Shape of electromagnetic waves?

[wil3]

Hello. In most texts I have read, EM radiation is depicted as sinusoidal in shape. I understand why this would be the case, as the oscillating fields are often the product of circular generators or alternating current, but is this always the case? For example, is the light we receive from stars sinusoidal as well? If so, why? Same with other sources of natural E&M, like atomic transitions that produce color.

I'm sure there is a very simple answer, but please help me understand this. Thank you.

 

[cepheid]

I seem to recall that any function whose dependence on position and time is of the form f(x±vt) will be a solution to the wave equation. In this case, the wave equation comes from Maxwell's equations for the electric and magnetic fields and hence dictates the nature of EM radiation.

The sinusoidal or "plane wave" solutions are significant because any other solution can be expressed as combinations of them. In general, of course, the light we receive is not monochromatic but consists of waves of many different frequencies, and the variation of the overall electric field as a function of time will depend on the amplitudes and phases of these various sinusoidal frequency components.

 

[Dr Lots-o'watts]

Although it may not actually be sinusoidal, it can always be expressed as a superposition of many (an infinity, with the first terms being more significant) sinusoidal waves.

 

[wil3]

Okay, so this is starting to look like what I was hoping the answer might be. Is the mathematical name for all of the superimposed sinusoids a Fourier series? I remember seeing a cool diagram of how to use a Fourier series to make a sawtooth wave.

Are Fourier series the equivalent of Taylor series, except that they represent the signal as a combination of sinusoidal terms rather than polynomials? I'm working from high school level calculus here, and so this is the best analog I can draw.

Thank you so much for these answers.

 

[Dr Lots-o'watts]

I wouldn't use the term "equivalent" but roughly, yes.

 

[Born2bwire]

I wouldn't use the term "equivalent" but roughly, yes.

Very roughly. A Taylor series is an expansion of a function that may not hold for all valid inputs to that function. A Fourier series, with certain exceptions (only continuous and periodic functions can be correctly expanded), is an expansion of a function using sine and cosine basis functions. However, unlike the Taylor's series, a Fourier series is valid across the same range as the original function.

 

[emxprt]

... Is the mathematical name for all of the superimposed sinusoids a Fourier series? Are Fourier series the equivalent of Taylor series, except that they represent the signal as a combination of sinusoidal terms rather than polynomials?

Other important differences: A Taylor series is a "local" approximations in that its accuracy is greatest near a single point. A Fourier series is a "global" approximation, whose overall accuracy is measured in terms of a global error measure.

Both types of series are examples of a general approach of representing a complicated function as a series of simpler functions. The Fourier series has several special properties that distinguish it and makes it more useful than many other types of series for certain problems: each component in a Fourier series is a valid solution to the wave equation (not so for the monomial components in a Taylor series), so that the propagation of each of these components can be tracked individually. Also, the Fourier components have the property that derivative operations on them are equivalent to multiplication by a constant. This makes them very simple and convenient for doing calculations associated with the wave equations, which involves derivative operators.