By Harold Elford Johns, John Robert Cunningham
Even though it follows the topical define that proved such a success in its previous variants, the Fourth version of this revered publication encompasses the entire advances and adjustments which were made due to the fact final it used to be revised. It not just offers new rules and knowledge, it shifts its emphases to appropriately replicate the unavoidably altering views within the box engendered through growth within the realizing of radiological physics.
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Extra resources for Physics of Radiology, Fourth Edition
Sample text
EXAMPLE 7 Consider a simple hannonic oscillator governed by (24) x+o/X=O. 3), (26) with t as a parameter. 3. Phase curves for the simple harmonic oscillator. A solution curve, plotted on the xy-plane, called the phase plane, is called a trajectory, which is oriented by the parameter t, and the direction of increasing t is indicated by the arrowheads. For a given set of initial conditions, the response of the system can be viewed as the motion of a point along a trajectory. Due to the uniqueness of solutions, at most one trajectory passes through each point in the xy-plane, with the exception of the equilibrium point which is at the origin x = y = o.
This result follows from the exact solution: First, note that, we have from equation (68), . (2 xx+yy= x +y 2)2 (69) from which, putting, (70) equation (69) becomes ~~(/)=/. /6. Limit cycle. which shows that r(t) grows with time; it actually goes to infinity when t = I 2{r(0)}2 ! This is an example of subcritical instability, which refers to the development of instability in a linearly-stable regime when the nonlinearities become sufficiently strong. Another type of fully nonlinear behavior which will be missed by a linearized analysis is that of a limit cycle.
L)t + 0 (94) . -I ) +-cos4> eFo 4>=-eA 8 2A (95) X I which is well behaved at A. = 1 !