1) square wave

Consider a square wave of length . Over the range , this can be written as

(1)

where is the Heaviside step function. Since , the function is odd, so , and

(2)

reduces to

			(3)
			(4)
			(5)
			(6)

The Fourier series is therefore

2) sawtooth

Consider a string of length plucked at the right end and fixed at the left. The functional form of this configuration is

(1)

The components of the Fourier series are therefore given by

			(2)
			(3)
			(4)
			(5)
			(6)
			(7)
			(8)
			(9)

The Fourier series is therefore given by

			(10)
			(11)

3) triangle

Consider a symmetric triangle wave of period . Since the function is odd,

			(1)
			(2)

and

			(3)
			(4)
			(5)
			(6)

The Fourier series for the triangle wave is therefore

(7)

Now consider the asymmetric triangle wave pinned an -distance which is ()th of the distance . The displacement as a function of is then

(8)

The coefficients are therefore

			(9)
			(10)
			(11)

Taking gives the same Fourier series as before.

4) Semi-circle

Given a semicircular hump

			(1)
			(2)

the Fourier coefficients are

			(3)
			(4)
			(5)

where is a Bessel function of the first kind, so the Fourier series is therefore