[permalink] [id link]
Substituting this solution into the wave equation above yields the paraxial approximation to the wave equation:
from
Wikipedia
Some Related Sentences
Substituting and solution
Substituting this assumed solution back into the differential equation gives
Substituting and into
Substituting this into ( 2 ) gives
Substituting into the previous expression for,
Substituting this into the definition of the derivative via a difference quotient and taking limits means that the higher order terms – and higher – become negligible, and yields the formula interpreted as
Substituting into the above formula for | F < sub > h </ sub >| yields a horizontal force to be:
Substituting the derivative of u < sub > ρ </ sub > into the expression for velocity:
Substituting v < sub > 1 </ sub > into the identity and removing common factors gives the numerical example cited above.
Substituting into the prior equation, we have
Substituting this into the gain equation, we obtain:
Substituting this into the gain equation and solving for:
Substituting these two back into the Euler-Lagrange equation () results in
Substituting into the Binet equation yields the equation of trajectory
Substituting this definition into the prior equation, we find the general gain equation:
Substituting both quantities into the above equation generates the formula:
Substituting this expression into the first equation gives after some rearrangement
Substituting into the quadratic form gives an unconstrained minimization problem:
Substituting this into ( 1 ) above, the period T of a rigid-body compound pendulum
Substituting these values into the above equation gives T = 2π.
Substituting the above 2 results ( for E and J respectively ) into the continuum form shown at the beginning of this section:
Substituting energy into the equation gives:
Substituting equation into equation gives, which can be substituted into equation so that:
Substituting Equation ( Q ) into Equation ( I ) and then Equation ( I ) into Equation ( V ) gives:
Substituting into the equation for the change in entropy:
Substituting and wave
Substituting the evanescent form of the wave vector k ( as given above ), we find for the transmitted wave:
Substituting this expression into the wave equation yields the time-independent form of the wave equation, also known as the Helmholtz equation:
Substituting this expression into the Helmholtz equation, the paraxial wave equation is derived:
Substituting this form into the wave equation, and then simplifying, we obtain the following equation:
Substituting and equation
which says that the length of this vector is proportional to the rest mass m. Substituting the operator equivalents of the energy and momentum from the Schrödinger theory, we get an equation describing the propagation of waves, constructed from relativistically invariant objects,
Substituting for the masses in the equation for the Roche limit, and cancelling out gives
Substituting the coordinates above and rearranging the equation gives
Substituting this last expansion into the differential equation, taking the inner product of the result with, and making use of orthogonality, one obtains
Substituting this into the previous equation, one gets
Substituting this into the previous equation, one gets
Substituting into the last equation yields, so, which implies that the stationary points are and.
Substituting and above
Substituting above yields the following natural frequency and damping factor
Substituting ω for W into the above velocity expression, and replacing matrix multiplication by an equivalent cross product:
Substituting from the expressions for v and v < sub > m </ sub > above gives:
Substituting the equations above, we get:
Substituting this value in for above and rearranging, our equation becomes:
Substituting x ( t ) from above:
Substituting from information theory, an equivalent form of the above definition is:
Substituting into the equation above yields
Substituting cosE as found above into the expression for r, the radial distance from the focal point to the point P, can be found in terms of the true anomaly as well:
Substituting into the equation above and solving for gives
Substituting this into the expression for the Bohr ratio mentioned above gives
Substituting this into the above we get
Substituting the above into the earlier equation for the shear rate of a Newtonian fluid flowing within a pipe, and noting ( in the denominator ) that d =
0.235 seconds.