What is a Wave?
Olivier Massin
Abstract: Are the waves we see moving at the surface of the sea processes, properties, substances, abstract entities, illusions… ? This paper argues first, following Simons (1987), that the temptation to categorize waves as processes —episodes that unfold over time— should be resisted, because processes, unlike waves, cannot move. On that basis, Simons argues that waves should instead be viewed as disturbances in a medium: dependent continuants continuously moving from a substance to another. I argue that the disturbance view fails to adequately account for the independent causal powers of waves, such as their ability to exert force. Instead, I defend the thesis that surface waves are fluent substances: material objects that persist through time (endure) while continuously exchanging their material constitution. This account resolves the problem of wave motion and offers a coherent framework for understanding the identity conditions of waves, distinguishing them from both static fluent objects (like rivers) and pure processes.
1. Introduction
We see waves moving across the ocean’s surface. Surfers track their approach to harness their momentum. Sailors navigate their swells to ensure safety. Engineers build fortifications to dissipate their kinetic energy and mitigate coastal erosion. Waves transport sediments and reshape the underwater topography: they form sandbanks and sculpt the seabed. But what are waves?
Sciences teach us a great deal about waves: their formation, propagation, dissipation, superposition, refraction, frequency, amplitude, power, force, the motion of the water molecules that underlie them, etc. Raichlen (2012) presents, in a remarkably clear and synthetic manner, the main aspects of scientific knowledge about waves. No less remarkably, throughout the entire book he avoids assigning any ontological category to waves: they are never described as processes, substances, perturbation, or properties. This is the question to be asked here: given our ordinary and scientific knowledge about waves, what is their ontological status?
To address that question, this paper focusses for a start on the ontology of surface water waves. First, I evaluate and reject the view that waves are processes (occurrents), arguing that this view fails to account for the fact that waves can move. Second, I assess Simons (1987)’s proposal that waves are disturbances of the water medium, and argues that this view though it does vindicates the motion of the waves, fails to account for the forces they exert. Third, I propose and defend the view that waves are fluent substances: independent continuants that move by continuously changing their constituent matter. In the fourth section, I address some objections.
2. The Process View
The temptation to classify waves as processes is strong. In contrast to substances, which are typically static and do not deform, waves on the sea’s surface are constantly moving and changing shape. When we throw a stone into a pond, we create circular waves: what we seem to have done is to initiate a process. A wave has a duration, it seems to ‘enfold’ between a point of origin and a point of dissipation, and it consists of phases of oscillation. Accordingly, to illustrate how a process ontology should be understood, Dupré and Nicholson (2018) spontaneously use the example of ocean waves.
The view waves are processes is often taken for granted, but rarely explicitly defended. One exception is Batchelor and Hastings (2012):
ordinary waves, such as ocean waves and sound waves, are processes. Alternatives to this view that we will discuss are that waves are substantial (objects), that they are categorical properties (qualities) like shapes, and that they are realizable properties (dispositions).
In defence of the view that waves are processes, we observe that processes are occurrents, and they are countable, temporally limited and superposable. (Batchelor and Hastings, 2012)
On this proposal, all waves are processes, and all processes are occurrents. Occurrents come in two kinds: instantaneous occurrents—sometimes called events—which lack temporal thickness (for example, beginnings and endings), and temporally extended occurrents, which are processes. Processes unfold over time: they persist by having distinct phases or temporal parts. Processes include conversations, cell meiosis, rugby matches, the fall of a leaf, the expansion of the universe, and—if Batchelor and Hastings are right—waves.
Although wave processualism is initially attractive, it faces an important objection first raised by Simons (1987, 308). Waves can move. Processes cannot move. Hence wave are not processes.
That waves can move is, we take it, a platitude about waves. A theory that denies the motion of waves would simply be changing the subject. The phenomenological, linguistic, and scientific evidence that waves move is overwhelming. Surfers wait for waves to approach them; we speak of the velocity of waves, and scientists measure and explain it.
The reason processes cannot move is the following (see Dretske, 1967; Hacker, 1982). Imagine you are eating breakfast on a train traveling from Venice to Florence. Your breakfast is a process composed of various temporal parts—sipping coffee, eating a croissant, and so on. As the train moves, you might drink your coffee near Padova and eat your croissant near Modena. Has the breakfast moved? The cup has moved, the train has moved, and you have moved—none of these are processes. But the breakfast itself has not moved. Rather, distinct phases of the breakfast process have taken place at distinct spatial locations. Processes may unfold across different places, but this does not constitute motion, for motion requires being entirely located at different places at different times, whereas the process is only partly located at different places at different times.[1]
3. The Disturbance View
The reason why processes cannot move is that they persist by having different temporal parts. For an entity to be able to move, it must persist without having temporal parts—that is, it must be wholly present at each moment of its existence. This is the very nature of continuants, the other chief category of temporal entities. Both continuants and processes persist through time, but continuants do so while lacking temporal parts. A football match, for instance, has a first and a second half—it is an occurrent. The ball, the goalposts, and the stadium, which do not have temporal phases, are continuants. The players’ lives, however, do have temporal phases—their youth, the summer of 2012, and so on. While the players are continuants, their lives are occurrents, more precisely, processes.
If waves are continuants because they move, what kind of continuants are they? A familiar distinction divides continuants into independent continuants – substances – and dependent continuants, which include states. Simons (1987) argues that waves form a special subclass of dependent continuants, which he calls disturbances. To say that a disturbance is a dependent continuant is to say that, unlike substances, which can exist on their own, a dependent continuant (like a knot in a rope, a smile on a face, or a shadow) requires a host substance. On this view, the water is the substance, and the wave is a parasitic entity, a structural modification travelling through it.
Although they are dependent continuants, disturbances are not accidents, Simons insists. Accidents cannot migrate from one substance to another: Eve’s smile cannot become Bob’s. But a single wave can, as it travels, depend successively on different portions of water. The categories discussed so far can be represented in the following tree:
The disturbance view has significant advantages. First, unlike the process view, it allows waves to move. Second, it does this while explaining a puzzling fact: when a wave moves, the water particles do not move with it. They stay roughly where they are, tracing small circular orbits, largely inferior to the wavelength[2]. What propagates is not matter but energy—or, in terms of forces, a dynamical imbalance: a travelling disequilibrium between gravitational and pressure forces, transmitted from one portion of water to the next. For that reason, treating waves as ordinary substances would be strange: substances do not leave their matter behind as they move.
Third, the disturbance view captures the affinity between waves and processes, preserving the grain of truth in the process view: waves depend on substances, but also on the processes unfolding within them. Here is Simons:
The wave is maintained by a process transferring motion from particle to particle of the medium, but it is not identical with this process: we cannot say that the process has a certain shape or velocity of propagation. But there is no ontological problem of the status of moving disturbances; they are simply a special and interesting kind of continuant: moments which continuously change their fundaments. (Simons, 1987: 308).
There is, however, an important problem for the disturbance view. Waves carry energy and exert forces. They shape coasts and seabeds, erode cliffs, propel surfers forward, destroy sea walls, power turbines, sometimes take lives. But it is doubtful that disturbances, depending on matter, but not containing matter, can do this. Waves are ontologically too thin, on the disturbance approach, to exert forces.
Simons (1987, pp. 308–309) argues convincingly that this category solves the problem of motion without committing us to the idea that the water itself moves with the wave. The disturbance theory, however, faces another problem. On that view, the water is not a part or internal constituent of the wave: the matter remains, as it were, external to it. A wave is merely a travelling form or deformation; it contains no matter. Yet it is plausible that only material entities—entities made of matter—can exert forces and carry energy. A disturbance is too thin for this: it has no mass; only the medium does. If we want to maintain that the wave strikes the cliff with considerable force, a force that can be measured, then the wave must, in some sense, include the matter that constitutes it at that instant. This consideration pushes us toward the substance view.
4. The Fluent Substances View
That waves can move and exert forces suggests they are substances—contra both the process view and the disturbance view. This proposal, however, faces an immediate objection: how can waves be substances if their matter does not travel with them?
We propose that waves are best understood as fluent substances. A fluent substance is an independent continuant made of matter. But unlike ordinary material objects, whose matter moves along with them, a fluent substance moves—or remains at rest—by continuously changing its material constitution. Instead of saying, with Simons, that waves are “moments which continuously change their fundaments,” we should say that waves are substances which continuously change their material constitution. Rather than treating waves as dependent continuants whose ever-shifting matter lies outside them, the substance view treats them as independent continuants whose continuously changing matter belongs to them.
Like the disturbance view, the substance view introduces a new ontological category. However, instead of positing a new kind of dependent continuant—disturbances, distinct from accidents—it posits a new kind of independent continuant: fluent substances, distinct from ordinary material substances. The advantage of this move is that it allows substances both to move—contrary to the process view—and to exert forces—contrary to the disturbance view.
Waves are not the only fluent substances. In their nicely entitled paper “The Water Falls, but the Waterfall Does Not Fall,” Galton and Mizoguchi (2009) consider another kind of fluent substance (the label is not theirs): streams, rivers, fountains, waterfalls, and the like. Unlike waves, these are fluent substances that remain in place precisely because water flows through them (they later add queues as another example of the kind). Interestingly, Galton and Mizoguchi contrast two views that closely mirror the distinction drawn here between the disturbance and substance conceptions of waves. On the first view—analogous to the disturbance view—rivers, streams, and the like are compared to conduits: forms (the riverbed) that depend on the moving water but do not include it. On the second view—analogous to the present substance view—the river contains water as a part, and the water’s flow is a process internal to the river. Galton and Mizoguchi do not adjudicate between these two views, but they note that the substance view is more general—since ocean currents, they observe, cannot be accommodated within a conduit/disturbance framework—and closer to the everyday notion of a river. I propose that, rather than two ways of viewing the same object, we have here two different objects: the conduit view captures not the river but the riverbed; only the substance view captures the river itself. Likewise, the disturbance view captures not the wave itself but only its form; only the substance view captures the entire wave—its form together with its matter.
Fluent substances are material substances that move—or remain at rest—by continuously changing their matter. Let us call those fluent substances that, like waves, move by changing their matter dynamic fluent substances, and those that, like waterfalls, remain at rest by changing their matter static fluent substances. We thus obtain the following typology:
One way to bring out the difference between bodies and fluent substances is to consider a variation on the Ship of Theseus thought experiment—call it the Wave of Theseus. Theseus is surfing a wave at time t. Suppose we tag every molecule constituting this wave at t (more on this below), and then, at t + 1, reassemble exactly those same molecules into a wave-shaped configuration. Which is Theseus’s wave: the one he continues to surf at t + 1, or the one we have artificially reassembled?
In the case of a fluent substance such as a wave—unlike in the case of a ship—there is no temptation to regard the reassembled wave as identical to the original. The reassembled wave is clearly a different wave. As it happens, this is not a far-fetched scenario: in ordinary conditions, successive regular waves are in fact constituted, to a large extent, by the very same water molecules. The molecules that constituted the wave at t typically constitute the following wave at t + 1. Yet despite this substantial material overlap, successive waves are clearly numerically distinct.
The identity of a wave is therefore tied to the continuity of its form and of the energy transfer it embodies, not to the retention of particular portions of matter. This contrasts sharply with the Ship of Theseus, where material continuity is at least a strong candidate for grounding identity. For waves, by contrast, material turnover is not merely compatible with persistence; it is essential to their nature as fluent substances.
5. The Boundaries of Waves
If waves are substances, what are their boundaries? How are we to delineate the water particles that belong to a wave from those that do not? The question is initially puzzling. Absent a clear answer, process theorists may take this to count in favor of the process view, invoking the familiar claim that occurrents, unlike substances, resist precise spatial localization. The question of their boundaries divides into two parts: (i) what are the surface boundaries of waves? and (ii) what are the underwater boundaries of waves?
(i) The surface boundaries of waves. The crest of a waves is the highest point (or line) of the wave. In two dimensions, the troughs of a waves are the two lowest point of the wave, on both side of its crest:
Cayley (1859) and Maxwell (1870) showed how to extend this approach to three dimensions by introducing slope lines (see Authorname, XXXX, for a presentation). This mathematical solution to the problem of the lower boundaries of mountains aligns with the intuitive idea that the lower limits of a mountain are given by the valley floors that border it. If correct, waves, like mountains, do have sharp, natural lower boundaries, pace Smith and Mark (2003). There is, however, a caveat: these natural boundaries depend on the chosen level of granularity, which may itself be somewhat arbitrary. If the level of granularity is sufficiently fine-grained, then the very first small stream encountered when descending the Matterhorn already marks its lower boundary. The same holds for waves: each flank of a large wave exhibits smaller secondary undulations; if we wish to disregard these, we must fix a higher level of granularity.
(ii) The underwater boundaries of waves. If waves include their matter and are not merely surface disturbances, as the substance view maintains, what are their underwater boundaries? One proposal—in two dimensions—would be simply to draw a straight line between the two troughs. This proposal should, however, be rejected. The motivation for including the water within the wave, to recall, was the wave’s capacity to exert forces and to transfer energy. That capacity does not depend solely on what happens within the height of the wave—the height being the vertical distance from trough to crest. It depends, instead, on the depth of the wave: the depth below the still water level at which the orbital motion of water particles becomes negligible, a depth that is typically about half the wavelength a depth that is typically about half the wavelength (i.e., the horizontal distance between successive crests).
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The first proposal, then, is that the lower boundary of a wave corresponds to its base –not its trough. The second proposal is that a wave is constituted by water particles over the course of a single orbital cycle; as soon as a particle begins a new orbit—when it reaches the lowest point of its trajectory—it becomes constitutive of a different wave. One might think of this on the model of football fans doing the wave in a stadium. A spectator belongs to one wave insofar as they stand up and sit back down once; they belong to the next wave when they stand up and sit back down again.
6. Objections and Responses
Objection 1: All substances are fluent and fluent substances just are processes. The fact that substances—especially living ones—continuously change their matter has led several authors to liken organisms to rivers, whirlpools, waterfalls, and flames. Nicholson (2018) offers a brief but illuminating historical survey of writers who, since the nineteenth century, have drawn such comparisons, and he endorses the analogy himself. On that basis, he rejects substantialism in favour of a processual view according to which “organisms are themselvesprocesses” (italics in original). Nicholson’s argument appears to be this:
P1 Waterfall and the like are processes.
P2 Organisms belong to the same category as waterfalls, insofar as their matter changes continuously.
C Organisms are processes.
The two premises are incompatible with the present proposal, which holds that (i) waterfalls are (static) fluent substances, and (ii) organisms and waterfalls fall into different ontological categories, namely bodies and fluent substances. Both premises are false. Concerning P1, Nicholson simply assumes that streams, flames, and waterfalls are processes. It is certainly true that they depend on processes. But this is not sufficient to make them processes themselves. As we have argued, they are substances, albeit fluent ones, as attested by the fact that they can move. This view is unlikely to be welcomed by processualists, since it entails that one class of paradigmatic examples they use to oppose substantivalism are, in fact, substances. Note, however, that this proposal remains compatible with a weaker version of processualism, according to which processes, rather than substances, are the ultimate constituents of reality. It is not, however, compatible with the stronger thesis that processes are the only constituents of reality. A world composed solely of fluent substances may well be a world in which processes are ontologically fundamental, but it is not, for all that, a world without substances.
The reason why P2 is false is this: even if it is true that organisms continuously change their matter, this is not sufficient to place them in the same category as fluent substances. On the present proposal, fluent substances are not merely substances whose matter is continuously replaced. Waves not only undergo continuous material turnover; they move by continuously changing their matter. Nicholson might reply that organisms, too, are able to move thanks to continual cellular renewal. But the connection between material change and motion is far tighter in the case of waves. For organisms, material turnover merely enables motion; it does not constitute it. For waves, by contrast, continuous material replacement is the very way they move—just as walking is the way quadrupeds move and slithering the way snakes move. A wave’s motion consists in this replacement. Organisms, by contrast, do not leave their cells behind as they go.
Objection 2: Electromagnetic Waves have no matter. A second objection is that the fluent-substance view of waves applies only to mechanical waves and cannot be generalized to electromagnetic waves, which lack constitutive matter. Unlike mechanical waves, electromagnetic waves do not propagate through a material medium but appear to travel through a vacuum. It would be surprising if mechanical waves were substances while electromagnetic waves belonged to an entirely different ontological category—such as processes or disturbances.
There is, however, another option. One may identify the matter of electromagnetic waves with electromagnetic fields. An electromagnetic wave would then correspond to a modification of the field, in much the same way as a water wave corresponds to a modification of the water. In quantum field theory, fields—electric and magnetic among others—can be regarded as fundamental; electromagnetic waves supervene on variations in these fields. While the fields themselves do not move, the waves that depend on them do propagate. One can think of this in terms of a metal ball that moves as a series of electromagnets are activated successively along its path. The magnets (and their fields) do not move, but the wave that the ball “surfs” does.
One might object that it is misleading to equate fields with the matter of electromagnetic waves, on the grounds that fields, unlike water, are not composed of particles. It lies beyond the scope of this paper to settle the question of what it is for something to count as material and whether atoms or elementary particles are essential to matter. It is worth noting, however, that on dynamical conceptions of matter, one of its essential features is its disposition to exert forces. Fields, by their very nature, are precisely dispositions to exert forces. From this perspective, the proposal to treat fields as the matter of electromagnetic waves is, on closer inspection, not as outlandish as it may initially appear.
7. Conclusion
The view that waves are processes fails to account for their motion. The view that they are mere disturbances fails to capture their robust causal powers. The most coherent metaphysics of waves posits them as dynamic fluent substances. This category allows us to treat waves as genuine parts of the furniture of the world—entities that move, strike, and persist—while acknowledging their unique dependence on a flux of underlying matter. The wave must change its matter to move, just as an organism must metabolize to survive.
References
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[1] More precisely requires an entity to be wholly (and exactly) located at different places at different times. By being exactly located at a place, we mean: (i) being entirely located at that place—that is, all parts of the entity are contained within it. (Indeed, being only partially located at different places at successive times is not sufficient for motion; in fact, most stationary objects satisfy that condition.) (ii) being pervasively located at that place—that is, no part of the place is free from the entity. Both conditions are necessary for defining motion: the fact that the Alps are partly located in Italy at t₁ and partly located in Austria at t₂ does not entail that they move. Likewise, the fact that Paris is non-pervasively located in France at t₁ and non-pervasively located in Europe at t₂ does not entail that Paris moves either (on the relatoin between entire, pervasive and exact location, see Parsons, 2007).
[2] See https://www.youtube.com/watch?v=NShUBfJQEHk for a visualisation.
