Mathematical Physics Vol 1

Chapter 7. Partial differential equations

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i = 1 , 2) to the axes of the Cartesian coordinate system x and y are ma x = − S 1 cos α + S 2 cos β = 0 ⇒ S 1 cos α = S 2 cos β ≡ S , ⇒ cos α = S S 1 , cos β = S S 2 , where we have used the assumption that there is no displacement in the direction of the x -axis ( a x = 0). ma u = − S 1 sin α + S 2 sin β ⇒ S 2 sin β − S 1 sin α = ρ V ∂ 2 u ∂ t 2 , ∂ 2 u ∂ t 2 is the projection of acceleration a u , and u = u ( x , t ) . Given that m = ρ V , and V = ∆ ℓ · P , where ρ is the density, ∆ ℓ the length, and P the cross-sectional area of the observed wire, and if we assume that P is the unit area, by dividing the second equation by S , we obtain S 2 S sin β − S 1 S sin α = ρ ∆ ℓ S ∂ 2 u ∂ t 2 . Using (7.7), we further obtain tg β − tg α = ρ ∆ ℓ S ∂ 2 u ∂ t 2 . (7.8) Given that ∂ u ∂ x x = tg α i ∂ u ∂ x x + ∆ x = tg β , and ∆ ℓ ≈ ∆ x , we obtain ∂ u ∂ x x + ∆ x − ∂ u ∂ x x ∆ x = ρ S ∂ 2 u ∂ t 2 . Observe now the limit value, when ∆ x → 0 lim ∆ x → 0 1 ∆ x ∂ u ∂ x x + ∆ x − ∂ u ∂ x x = lim ∆ x → 0 ρ S ∂ 2 u ∂ t 2 . As the left hand side represents the second partial derivative for x (by definition), and the function under the limit value on the right hand side does not depend on ∆ x , we finally obtain (7.9) where c 2 = ρ / S . The relation (7.9) represent the so-called one-dimensional wave equation . (7.7) ∂ 2 u ∂ x 2 = c 2 ∂ 2 u ∂ t 2 ,

Example 236 Find the differential equation of the family of spheres of radius r and center in the x = y plane.