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A single equation can be developedby substitutingEqs (1629) and (1630) into Eq (1631) to give -v: r'' -'t2 't '{i-xt '\'i ftr,1-)l,r,rr) f f(x v,) rl - \'2 )'r - " X2 - Xt t,r _ -r': )Z (16 34 -{i - rr ),i - lr ^, )'r , -ri -t: ]i rl - rt ): -r'r 12 -{l )z _ )r
I 66 MULTIDIMENSIONAL INTERPOLATION
EXAMPLE 6 16 Bilineorlnterooloiion
Problem Stotement Supposeyou have measured temperatures a number of coordiat nateson the surfaceof a rectangular heatedplate:
T ( 2 , 1 :)6 0 T(2,6): 55
: T(:9,1) 5l 5 T(9,6):10
interpolation estimate temperature : 525and1',- 48 Usebilinear to the atrr gives Solution Substituting values Eq(1632) these into : "/ ( ) j ) _ { 6 ) 5 2 5 - 9 4o 8 - 6 - u 2-9 l-6 5 2 5 - 2 4)/8 - \ - -6 + 9-2 t-6 s-}s-24R-l
525-948-l ss + -;; + -;; u-,
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| 662 Multidimensionol Interpolotion in MATLAB
piecewiseinterpolaMATLAB has two builrin functionsfor two- and three-dimensional tion: interp2 and interp3 As you might expectfrom their names, thesefunctionsoperFor example,a simplerepresentation ate in a similar fashionto interpl (Section 1652) o f t h es y n t a x f i n e o 2 i s o
zi = Lt\r-e(p2 lx, y, z, xi, yi, 'mechod' )
where x and y - matrices containing the coordinatesof the points at which the values in the matrix z are given, zi : d matrix containingthe resultsof the interpolationas evaluated at the points in the matricesxr and yr, and method: the desiredmethodNote that the methodsareidenticalto thoseusedby in*'erpi; that is, linear, nearesf , spline, a n dc u b i c As with interpl, if the,nethod argument omitted,the defaultis linearinterpolation is For example,interp2 can be usedto make the sameevaluation in Example 166as as
>> X= >> y= >>
zlnl-
2 9); 1 6l; 6A 51 5;55 701; erp2(x,y,2,525,48)
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AND PIECEWISE INTERPOLATION SPLINES
, H E A T R A N S F E R
Bockground Lakes in the temperatezone can becomethermally stratified duringthe summerAs depictedin Fig 1611,warm, buoyantwater near the surfaceoverliescolder, denserbottom water Such stratification effectively divides the lake vertically into two layers: the epilitnnion and the hypolimnion, separated a plane called the thermocline by Thermal stratification has great significancefor environmentalengineersand scientists studying such systemsIn particular,the thermoclinegreatly diminishesmixing between the two layersAs a result, decompositionof organic matter can lead to severedepletionof oxygen in the isolated bottom waters The location of the thermoclinecan be defined as the inflection point of the temperatur d e p t hc u r v e - t h a t i s t h e p o i n t a r w h i c h d 2 T / d z 2 : 0 l t i s a l s o t h e p o i n t a t w h i c ht h e absolutevalue of the first derivative or gradient is a maximum The temperaturegradient is important in its own right becauseit can be usedin coniunction with Fourier's law to determinethe heatflux acrossthe thermocline: J:-DpC, dT az
(r633)
where "f : heat flux [call(cm' s)], cv: an eddy diffusion coefficient (cm'ls), p : density (31 g/cm3), and C: specific heat[: I call(g C)] In this case study, natural cubic splines are employed to determinethe thermocline gradientfor PlatteLake, Michigan (Table 163)The latter is also depth and temperature usedto determinethe heatflux for the casewherea = 001 cm2/s
I6 FIGURE I I Temperoiure depth versus for during summer Plotte Michigon Loke,
r ('c)
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