With the notation for the group velocity Vg(k)Vg(k) and the inverse K1(ν)K1(ν) such that ν=Ω1(K1(ν))ν=Ω1(K1(ν)), it follows that dν=Vg(K1(ν))dk, and hence sˇ(ω)=∫S¯1(K1(ν),ω)Vg(K1(ν))dνi(ν−ω)Assuming that S¯1(K1(ν),ω)/Vg(K1(ν)) is an analytic function in the complex ν-planeν-plane, Cauchy׳s principal value theorem leads to the result that equation(8) sˇ(ω)=2πS¯1(K1(ω),ω)Vg(K1(ω))and hence equation(9) S¯1(K1(ω),ω)=12πVg(K1(ω))sˇ(ω)This www.selleckchem.com/products/DAPT-GSI-IX.html is the source condition  , the condition that S  1 produces the desired elevation s(t)s(t) at x  =0. This condition shows that the function ω→S¯1(K1(ω),ω) is uniquely   determined by

the given time signal. However, the function S¯1(k,ω) of 2 independent variables is not uniquely determined; it is only uniquely defined for points (k,ω)(k,ω) that satisfy the dispersion relation. Consequently, the source function S1(x,t)S1(x,t) is not uniquely defined, and the spatial dependence can be changed when combined with specific changes in the time dependence. To illustrate this, and to obtain some typical and practical results, consider sources of the form S1(x,t)=g(x)f(t)S1(x,t)=g(x)f(t)in Bafilomycin A1 mw which space and time are separated: g   describes the spatial extent of the source, and f   is the so-called modified influx signal. Then S¯1(k,ω)=g^(k)fˇ(ω)

and the source condition for the functions f and g together is written as g^(K1(ω))fˇ(ω)=12πVg(K1(ω))sˇ(ω)Clearly, the functions f and g are not unique, which is illustrated for two special cases. Point generation: A source that is concentrated at x=0x=0 can be obtained using the Dirac delta-function

δDirac(x)δDirac(x). Then taking S1(x,t)=δDirac(x)f(t)S1(x,t)=δDirac(x)f(t), it follows (using δ^Dirac(k)=1/2π) that S¯1(k,ω)=fˇ(ω)/2π. The source condition then specifies the modified influx signal f(t)f(t) equation(10) S1(x,t)=δDirac(x)f(t)withfˇ(ω)=Vg(K1(ω))sˇ(ω)Observe Cell press that in physical space, the modified signal f  (t  ) is the convolution between the original signal s  (t  ) and the inverse temporal Fourier transform of the group velocity ω→Vg(K1(ω))ω→Vg(K1(ω)). As a final remark, notice that the area extended and the point generation are the same for the case of the non-dispersive shallow water limit for which Ω1(k)=c0kΩ1(k)=c0k and Vg(k)=c0Vg(k)=c0 (which then coincides with the phase velocity). In that case S¯1(K1(ω),ω)=c0sˇ(ω)/2π and the familiar result for influxing of a signal s(t)s(t) at x=0 is obtained ∂tη=−c0∂xη+c0δDirac(x)s(t)∂tη=−c0∂xη+c0δDirac(x)s(t) For the uni-directional equations in the previous subsection the solution is uniquely determined by the specification of the elevation at one point.