By Professor Dr. Dietrich Stauffer, Professor Dr. H. Eugene Stanley (auth.)
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Example text
11. Two coupled one-dimensional oscillators between two fixed walls. All three spring constants are equal a) Two Coupled Oscillators. Let two point masses of mass m be connected to one another by a spring, and connected to two rigid walls, each by a further spring (Fig. 11). The three springs all have the force constant K. Let the system be one-dimensional, the coordinates Xl and X2 giving the displacements of the two point masses from their rest positions. Then the Hamilton function, with Vi = dxi/dt, is: H = (m/2) [v?
Anybody who dives in the sea for treasure or coral must therefore surface very slowly, as the sudden lowering of pressure would allow dangerous bubbles to grow in the blood vessels. 59) 1. 59) gives three equations for four unknowns, v and (!. If the flow is compressible we need also to know how the density depends on the pressure. (P =0)(1 + ",P), the compressibility '" being defined thereby. In an elastic solid the stress tensor 0" is no longer given by a unique pressure P, and instead of a unique compressibility we now need many elastic constants C.
The general oscillation is a superposition (addition, or linear combination) of these two normal oscillations. The essential aspects of the harmonic vibrations in a solid body are therefore represented by this simple example; the next section does the same, only in a more complicated case. b) Normal Vibrations in the Solid State. We now calculate in a similar way the vibration frequencies of the atoms in a solid body which has atoms of mass m and only one type. Let the equilibrium position of the i-th atom be and let qi = i be the displacement from eqUilibrium.