u(x_1, x_2, t) = 1/2 pi c doubleintegral_D g(y_1, y_2)/(c^2 t^2 - (y_1 - x_1)^2

Question

u(x_1, x_2, t) = 1/2 pi c doubleintegral_D g(y_1, y_2)/(c^2 t^2 - (y_1 - x_1)^2 - (y_2 - x_2)^2)^1/2 dy_1 dy_2 + partial differential/partial differential t [1/2 pi c doubleintegral_D f(y_1, y_2)/c^2t^2 - (y_1 - x_1)^2 - (y_2 - x_2)^2)^1/2 dy_1 dy_2], where D is the disk {(y_1 - x_1)^2 + (y_2 - x_2)^2 lessthanorequalto c^2t^2}. This Poisson's formula shows, in contrast to Kirchhoff's formula, that the value of u(x_1, x_2, t) does depend on the values of f and g on all of D, and not just on the light cone, (y_1 - x_1)^2 + (y_2 - x_2)^2 = c^2t^2. This means that Huygens's principle is false in two dimensions.

Explanation / Answer

I assume that you need clarification.
The formula for displacement shown above is Poisson's formula and the value of u(x1,x2,t) is dependent on the double integration over the region D where D is the region bounded by the disk
(y1-x1)2 + (y2-x2)2=ct
The inequality signifies the interior of the above disk.
Thus, 'u' depends on the values of f and g at points within the 'light cone' as integration is done for the region in the interior of the cone.

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