Wave Propagation in Linear Media | re-examined

]
];
8 Application of the quadrature routine
GaussTransNeg,x,t,w,k,n,opts],
_, Message[Phi::invalidpart,pt]],
_,
Message[Phi::invalidshape,sh]
PhiTrans[x_, t_, w_, k_, opts___Rule] :=
If[(Shape/.foptsg/.Options[Phi]) === Gauss,
Message[Phi::missingval],
PhiTrans[x,t,w,k,1,opts]];
Remark (Input checking) Apart from checking the option values, the function provides
other tests of the user's input, too. The parameter n, for example must be greater than
zero, which is expressed with the conditional pattern matching statement n ?Positive.
Therefore the function evaluates only if n satis es this requirement. Otherwise no error
message is generated, but the function is left unevaluated. The same is done with the
additional arguments that match the pattern opts Rule only if they have the syntax of
a rule, that is if they look like lhs->rhs. However, this does not encompass a validity
check for the option names or their values.
The modules PhiTrans and PhiEvan compute the wave function inside the barrier, PhiInc and
PhiRef return the incident and re ected wave outside the barrier. A still more comfortable
driver is the function Phi that decides (based upon the spatial coordinate given by the user)
which module to call. For the boundary case X =0it takes the functions that are de ned
for the interior of the barrier (which must, of course, yield the same results as those for the
outside). This function also traps the somewhat inconvenient case X = T = 0 for rectangular
waves and sets the return value to the exactly known result 1=2.
Phi[x_, t_, w_, k_, n_?Positive, opts___Rule] :=
Which[t == 0 && x == 0 && Shape/.foptsg/.Options[Phi] === Rect, 0.5,
t < 0, 0,
x >= 0, PhiEvan[x,t,w,k,n,opts] + PhiTrans[x,t,w,k,n,opts],
x < 0, PhiInc[x,t,w,k,n,opts] + PhiRef[x,t,w,k,n]];
Phi[x_, t_, w_, k_, opts___Rule] :=
If[(Shape/.foptsg/.Options[Phi]) === Gauss,
Message[Phi::missingval],
Phi[x,t,w,k,1,opts]];
All the driver functions dealing with the wave function inside the barrier also exist in a version
that computes the gradient of the wave function, namely PhiGradTrans, PhiGradEvan, and
PhiGrad. They are implemented only for triangular and Gaussian waves. For a rectangular
shape, the integral would diverge for the boundary and for T =0.
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