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Thread: Sine Wave

  1. #1 Sine Wave 
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    Hi All,

    maybe it is a silly question but i have to ask it. Why a sinusoidal wave is single frequency? I'm after an explanation without referring to Fourier transform or series.

    Thanks,
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    It is by definition.
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    Quote Originally Posted by mmgbmm View Post
    Hi All,

    maybe it is a silly question but i have to ask it. Why a sinusoidal wave is single frequency? I'm after an explanation without referring to Fourier transform or series.

    Thanks,
    Yours is not a silly question at all. However, the restrictions you place on the answer (no reference to Fourier) prevent us from saying anything other than what mathman said -- "it's by definition."

    Note that three parameters fully describe a sine wave: Amplitude, frequency and phase. Each of these parameters is a single number. So, mathematically -- by definition, if you will -- a sine wave has just a single frequency.

    Now, if you're willing to relax your constraints on us a bit, Fourier showed (with help from folks like Dirichlet) that we can represent a waveform as the sum of appropriately weighted and delayed sinusoids. That's not the only possible representation (see, e.g., Walsh functions and wavelet transforms), but setting that aside, the sine wave assumes the role of a sort of representational atom, if you will, in Fourier series/transforms. Since it's therefore not "made of" anything more elemental, there is only one frequency to speak of in describing it.
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    You can also think of it mechanically. If something oscillates with simple harmonic motion, or rotates at constant speed, it generates a sine wave:
    The Graph of the Sine Function
    Simple harmonic motion - Wikipedia, the free encyclopedia
    You can do everything right, strictly according to procedure, on the ocean and it'll still kill you, but if you're a good navigator at least you'll know where you were when you died.
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    Quote Originally Posted by tk421 View Post
    That's not the only possible representation
    Quote Originally Posted by tk421 View Post
    Since it's therefore not "made of" anything more elemental, there is only one frequency to speak of in describing it.

    The problem is that these two statements actually contradict each other. Given that there are many types of integral transformations by which a function (including the sinusoids) can be decomposed into a set of basis functions, what is it that makes the Fourier transformation and hence the sinusoids so special?
    A tensor equation that is valid in any coordinate system is valid in every coordinate system.
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    Quote Originally Posted by KJW View Post
    The problem is that these two statements actually contradict each other.
    Yes, which is precisely why I added the bit about "setting that aside."


    Given that there are many types of integral transformations by which a function (including the sinusoids) can be decomposed into a set of basis functions, what is it that makes the Fourier transformation and hence the sinusoids so special?
    What makes sinusoids special as basis functions is that sinusoids (exponentials, really) are eigenfunctions of linear systems, which dominate much of our physical world (although, of course, nonlinearity is extremely important). Why linearity should dominate is a deeper question than I can answer.
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    Thanks to all the replies.

    What KJW stated is exactly what I was thinking about. Sine and cosine decomposition is just one of many possible orthogonal bases that we can use.

    Quote Originally Posted by tk421 View Post

    What makes sinusoids special as basis functions is that sinusoids (exponentials, really) are eigenfunctions of linear systems, which dominate much of our physical world (although, of course, nonlinearity is extremely important). Why linearity should dominate is a deeper question than I can answer.
    Now, this is a very interesting approach. I need to think about it for a while
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    Quote Originally Posted by mmgbmm View Post
    Thanks to all the replies.

    What KJW stated is exactly what I was thinking about. Sine and cosine decomposition is just one of many possible orthogonal bases that we can use.



    Now, this is a very interesting approach. I need to think about it for a while
    Just consider how often we use linear equations to model physical phenomena, and the practical rationale for using sines becomes more evident.

    That said, some of my colleagues have argued that the reason we use linear equations so much is that they're relatively simple, and simplicity breeds. Use of sinusoids therefore may just be an artifact of our attraction to simple models.

    Although I agree with some of that argument, I take a somewhat different view, noting that Maxwell's equations and Schrodinger's equation are linear (the Einstein field equations of GR, however, are not, although linearized approaches are often used to enable derivation of approximate solutions). Thus, linear descriptions underpin much of what we encounter.
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  9. #9  
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    Quote Originally Posted by tk421 View Post
    Yes, which is precisely why I added the bit about "setting that aside."

    I know. It wasn't a criticism. I was just pointing that the common view that sinusoids are THE fundamental basis functions requires a deeper explanation.

    Quote Originally Posted by tk421 View Post
    What makes sinusoids special as basis functions is that sinusoids (exponentials, really) are eigenfunctions of linear systems

    That can't be the reason. The Sturm-Liouville equation is also linear, yet is general enough to produce a wide range of basis functions (as eigenfunctions).


    Quote Originally Posted by tk421 View Post
    which dominate much of our physical world

    It is my view that physics should not be used to explain mathematics. Quite the contrary, I seek a mathematical explanation of physics.


    As far as I can see at the moment, the one thing that sets the sinusoids apart from the other basis functions is that the linear second-order differential equation that defines them is independent of the independent variable.
    A tensor equation that is valid in any coordinate system is valid in every coordinate system.
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