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Harmonics formed on waves are generated as component frequencies of a fundamental frequency of the wave.

The fundamental & the higher frequencies(harmonics) generate periodic signals from the original wave. And every periodic signal can be written as a sum of the variuos harmonics using the Fourier series.

Hence, to find the various harmonics using the fourier series, we can use…

nth harmonic : (a_ncosx+b_nsinx)

where,

&

where p is the number of unique values of the function y. The following example will make things a bit more clear…

Example : y is a function of x periodic with period 2pi. Some experimental values of y are given below calculated for certain values of x. Expand y to 2 harmonics.

Solution :

Clearly, in the above, p=6,

& We simply need to find:

1st harmonic + 2nd harmonic = (a₁cosx+b₁sinx) + (a₂cos2x+b₂sin2x)

So, all we need is a₁, b₁, a₂ &
b₂

for which we use the formula mentioned above:

&

where x_i=0, 60, 120… & so on.

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Here are some important properties of the Laplace Transform F(s) being the Laplace transform of f(t).

Initial Value Theorem
Final Value Theorem
periodic with a period T

Above, F₁(s) is the Laplace transform of f(t) for the first cycle.

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In the previous post on Fourier Series, we looked at functions periodic with period 2pi. Now, we’ll take a look at Fourier series for functions having an arbitary period, lets say, some period 2L.

The general formulas we would need for finding the Fourier series are as follows…

The Series:

The Constants:

These apply when the period given in the question are [-L, L]. We would modify the limits of integration in the above depending on the given interval.

Further, for functions periodic with a period 2pi, we only need to put L=pi in the above formulas. You can find those formula here.

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Once you know the basic Fourier series & the Euler’s formula, you may find the Fourier series of a function…

Lets say we have a function,

on the interval…

We have here a simple saw tooth function which we’ve made periodic with a period 2pi. The function would look something like so…

To start with, we have the interval correpsonding to c to c+2pi which gives c=-pi.

Hence, the values of the coefficients would now be given by…

Hence, on integration we obtain…

This would generate the following Fourier series for our function f(x)…

The procedure to find a Fourier Series cuts down to only a few steps if the function is odd or even. We would look particular cases to find Fourier Series for Odd/Even Functions.

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One of the equations you would come across during higher studies would be the heat equation. Its another partial differential equation which we can solve using something called the Fourier series… Thanks to Joseph Fourier who introduced this series for the same purpose.

Using the Fourier Series, we simply decompose a particular real-valued function into terms containing the sine & cosine functions.

This function, lets say f(x) is taken to be periodic with a period 2pi. We could further extent the definition of the Fourier Series for functions with arbitrary periods.

The Fourier Series of a function f(x) could we written down as…

In the above decomposition, are called Fourier Coefficients or Euler’s Coefficients.

The former because they appear in the Fourier Series & the latter because we obtain their values using the Euler’s Formula.

Now the Euler’s Formula gives the the values of the Euler’s coefficients & for a function with period 2pi, we have…

Now, once you know the values of the coefficients(which would come out to be functions of n) you could find out the complete sum & hence, the complete Fourier series of the function f(x).

The value of ‘c’ here would depend on the interval specified with the function f(x).

We can perhaps do an example of the above to make things make sense…..

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