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Initial Condition Example 3

Let

f(x, 0) = ⎧⎨⎩ Hx if 0 ≤ x ≤ L ⁄ 2 H(1 − x) if L ⁄ 2 ≤ x ≤ L

This well-known initial condition results in a linear increase from 0 to H of the deflection as x moves from 0 to L ⁄ 2, with a decrease in the deflection from H to 0 as x changes from L ⁄ 2 to L.

Once again applying orthogonality results in the equation

(13) ^∞⎲⎳_n = 1^∞⎲⎳_m = 1A_n^L⌠⌡₀sin⎛⎝(nπx)/(L)⎞⎠sin⎛⎝(mπx)/(L)⎞⎠dx = ^∞⎲⎳_m = 1H^L ⁄ 2⌠⌡₀xsin⎛⎝(mπx)/(L)⎞⎠dx + ^∞⎲⎳_m = 1H^L⌠⌡_L ⁄ 2(1 − x)sin⎛⎝(mπx)/(L)⎞⎠dx

The left hand side of the equation results in an orthogonal solution. The terms on the right hand side are not orthogonal, but can be integrated. For odd n terms (when n = m), then A_n = 0, so half of the infinite series disappears. For even n terms (when n = m), then A_n = ( − 1)ⁿ(H)/(n²π²). Thus, the solution of the equation is

(14) f(x, t) = (H)/(π²)^∞⎲⎳_n = 1( − 1)ⁿn²sin⎛⎝(nπx)/(L)⎞⎠e^{− (ncπ ⁄ L)²t}

EXAMPLE 2: WAVE EQUATION

See the section on the vibrating string, which is based on the wave equation.

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