By David Morin
A publication on wave physics, designed for school sophomores.
Last up to date: January 2011.
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Additional resources for Waves
Let’s see why this is true. If we apply the x0 (0) and x1 (0) boundary conditions to Eq. (55), we obtain x0 (0) = C1 and x1 (0) = C1 cos θ + C3 sin θ. Since we are assuming that ω (and hence θ) is given, these two equations determine C1 and C3 . But C1 and C3 in turn determine all the other xn (0) values via Eq. (55), because the sin ωt terms are all zero at t = 0. So for a given mode, x0 (0) and x1 (0) determine all the other initial positions. In a similar manner, the x˙ 0 (0) and x˙ 1 (0) values determine C2 and C4 , which in turn determine all the other initial velocities.
This is therefore the chapter in which we will make the transition from the oscillations of one particle to the oscillations of a continuous object, that is, to waves. The outline of this chapter is as follows. 1 we solve the problem of two masses connected by springs to each other and to two walls. We will solve this in two ways – a quick way and then a longer but more fail-safe way. We encounter the important concepts of normal modes and normal coordinates. We then add on driving and damping forces and apply some results from Chapter 1.
TWO MASSES 3 We can now solve for x1 (t) and x2 (t) by adding and subtracting Eqs. (3) and (5). The result is x1 (t) = As cos(ωs t + φs ) + Af cos(ωf t + φf ), x2 (t) = As cos(ωs t + φs ) − Af cos(ωf t + φf ). (6) The four constants, As , Af , φs , φf are determined by the four initial conditions, x1 (0), x2 (0), x˙ 1 (0), x˙ 1 (0). The above method will clearly work only if we’re able to guess the proper combinations of the F = ma equations that yield equations involving unique combinations of the variables.