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Line 188: − f(xf, tf)+ − We know that f(stuff) is nothing more than the + − + − waves and the amplitude function. + − Calculation of g proves more challenging. We know + − + − d2y/dt2 - c2d2y/dx2 = 0+ − − However, this is for a uniform, sourceless wave. − − Like most generalizations, this is an unrealistic − − situation. What we need is a function that can − − generate a brief pulse. This sounds like a delta − − function, so we will use this. − − d2g/dt2 - c2d2g/dx2 = (d)(x-xi)(t-ti)
Target Diamond Structural Analysis (view source)
Revision as of 13:55, 30 April 2009
, 13:55, 30 April 2009→Interference
<math>\int{dti} \int{f(x_i,t_i)g(\Delta x, \Delta t) dx_i dy_i}</math>
<math>\int{dti} \int{f(x_i,t_i)g(\Delta x, \Delta t) dx_i dy_i}</math>
We know that
<math>f(x_i,t_i)\,</math>
product of the phases of the incoming and outgoing
is nothing more than the product of the phases of the incoming and outgoing waves and the amplitude function. Because this equation only takes into account one incoming and one outgoing wave, we will need to recalculate the later equations using different f-functions for all outgoing waves and sum these results. Fortunately, as we discovered above, only the first and second outgoing waves are relevant.
Calculation of g proves more challenging. We know the wave equation
<math>\frac{d^2y}{dt^2} - \frac{c^2d^2y}{dx^2} = 0</math>
the wave equation
However, this is for a uniform, sourceless wave. Like most generalizations, this is an unrealistic situation in the real world. What we need is a function that can generate a brief pulse. This sounds like a delta function.
<math>\frac{d^2g}{dt^2} - \frac{c^2d^2g}{dx^2} = \delta(\Delta x)\delta(\Delta t)</math>
This is not an easy equation to solve without
This is not an easy equation to solve without