Mathematical Methods for Physics and Engineering - Matematica.NET

20.1 IMPORTANT PARTIAL DIFFERENTIAL EQUATIONSuTθ 2∆sθ 1Txx +∆xxFigure 20.1tension T .The forces acting on an element of a string under uniformDividing both sides by ∆x we obtain, for the vibrations of the string, the one-dimensionalwave equation∂ 2 u∂x = 1 ∂ 2 u2 c 2 ∂t , 2where c 2 = T/ρ. ◭The longitudinal vibrations of an elastic rod obey a very similar equation tothat derived in the above example, namely∂ 2 u∂x 2 = ρ ∂ 2 uE ∂t 2 ;here ρ is the mass per unit volume and E is Young’s modulus.The wave equation can be generalised slightly. For example, in the case of thevibrating string, there could also be an external upward vertical force f(x, t) perunit length acting on the string at time t. The transverse vibrations would thensatisfy the equationT ∂2 u∂x 2 + f(x, t) u=ρ∂2 ∂t 2 ,which is clearly of the form ‘upward force per unit length = mass per unit length× upward acceleration’.Similar examples, but involving two or three spatial dimensions rather than one,are provided by the equation governing the transverse vibrations of a stretchedmembrane subject to an external vertical force density f(x, y, t),( ∂ 2 )uT∂x 2 + ∂2 u∂y 2 + f(x, y, t) =ρ(x, y) ∂2 u∂t 2 ,where ρ is the mass per unit area of the membrane and T is the tension.677