Derive a wave equation for an n mass coupled system

[Bobby Garret]

1. Derive the wave equation for longitudinal vibrations in an extended 1-D system of masses and springs. The average distance between masses is D [m], the spring constants are K [kg/s2 ], and the masses are M [kg]. b) Determine the wave speed c as a function of D, K, and M. Verify that it has the right units.2. fn = f(xn), fn+1 = f(xn + D). i was given this as a hint.3. ∂^2y /∂x^2 = (1/c^2) ∂^2y /∂t^2. I can derive it like this. But I have no clue what my partials are supposed to be. How do i find an equation to fit into this form?

 

[jbsteele]

a) The wave equation for longitudinal vibrations in an extended 1-D system of masses and springs is given by:∂^2y /∂x^2 = (1/v^2) ∂^2y /∂t^2where v is the wave speed and y is the displacement of the mass from its equilibrium position.b) The wave speed c can be determined as a function of D, K, and M using the following equation:c = (K/M)^1/2 * DThis equation has the right units of m/s.