Fourier Series--Triangle Wave

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

			(1)
			(2)

and

		$2/L{int_0^(L/2)x/(L/2)sin((npix)/L)dx+int_(L/2)^L[1-2/L(x-1/2L)]sin((npix)/L)dx}$	(3)
			(4)
		$(32)/(pi^2n^2){0 n=0, 4, ...; 1/4 n=1, 5, ...; 0 n=2, 6, ...; -1/4 n=3, 7, ...$	(5)
		$8/(pi^2n^2){(-1)^((n-1)/2) for n odd; 0 for n even.$	(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

$f_m(x)={(mx)/L for 0<=x<=L/m; 1-m/((m-1)L)(x-L/m) for L/m<=x<=2L-L/m; m/L(x-2L) for 2L-L/m<=x<=2L.$

(8)

The coefficients are therefore

Taking gives the same Fourier series as before.

See also