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Thread: Can "wave" be given a precise mathematical definition?

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    Can "wave" be given a precise mathematical definition?

    Is there a precise mathematical definition for a "wave"? I think not. Are there ongoing attempts to formulate one? Or is it accepted that "wave" should remain a mathematically ambiguous concept?

    One could make the attempt: "A wave is any solution to the wave equation". However, although the phrase "the wave equation" is used to refer to a particular equation (https://en.wikipedia.org/wiki/Wave_equation) there are other partial differential equations whose solutions are also regarded as waves. If a "wave" is defined as a solution to some "wave equation", what is the definition of a "wave equation"?

    According to http://www.scholarpedia.org/article/...onlinear_waves
    Now, whilst most people have a general notion of what a wave is, based on their everyday experience, it is not easy to formulate a definition that will satisfy everyone engaged in or interested in this wide ranging subject. In fact, many technical works related to waves eschew a formal definition altogether and introduce the concept by a series of examples; for example, Physics of waves [Elm-69] and Hydrodynamics [Lam-93]. Nevertheless, it is useful to at least make an attempt and a selection of various definitions from normally authoritative sources is given below:
    Evaluating those definitions, the article says:
    The variety of definitions given above, and their clearly differing degrees of clarity, confirm that 'wave' is indeed not an easy concept to define!

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    I'm far from an expert in mathematics, but I wonder if the difficulty is partly that the definition of a "wave" is not precise in the first place. My impression is that a "wave" means "the transmission of energy without the movement of matter." For example, there are lateral waves and longitudinal waves, and I imagine the mathematical description will vary depending on what is going on in the system.
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    Quote Originally Posted by Jens View Post
    I'm far from an expert in mathematics, but I wonder if the difficulty is partly that the definition of a "wave" is not precise in the first place.
    Do you mean that the definition of "wave" in physics and common speech is not precise - hence we have no guide to formulating a mathematical definition? I agree with that thought.

    My impression is that a "wave" means "the transmission of energy without the movement of matter." For example, there are lateral waves and longitudinal waves, and I imagine the mathematical description will vary depending on what is going on in the system.

    Ocean waves and sound waves involve the movement of matter - but it doesn't move great distances.

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    Quote Originally Posted by tashirosgt View Post
    Do you mean that the definition of "wave" in physics and common speech is not precise - hence we have no guide to formulating a mathematical definition? I agree with that thought.




    Ocean waves and sound waves involve the movement of matter - but it doesn't move great distances.
    More precisely, the matter moves shorter distances than distance moved by the wave and its energy.

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    Quote Originally Posted by Colin Robinson View Post
    More precisely, the matter moves shorter distances than distance moved by the wave and its energy.

    That thought might lead somewhere, but there are some roadblocks.

    In electromagnetic waves, there need be no material medium involved.

    Whether there must be a net movement of energy is an interesting question. Does plotting the y-coordinate of an idealized frictionless pendulum vs time give a graph that is a "wave"?

    In general, how do we assign a position to energy? Need energy have a position? I agree one can say "33 joules at location (3,-7,12)", but does this mean anything? For example, if we consider the gravitational potential energy between two masses, is that energy located in particular places?

    If we use energy as the motivation for a mathematical definition of wave, how do we introduce it? As far as I can see, the general setting for a "wave" would be a "field" in the physical sense of the term "field" ( not in the algebraic sense of "field"). A general mathematical structure for a "field" would be a function u(x,t), where t is a real number. The variable x may be a vector. The value of u (for a particular field) may be a scalar, a vector, or a more complicated mathematical structure such as an operator. We would attempt to define "wave" by saying which "fields" are fields with waves in them. How can something analogos to "energy" be brought into the picture?

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    Quote Originally Posted by tashirosgt View Post
    Is there a precise mathematical definition for a "wave"?
    Can you clarify what you mean by a "mathematical definition"? A wave is a physical phenomenon. We can describe it mathematically, but by itself it is not a mathematical construct.
    "I'm planning to live forever. So far, that's working perfectly." Steven Wright

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    Quote Originally Posted by Noclevername View Post
    Can you clarify what you mean by a "mathematical definition"? A wave is a physical phenomenon. We can describe it mathematically, but by itself it is not a mathematical construct.
    (I assume you know what is meant by a mathematical definition in general!) As to the case at hand. If I were to ask for a mathematical definition of "velocity", it could be answered by citing the mathematical definition of derivative. That mathematical definition can be applied to situations where the concept of "instantaneous rate of change" is needed. the velocity of an object being one particular example. Is there a similar situation for the concept of "wave"?

    Another way of looking at it: Suppose we looked at all examples of physical phenomena people have chosen to call "waves". What general rules are they following to make such a classification? If Alice says "This is a wave" and Bob says "No, it isn't", what criteria are used to settle the dispute?

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    Quote Originally Posted by tashirosgt View Post
    Do you mean that the definition of "wave" in physics and common speech is not precise - hence we have no guide to formulating a mathematical definition? I agree with that thought.
    Yes, basically. I mean, there are mathematical waves, for example a sine wave, and you can define that easily as x = sin(y). And it makes a wave. But an ocean wave is really an aggregate of individual particles moving in complex ways.

    Quote Originally Posted by tashirosgt View Post
    Ocean waves and sound waves involve the movement of matter - but it doesn't move great distances.
    Yes, what is meant by "movement" is "net movement." Typically, the individual particles oscillate while energy is moved. But that also brings up another issue of what a wave is. We sometimes use the word "wave" to refer to things that are not really waves in the way I defined it, i.e. tidal waves and shock waves, where matter does move a lot.
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    Quote Originally Posted by Noclevername View Post
    Can you clarify what you mean by a "mathematical definition"? A wave is a physical phenomenon. We can describe it mathematically, but by itself it is not a mathematical construct.
    I think a wave can be both. A sine wave is a mathematical shape (which also occurs physically as an approximation), and we called it a sine wave.
    As above, so below

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    Quote Originally Posted by tashirosgt View Post
    Another way of looking at it: Suppose we looked at all examples of physical phenomena people have chosen to call "waves". What general rules are they following to make such a classification? If Alice says "This is a wave" and Bob says "No, it isn't", what criteria are used to settle the dispute?
    Maybe someone with a stronger physics or mathematics background would disagree, but I think that to a large extent it's like, if it looks like a wave it's a wave. We often say "in a medium," but as you rightly pointed out, EM waves don't require a medium in the conventional sense. So I think that some waves can be described fairly simply mathematically, like a sine wave, but a sound wave is quite complicated, because it's individual air molecules interacting with others, and you can try to model it mathematically but it becomes a many-bodies problem.
    As above, so below

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    So it seems like more of a semantics problem than it is a mathematical one.
    "I'm planning to live forever. So far, that's working perfectly." Steven Wright

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    Quote Originally Posted by Noclevername View Post
    So it seems like more of a semantics problem than it is a mathematical one.
    Yeah, I think that (like many things) it is basically an issue of semantics. I think the concept of a "wave" is fairly nebulous.
    As above, so below

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    Quote Originally Posted by Noclevername View Post
    So it seems like more of a semantics problem than it is a mathematical one.
    I agree, but worrying about semantics problems can be productive. Although just throwing together words or symbols doesn't necessarily produce a sensible idea, attempting to assign precise ideas to words and symbols can be enlightening.

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