Mechanics

Download Higher Mechanics (1st edition, 1920) by Horace Lamb, Sc.D., LL.D., F.R.S. PDF

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By Horace Lamb, Sc.D., LL.D., F.R.S.

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Additional resources for Higher Mechanics (1st edition, 1920)

Example text

W ∂y = 0. 1 ln |awk + by| + Φ(w). b ∂w ∂w + (aw k + bxm) = cxn. 45 with f (w) = awk , g(x) = bxm , and h(x) = cxn . 51. ∂w ∂x + (aw k + bxm) ∂w ∂y = cw n. 46 with f (w) = awk , g(x) = bxm , and h(w) = cwn . ∂w ∂w + axny mw k = 0. ∂x ∂y General solution: a 1 y 1–m – xn+1 wk = Φ(w) 1–m n+1 a ln |y| – xn+1 wk = Φ(w) n+1 1 y 1–m – awk ln |x| = Φ(w) 1–m ln |y| – awk ln |x| = Φ(w) 52. 53. if n ≠ –1, m = 1; if n = –1, m ≠ 1; if n = –1, m = 1. ∂w ∂w +k = 1. ∂x ∂y du = x + Φ(y – kx), –n cu + a + bk (ax + by + cw)n General solution: 54.

X y= f G(t) – G(x) + w dt + Φ w – G(x) , where G(x) = g(x) dx. 2. EQUATIONS WITH TWO INDEPENDENT VARIABLES CONTAINING ARBITRARY FUNCTIONS Page 27 27 ∂w ∂w + f (w) = g(y). ∂x ∂y General solution: 38. y x= ψ G(t) – G(y) + F (w) dt + Φ F (w) – G(y) , y0 where G(y) = g(y) dy and F (w) = parametrically by ψ = f (w) dw. The function ψ = ψ(z) is defined 1 , z = F (w). f (w) ∂w ∂w + f (w) = g(y – ax). ∂x ∂y This is a model equation describing nonlinear waves issuing from a moving source (the variables x and y play the role of time and the spatial coordinate, respectively, and a is the source velocity).

X ∂y General solution: Φ w, y – awk x – bx = 0 47. Φ w – cx, 48. ∂w ∂x + (aw k + bx) ∂w ∂y awk+1 + bw – cy k+1 =0 if c = 0, if c ≠ 0. = 0. General solution: y = axwk + 12 bx2 + Φ(w). 49. ∂w ∂x + (aw k + by) General solution: x = 50. ∂w ∂y = 0. 1 ln |awk + by| + Φ(w). b ∂w ∂w + (aw k + bxm) = cxn. 45 with f (w) = awk , g(x) = bxm , and h(x) = cxn . 51. ∂w ∂x + (aw k + bxm) ∂w ∂y = cw n. 46 with f (w) = awk , g(x) = bxm , and h(w) = cwn . ∂w ∂w + axny mw k = 0. ∂x ∂y General solution: a 1 y 1–m – xn+1 wk = Φ(w) 1–m n+1 a ln |y| – xn+1 wk = Φ(w) n+1 1 y 1–m – awk ln |x| = Φ(w) 1–m ln |y| – awk ln |x| = Φ(w) 52.

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