantly lower than the initial wave intensity at earlier points along its path. Surely, Feynman was joking? The absorber is supposed to dissipate the energy of the incoming waves and prevent their reflection. Thus, the ideal place to measure the wave intensity should be farther away from the absorber, where the waves have a higher intensity. However, the placement of a detector in the water can create reflected waves, which could interfere with the incoming waves and distort the measurement results.

 

Water waves are an example of waves that involve a combination of both longitudinal and transverse motions.  Generally speaking, the height (vertical displacement) of water waves can be represented mathematically using complex functions or simply sine functions. In other words, the amplitude of the wave is a real number if it is expressed using a sine function, but the amplitude can be a complex number when it is expressed as a complex function. Importantly, the amplitude of water waves does not remain constant when they are moving in the direction of the gradual sand “beach.” It is better to use high-speed cameras combined with image processing software to accurately capture and analyze the wave amplitudes and intensities in real time.

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3. Interference pattern:

“The intensity I₁₂ observed when both holes are open is certainly not the sum of I₁ and I₂. We say that there is ‘interference’ of the two waves (Feynman et al., 1963, p. 37–4).”

  

Perhaps Feynman could have explained how the separation between two slits may affect the interference pattern. When the two slits are closer to each other, the overlapping water waves form a higher central peak as shown in the middle of the graph (See Fig 37-2 below). In his Cornell lecture, the two slits are drawn further apart, thus there is no central peak due to the reduced overlapping in the middle, but there are two separate peaks corresponding to each of the two slits (See Fig 29 below). Interestingly, another difference between the two interference patterns is that one decreases to almost zero as the distance x increases, but the one drawn in Cornell remains at almost half of the maximum intensity. Perhaps the two wave intensities were drawn differently because Feynman considered the detector was at the absorber in Fig 37-2, but the detector was far from the absorber in Fig 29.

 

For the purpose of visualizing interference patterns, Feynman has chosen water waves instead of light waves. In Feynman’s words, “the next waves of interest, that are easily seen by everyone and which are usually used as an example of waves in elementary courses, are water waves. As we shall soon see, they are the worst possible example, because they are in no respects like sound and light; they have all the complications that waves can have (Feynman et al., 1963, p. 51–7).” Thus, we should not expect the diffraction pattern of water wave from a single slit to be accurately represented by Fig 37-2 or Fig 29 as shown above. To be precise, the diffraction pattern of light waves through a single slit may be represented by the diagram (see below), but some may assume the water waves can have a similar diffraction pattern as light waves.

 

Source: Single Slit Diffraction: Definition, Formula, Types and Examples (geeksforgeeks.org)

The intensity I₁₂ observed when both holes are open is certainly not the sum of I₁ and I₂. We say that there is “interference” of the two waves. At some places (where the curve I₁₂ has its maxima) the waves are “in phase” and the wave peaks add together to give a large amplitude and, therefore, a large intensity. We say that the two waves are “interfering constructively” at such places. There will be such constructive interference wherever the distance from the detector to one hole is a whole number of wavelengths larger (or shorter) than the distance from the detector to the other hole (Feynman et al., 1963, p. 37–4).”

 

It should be worth mentioning that constructive interference is more than just an addition of two peaks; it is also about the combining of whole waveform (including peaks and troughs). Furthermore, it affects the rate of flow of energy, i.e., it is not only an increase in amplitude. We may define constructive interference as a phenomenon involving maximum fluctuation in intensity depending on the energy from both waves adding up, which leads to a greater energy flow in the regions where the waves reinforce each other. On the contrary, destructive interference refers to minimal fluctuation in intensity due to the energy contributions from two waves cancelling out. In essence, the wave energy is continuously being redistributed by interfering waves, but the intensity at regions of constructive interference could momentarily reach zero due to zero displacement for both waves, which depends on their phase relationship.

 

Review Questions:

1. How would you explain the idealized conditions of the double-slit experiment involving water waves?

2. How would you measure the wave intensity (at the absorber) in the context of double-slit experiment involving water waves?

3. How would you explain the interference pattern formed by the water waves?

 

The moral of the lesson: The interference pattern formed by water waves can be explained by the superposition of two waves, where the wave intensities combine non-linearly, creating regions of constructive interference (maximum fluctuation in intensity) and destructive interference (minimal fluctuation in intensity).

 

References:

1. Feynman, R. P. (1965). The character of physical law. Cambridge: MIT Press Feynman,

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.