is the particle displacement m u03c1 is the density of the bar kg m 3 s is the longitudinal wave speed in the bar m sec x is the spatial location m t is the time sec a is the area of the bar m 2 f is applied force n and u03b4 is the dirac delta function m 1 the wave speed s is equal to the square root of the quantity of the modulus of elasticity e n m 2 divided by the density s u221a e u03c1 the wave equation assumes a uniform area and negligible internal loss in the bar the free boundary at x 0 can be modeled as equ3 the boundary condition at x l is obtained by matching the force at the end of the bar to the viscous dissipative force in the damper this expression is equ4 where c is the viscous damping coefficient n sec m when c is equal to zero or infinity the boundary at x l reflects all the wave energy and the system response is composed only of standing waves when c is equal to a u221a u03c1e the boundary at x l absorbs all the wave energy the system response is composed only of propagating waves all other values of c exhibit some combination of standing and propagating wave.