Fourier Series--Triangle Wave

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FourierSeriesTriangleWave

Consider a symmetric triangle wave T(x) of period 2L. Since the function is odd,

a_0 =

(1)

a_n = 0,

(2)

and

b_n = [画像: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)

= [画像:(32)/(pi^2n^2)cos(1/4npi)sin^3(1/4npi)]

(4)

= [画像:(32)/(pi^2n^2){0 n=0, 4, ...; 1/4 n=1, 5, ...; 0 n=2, 6, ...; -1/4 n=3, 7, ...]

(5)

= [画像: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

[画像: f(x)=8/(pi^2)sum_(n=1,3,5,...)^infty((-1)^((n-1)/2))/(n^2)sin((npix)/L). ]

(7)

FourierSeriesTriangleWaves

Now consider the asymmetric triangle wave pinned an x-distance which is (1/m)th of the distance L. The displacement as a function of x is then

[画像: 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

a_0 =

(9)

a_n =

(10)

b_n = [画像:-(2(-1)^nm^2)/(n^2(m-1)pi^2)sin[(n(m-1)pi)/m].]

(11)

Taking m=2 gives the same Fourier series as before.

See also

Fourier Series, Fourier Series--Sawtooth Wave, Fourier Series--Square Wave, Triangle Wave

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Cite this as:

Weisstein, Eric W. "Fourier Series--Triangle Wave." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FourierSeriesTriangleWave.html

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