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This page illustrates the computation using discretized computer simulation. Consider a computational time step Dt [s] and a position stepsize dr [m].

Consider the pressure observed at the center of a sphere in which energy is deposited uniformly at time zero. The velocity potential f at time t would be proportional to the energy deposition W_o deposited in a spherical shell at a distance r = c_st from the center. There is also a 1/r dependence to f . For this simple example, a simple expression describes the time-resolved velocity potential:

The pressure is related to the time derivative of f:

The convention describes -f as proportional to W_o, and P as proportional to -df/dt, so P is proportional to W_o.

IN SUMMARY, the pressue P is proportional to GW_o, where G is called the Grueneisen coefficient, a dimensionless factor that describes the fraction of thermal energy deposition [J/m³] that converts to pressure [J/m³]. This holds true until time t = radius/c_s where radius is the radius of the sphere. At times after this, there is no velocity potential because there is no W_o deposited beyond the sphere. f goes to zero which is constant, so pressure goes to zero too.

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