Cylindrical and spherical coordinates

V

[Φ(R) ∆Ψ(R) + ∇Φ(R) · ∇Ψ(R)] dV =

 

Sg

Φ(R)∂Ψ(R)

∂n dS.

(2.200) We have used (2.146) for n· ∇Ψ(R).

Green’s second formula is obtained if the above procedure is applied to

v(R) =Ψ(R)∇Φ(R), (2.201)

subtracting the result from (2.200):

  

V

[Φ(R) ∆Ψ(R) − Ψ(R) ∆Φ(R)] dV

=

 

S_g



Φ(R)∂Ψ(R)

∂n − Ψ(R)∂Φ(R)

∂n



dS. (2.202)

Now, we simply have to provide a physical meaning for the fields Φ and Ψ and to interpret (2.202) in terms of wave theory to obtain Huygens’ principle as a mathematical formulation: It is the ∆-operator appearing in the wave equation (2.181) and in both Green formulas suggesting this.

2.2.4 Cylindrical and spherical coordinates

In isotropic materials, phase surfaces of waves emanating from a point source are spherical; in general, the amplitude is direction dependent. Insofar, the mathematical characterization of these wave fronts does not fit into the

cartesian coordinate system that has only been used until now; the utilization of spherical coordinates is mandatory (transducer sound fields originate from the superposition of spherical waves)! Additionally, cylindrical coordinates are often useful, for instance, to characterize a specimen like a pipe mathemat-ically. Therefore, we briefly refer to essential differences of such orthogonal curvilinear coordinates as compared to cartesian coordinates.

Circular cylindrical coordinates r,ϕ, z are nothing but polar coordi-nates r,ϕ in the xy-plane combined with the cartesian component z. Cartesian coordinates are spanned by a trihedron of orthogonal unit vectors e_x, e_y, e_z; (scalar) vector components result from the projection (scalar products) of a vector to the orthonormal trihedron vectors, and therefore the definition of a similar orthonormal trihedron for cylindrical coordinates is appropriate. We refer to Figure 2.15: For simplicity, we only sketch the xy-plane—the unit vec-tor e_zcharacterizes the cylinder coordinate z—and identify a point • in this plane through the radial coordinate r and the angular coordinateϕ, counted from the x-axis; we have 0≤ r < ∞ and 0 ≤ ϕ ≤ 2π. The relation between r,ϕ and x, y is given by coordinate transform equations (2.1):

x = r cosϕ, (2.203)

y = r sinϕ. (2.204)

The cartesian x- and y-coordinates are spanned by e_x and e_y, and because the pertinent x- and y-coordinate lines are straight, the unit vectors e_xand e_y

r

O y

x ϕ

ϕ ϕ

ex

ex

er

e_y ey

e_ϕ

FIGURE 2.15

Orthogonal unit vectors for circular cylindrical coordinates.

always have the same direction, which also holds for the point with coordinates r,ϕ: ex and e_y indicate for any point in the xy-plane the direction of varia-tion of the respective coordinate. Unit vectors e_r and e_ϕ for the cylindrical coordinates r andϕ similarly should point into those directions of pertinent coordinate variations. Consequently, e_r points into radial direction and e_ϕ into the direction tangential to a circle with radius r, of course in the direc-tion of increasing ϕ. We stated that: A vector is defined by its length and direction, both parameters can be computed given the cartesian components of the vector; this must also be true for the unit vectors e_rand e_ϕ. We obtain their cartesian components through projection to the unit vectors e_x and e_y: e_r= (e_r· ex) e_x+ (e_r· ey) e_y, (2.205) e_ϕ = (e_ϕ· ex) e_x+ (e_ϕ· ey) e_y. (2.206) In Figure 2.15, we immediately read off these projection:³³

e_r= cosϕ ex+ sinϕ ey, (2.207) e_ϕ=−sin ϕ ex+ cosϕ ey, (2.208) if we assume per definitionem that e_rand e_ϕare unit vectors; yet, with (2.23), we immediately prove this fact. The calculation of

e_r· e_ϕ= 0 (2.209)

confirms orthogonality of e_rand e_ϕ; trivially, e_zis orthogonal to both. Appar-ently, with e_r, e_ϕ, e_z, we have found the right-handed orthonormal trihedron for circular cylindrical coordinates! The spatial dependence of this trihedron, in this case, the dependence on ϕ, represents the essential difference with regard to cartesian coordinates.

With e_r, e_ϕ, e_z, the components Ar, A_ϕ, Az of a vector A in cylindrical coordinates can be defined:

A = A_re_r+ A_ϕe_ϕ+ A_ze_z, (2.210) where

Ar= A· er,

A_ϕ= A· e_ϕ, (2.211)

Az= A· ez. (2.212)

With A = Axe_x+ Aye_y+ Aze_z and (2.207) and (2.208), we immediately obtain equations to transform cartesian components Ax, Ay, Az into circular cylindrical components A_r, A_ϕ, A_z:

A_r= A_xcosϕ + Aysinϕ,

A_ϕ= − Axsinϕ + Aycosϕ, (2.213) Az= Az,

33Clearly, Equations 2.207 and 2.208 can be formally derived from the coordinate trans-form equations (2.203) and (2.204) (Langenberg 2005).

and the matrix notation of these equations

directly reveals how to obtain the transform of circular cylindrical components Ar, A_ϕ, Az into cartesian components Ax, Ay, Az; the coefficient matrix has to be inverted. The property of orthogonality of this matrix yields the inverse to be equal to the transpose:

⎛ With the help of this matrix, we can also show that the value of the scalar product of two vectors A and B is independent of the coordinate system:

ArBr+ A_ϕB_ϕ+ AzBz= AxBx+ AyBy+ AzBz. (2.216) The elastodynamic energy densities are defined as scalar product of two vec-tors and the double contraction of two second rank tensors, respectively (Sec-tion 4.3), and therefore their independence from the coordinate system is ensured. Here, we meet the cue: tensors in other than cartesian coordinates.

For example, the rϕ-component of a tensor of second rank D is defined by:³⁴ D_rϕ= e_r· D · e_ϕ

= D : e_ϕe_r; (2.217)

as a consequence, the following transform equation corresponding to (2.214) is obtained: Applying the summation convention to (2.218), we can rapidly show that the double contraction of two second rank tensors is also independent of the coordinate system (the double contraction is, just like the scalar product, only a number).

34Numbering cylindrical coordinates r, ϕ, z in terms of ξi, i = 1, 2, 3, we obtain all tensor components as

D_ξiξj = e_ξi· D · e_ξj, i, j = 1, 2, 3;

the short-hand notation D_ξiξj = Dij requires the understanding of the underlying coordi-nate system.

In principle, all facts are at hand to investigate consequences of coordi-nate changes for the analysis of scalar, vector, and tensor fields. The essential tool of this analysis is the del-operator whose components possess a physical dimension, namely the unit m⁻¹, under the assumption that x, y, z are (carte-sian) coordinates with unit m (meter) (Equation 2.139). In case of cylindrical coordinates, ∂/∂r and—of course—∂/∂z exhibit this unit, yet ∂/∂ϕ does not.

Therefore, we must supply the unit m to the differential variation along the ϕ-coordinate line, replacing ∂ϕ by the differential arc length variation ∂s = r∂ϕ on a circle with radius r. Consequently, the del-operator in circular cylin-drical coordinates reads as

As a matter of fact, the same representation is mathematically obtained if the so-called scale factors of the orthogonally curvilinear cylindrical coordinates are introduced.³⁵ With (2.219) and (2.210), it is finally clear what we have to cope with doing analysis in other than cartesian coordinates; for instance, calculation of the divergence of a vector field A(R) = A(r,ϕ, z) in cylindrical coordinates requires the computation of

∇ · A(R) =

35For circular cylindrical coordinates, the scale factors read as hr= 1,

hϕ= r, hz= 1.

according to (2.150)—we explicitly refer to the dependence of the unit vectors e_r(ϕ), e_ϕ(ϕ) upon ϕ which, therefore, must be differentiated too—: It is often mentioned that the divergence-∇-operator is written as

∇ = er

yet, this is only true if it is agreed upon that (2.221) is only applied to the scalar components Ar, A_ϕ, Az. With that in mind, it is correct to state that the gradient-∇-operator exhibits a different representation than (2.221), and the curl-∇-operator does not at all have a component representation in other than cartesian coordinates. Yet, consequently staying with (2.219) thus always agreeing to differentiate the vector components—compare (2.210)—we even obtain and correct results for all other∇-applications. Corresponding formulas are listed in the Appendix.

Spherical coordinates: As already mentioned, ultrasonic radiation fields exhibit demonstrative features only in spherical coordinates. As it is obvious from the simpler example of cylindrical coordinates, it is basically sufficient to know the coordinate transform equations and, already derived from them, the cartesian component representation of the orthonormal trihedron. Coordinate transform equations can be taken from Figure 2.16: The polar coordinate r in the xy-plane depends on the magnitude of the vector of position, the spherical coordinate R, via

r = R sinϑ, (2.223)

where ϑ denotes the coordinate “polar angle”; in connection with (2.203), (2.204), and another look at Figure 2.16, we obtain

x = R sinϑ cos ϕ,

y = R sinϑ sin ϕ, (2.224)

z = R cosϑ;

the spherical coordinate ϕ is called “azimuth angle”. The orientation of the right-handed orthonormal trihedron ordered according to e_R, e_ϑ, e_ϕcan also be extracted from Figure 2.16, as well as the projections to cartesian coordinates:

O

x

y z

R

eR

e_ϕ

e_ϑ

r ϑ

ϕ

FIGURE 2.16

Orthonormal trihedron of spherical coordinates R,ϑ, ϕ.

e_R= (e_R· ex) e_x+ (e_R· ey) e_y+ (e_R· ez) e_z

= sinϑ cos ϕ ex+ sinϑ sin ϕ ey+ cosϑ ez, e_ϑ= (e_ϑ· ex) e_x+ (e_ϑ· ey) e_y+ (e_ϑ· ez) e_z

= cosϑ cos ϕ ex+ cosϑ sin ϕ ey− sin ϑ ez, (2.225) e_ϕ= (e_ϕ· ex) e_x+ (e_ϕ· ey) e_y+ (e_ϕ· ez) e_z

= − sin ϕ ex+ cosϕ ey. We explicitly refer to

e_R= ˆR, (2.226)

that is to say, the vector of position has the component representation R= R sinϑ cos ϕ ex+ R sinϑ sin ϕ ey+ R cosϑ ez (2.227) in the cartesian orthonormal trihedron.

The system of Equations 2.225 defines the transform matrix for vector and tensor components, i.e., the transformation of the cartesian components Ax, Ay, Az of a vector A into its spherical components

A= A_Re_R+ A_ϑe_ϑ+ A_ϕe_ϕ (2.228)

according to:

Again, the inverse of the transform matrix is equal to its transpose, immedi-ately yielding the inversion of (2.229) and the transform equation for tensor components similar to (2.218).

The same arguments as in the cylinder coordinate paragraph lead us to the representation of the del-operator in spherical coordinates:³⁶

∇ = eR

single and multiple gradients, divergences, and curls can then be calculated;

the respective formulas may be taken from the Appendix.

2.3 Time and Spatial Spectral Analysis with

In document Karl-Jorg Langenberg, Rene Marklein, Klaus Mayer. Ultrasonic Nondestructive Testing of Materials - Theoretical Foundations 2012. (Page 72-79)