Mathematical Physics Vol 1

7.3 Linear and quasilinear first order PDE

341

and thus the respective projections of these forces on the z – axis are T ∆ y ( sin x β − sin x α ) ≈ T ∆ y ( tg x β − tg x α )= T ∆ y ∂ u ( x + ∆ x , y ) ∂ x −

∂ x ≈

∂ u ( x , y )

∂ 2 u ∂ x 2

≈ T ∆ x ∆ y

+ O 1 ( ∆ x , ∆ y ) ,

where

lim ∆ x → 0 ∆ y → 0

O 1 = 0 .

Similarly, for the other force we obtain T ∆ x ( sin y β − sin y α ) ≈ T ∆ y ( tg x β − tg y α )= T ∆ x

∂ y ≈

∂ u ( x , y + ∆ y ) ∂ y

∂ u ( x , y )

−

∂ 2 u ∂ y 2

≈ T ∆ x ∆ y

+ O 2 ( ∆ x , ∆ y ) ,

where

lim ∆ x → 0 ∆ y → 0

O 2 = 0 .

With these constraints, the equation (7.12) becomes

= T

∆ y

∂ 2 u ∂ t 2

∆ u y

∆ u x ∆ x

ρ ∆ x ∆ y

∆ x ∆ y + O 1 + O 2 .

+

Observe now the limit process, when ∆ x → 0 and ∆ y → 0. As a result, we obtain ∂ 2 u ∂ t 2 = T ρ ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 . We have thus obtained the partial differential equation for the oscillating membrane ∂ 2 u ∂ t 2 = c 2 ∆ u , (7.13) where T ρ = c 2 , and the operator ∆ is the Delta operator. 4

R Note that the constant c introduced in this way has the dimension of velocity, and the equation itself is also known as the wave equation . Unlike equation (7.9), which is called one-dimensional, this one is two-dimensional, because the function u depends on three variables, the first two being coordinates x and y , while the third variable is time t .

7.3 Linear and quasilinear first order PDE

A linear first order partial differential equation has the form

n ∑ i = 1

∂ u ∂ x i

a i

+ bu + c = 0 ,

(7.14)