Appendix A Curvilinear Coordinates A.1 Spherical Coordinates When p is known in terms of the spherical coordinates of the location at which is observed, the most expedient evaluation of ∇ p uses a form in terms of the radial, polar, and azimuthal components. This form is more complicated in appearance than merely the derivative of a component with respect to a coordinate because incrementing the angular variables alters the direction of the unit vectors. There are several approaches by which these effects may be described in a derivation of the gradient ∇ p and the Laplacian ∇2 p in spherical coordinates. We shall employ a direct approach based on the coordinate transformation to derive ∇ p, and then switch to a slightly more subtle approach based on the properties of the spherical coordinate unit vectors to derive ∇2 p. A.1.1 Transformations The derivation of a gradient in spherical coordinates begins by considering the pres- sure to be a function of those coordinates. In turn, the spherical coordinates are treated as functions of the Cartesian coordinates. Doing so calls for the chain rule for partial differentiation. It is helpful to have the transformations at hand. The coordinates are defined in Fig. A.1. The Cartesian coordinates corresponding to a given a set of spherical coordinates are x = r sin ψ cos θ y = r sin ψ sin θ (A.1.1) z = r cos ψ The inverse transformation gives the spherical values for a given set of Cartesian coordinates. It is © Springer International Publishing AG 2018 665 J.H. Ginsberg, Acoustics—A Textbook for Engineers and Physicists, Volume II, DOI 10.1007/978-3-319-56847-8 666 Appendix A: Curvilinear Coordinates Fig. A.1 Spherical z coordinates and associated Constant - unit vectors r and e r - Constant e and - r e y Constant Equator r and x 1/2 r = x2 + y2 + z2 z ψ = −1 cos 1/2 x2 + y2 + z2 (A.1.2) y y x θ = −1 = −1 = −1 tan sin / cos / x x2 + y2 1 2 x2 + y2 1 2 We also will have need for the definition of spherical unit vectors. Each vector is the change of the position vector from the origin when the corresponding coordinate is increased by a unit value, with the others held constant. The result of doing so is e¯r = sin ψ cos θe¯x + sin ψ sin θe¯y + cos ψe¯z e¯ψ = cos ψ cos θe¯x + cos ψ sin θe¯y − sin ψe¯z (A.1.3) e¯θ =−sin θe¯x + cos θe¯y A.1.2 Gradient The Cartesian components of a gradient are the derivative with respect to the corre- sponding coordinate. The chain rule is needed to evaluate these derivatives when p is known in terms of r, ψ, and θ,sowehave ∂ ∂ ∂ ∂ ∂ψ ∂ ∂θ p = p r + p + p ∂x ∂r ∂x ∂ψ ∂x ∂θ ∂x ∂ p ∂ p ∂r ∂ p ∂ψ ∂ p ∂θ = + + (A.1.4) ∂y ∂r ∂y ∂ψ ∂y ∂θ ∂y ∂ ∂ ∂ ∂ ∂ψ ∂ ∂θ p = p r + p + p ∂z ∂r ∂z ∂ψ ∂z ∂θ ∂z Appendix A: Curvilinear Coordinates 667 The transformation from Cartesian to spherical coordinates is used to fill in the derivatives appearing in these equations. Implicit differentiation is useful for the angles. The derivatives of Eq. (A.1.2) with respect to x proceed as follows: ∂r ∂ / x = 2 + 2 + 2 1 2 = x y z / ∂x ∂x x2 + y2 + z2 1 2 ∂ ∂ψ ∂ z ∂ z ( ψ) ≡− ψ = = cos sin / ∂x ∂x ∂x r ∂x x2 + y2 + z2 1 2 =− xz / x2 + y2 + z2 3 2 ∂ 1 ∂θ ∂ y y (tan θ) ≡ = =− (A.1.5) ∂x (cos θ)2 ∂x ∂x x x2 Derivatives with respect to y and z are obtained by similar operations. We seek a description of ∇ p in which only variables associated with spherical coordinates appear. The transformations from Cartesian to spherical coordinates, along with x2 + y2 + z2 = r 2, are used to eliminate x, y, and z from the preceding. Doing so leads to ∂r ∂r ∂r = sin ψ cos θ, = sin ψ sin θ, = cos ψ ∂x ∂y ∂z ∂ψ cos ψ cos θ ∂ψ cos ψ sin θ ∂ψ sin ψ = , = , =− (A.1.6) ∂x r ∂y r ∂z r ∂θ sin θ ∂θ cos θ ∂θ =− , = , = 0 ∂x r sin ψ ∂y r sin ψ ∂z These expressions are substituted into Eq. (A.1.4), which then are used to form the gradient in Cartesian components. This step yields ∂ p 1 ∂ p 1 ∂ p sin θ ∇ p = ψ θ + ψ θ − e¯ ∂ sin cos ∂ψ cos cos ∂θ ψ x r r r sin ∂ p 1 ∂ p 1 ∂ p cos θ + ψ θ + ψ θ + e¯ ∂ sin sin ∂ψ cos sin ∂θ ψ y (A.1.7) r r r sin ∂ p 1 ∂ p + cos ψ − sin ψ e¯ ∂r r ∂ψ z The last step is to convert this expression into spherical coordinate components. By definition, a component is the projection of a vector onto a coordinate axis, so we may write ∇ p ≡ (∇ p ·¯er ) e¯r + ∇ p ·¯eψ e¯ψ + (∇ p ·¯eθ) e¯θ (A.1.8) Substitution of the unit vectors in Eq. (A.1.3) and the gradient on Eq. (A.1.7) ulti- mately reduces to 668 Appendix A: Curvilinear Coordinates ∂ p 1 ∂ p 1 ∂ p ∇ p = e¯ + e¯ψ + e¯θ (A.1.9) ∂r r r ∂ψ r sin ψ ∂θ One device for remembering this expression is that dr is the displacement when r is incremented infinitesimally, rdψ is the meridional displacement for an infinitesimal increment of ψ, and (r sin ψ) dθ is the azimuthal displacement when θ is incremented. A.1.3 Laplacian The final form of the gradient in spherical coordinates is understandable as changes in the field when the position is shifted infinitesimally in each coordinate direction. The Laplacian is less intuitive and somewhat more difficult to derive. One approach for deriving ∇2 p parallels that used to describe ∇ p.Specifically, it applies the chain rule to convert (∂/∂x)(∂ p/∂x), (∂/∂y)(∂ p/∂y), and (∂/∂z)(∂ p/∂z) to derivatives with respect to r, ψ, and θ. The derivation that follows proceeds differently. It uses Eq. (A.1.9) to describe both terms in ∇2 p ≡∇·∇p, so that ∂ ∂ ∂ ∂ 2 1 1 p ∇ p = e¯r + e¯ψ + e¯θ · e¯r ∂r r ∂ψ r sin ψ ∂θ ∂r 1 ∂ p 1 ∂ p (A.1.10) + e¯ψ + e¯θ r ∂ψ r sin ψ ∂θ Neither the components of ∇ p nor the unit vectors are constant. For this reason, the derivatives must be evaluated prior to taking dot products. In other words, ∂ ∂ ∂ ∂ 2 p 1 p 1 p ∇ p =¯er · e¯r + e¯ψ + e¯θ ∂r ∂r r ∂ψ r sin ψ ∂θ 1 ∂ ∂ p 1 ∂ p 1 ∂ p + e¯ψ · e¯r + e¯ψ + e¯θ (A.1.11) r ∂ψ ∂r r ∂ψ r sin ψ ∂θ 1 ∂ ∂ p 1 ∂ p 1 ∂ p + e¯θ · e¯ + e¯ψ + e¯θ r sin ψ ∂θ ∂r r r ∂ψ r sin ψ ∂θ Further progress entails evaluation of the derivatives of the unit vectors with respect to each spherical coordinate. These operations are applied to the unit vectors in Eq. (A.1.3). For example, ∂e¯ψ =− ψ θe¯ − ψ θe¯ − ψe¯ ∂ψ sin cos x sin sin y cos z (A.1.12) It will be easier to evaluate the dot products in Eq. (A.1.11) if the unit vector deriv- atives are expressed in terms of e¯r , e¯ψ, and e¯θ. Toward that end we form the dot product of each unit vector with the derivative of each unit vector. For the preceding description of ∂e¯ψ/∂ψ, these operations yield Appendix A: Curvilinear Coordinates 669 ∂e¯ψ ∂e¯θ e¯ · = ψ θe¯ + ψ θe¯ + ψe¯ · =− r ∂ψ sin cos x sin sin y cos z ∂ψ 1 ∂e¯ψ ∂e¯ψ e¯ψ · = − ψ θe¯ − ψ θe¯ − ψe¯ · = ∂ψ cos cos x cos sin y sin z ∂ψ 0 (A.1.13) ∂e¯ψ ∂e¯ψ e¯θ · = − θe¯ + θe¯ · = ∂ψ sin x cos y ∂ψ 0 These components are used to construct the vector representation of ∂e¯θ/∂θ according to ∂e¯ψ ∂e¯ψ ∂e¯ψ ∂e¯ψ = e¯ · e¯ + e¯ψ · e¯ψ + e¯θ · e¯θ =−¯e ∂ψ r ∂ψ r ∂ψ ∂ψ r (A.1.14) The other derivatives are found by a similar process. The full set of results is ∂ ¯ ∂ ¯ ∂ ¯ er = ¯, eψ = ¯, eθ = ¯ ∂ 0 ∂ 0 ∂ 0 ∂ ¯ r ∂ ¯ r r∂ ¯ er eψ eθ ¯ =¯eψ, =−¯e , = ∂ψ ∂ψ r ∂ψ 0 (A.1.15) ∂e¯r ∂e¯ψ ∂e¯θ = ψ e¯θ, = ψ e¯θ, =− ψe¯ − ψe¯ψ ∂θ sin ∂θ cos ∂θ sin r cos Now that derivatives of the unit vectors have been characterized, we may proceed to carry out the derivatives in Eq. (A.1.11). Some terms are zero because the unit vectors form an orthonormal set, and others vanish because some partial derivatives of a unit vector are zero. What remains leads to ∂2 p 2 ∂ p 1 ∂2 p cot ψ ∂ p 1 ∂2 p ∇2 p = + + + + (A.1.16) ∂r 2 r ∂r r 2 ∂ψ2 r 2 ∂ψ r 2 (sin ψ)2 ∂θ2 A.1.4 Velocity and Acceleration To describe the velocity and acceleration of a particle, it is necessary to account for the time dependence of the spherical coordinates. The position of a particle depends explicitly on the value of r because x¯ = re¯r .