Assam Don Bosco University Waves in 3D Parag Bhattacharya Department of Basic Sciences School of Engineering and Technology

Vector Analysis

Triple products: 1) Scalar triple product 2) Vector triple product Note: Remember BAC-CAB rule

Quadruple products: 1) Scalar quadruple product 2) Vector quadruple product

Vector Differential Calculus: The operator ∇ (“del”) is defined as: There are 3 ways by which ∇ can operate: A) On a scalar function T –> ∇ T, i.e., as gradient operation B) On a vector function v 1) Via dot product –> ∇v, i.e., as divergence operation 2) Via cross product –> ∇ ×v, i.e., as curl operation

The Gradient operation: Given a scalar function depending on the 3 spatial coordinates, T(x, y, z), its gradient at any point is a vector given as: The Divergence operation: Given a vector function depending on the 3 spatial coordinates, v(x, y, z), its divergence at any point is a vector given as:

The Curl operation: Given a vector function depending on the 3 spatial coordinates, v(x, y, z), its curl at any point is a vector given as:

Product Rules: 1) Distributive property 2) Scalar multiple If λ is a scalar constant

Product Rules: 3) 4) 5) 6) 7) 8)

Quotient Rules: 1) 2) 3)

Second Derivatives: 1) The Laplacian operator ∇ 2 2) 3) 4) Note: The above definition of the Laplacian is for scalar functions only. Note: Equation (4) above is used to define the Laplacian of a vector function as:

Second Derivatives: 5) The Laplacian for vector functions Two definitions: Definition 1: Definition 2:

Directional Cosines: X Y Z k θzθz θxθx θyθy Cosines of the angles a vector makes with the X, Y and Z axes. Given a vector k: Its directional cosines are: making angles θ x, θ y and θ z with the X, Y and Z axes respectively.

Curvilinear Coordinate Systems

Spherical Polar Coordinates (SPC) r θ Φ X Y Z O i j k r (x, y, z) (r, θ, Φ) x i + y j + zk

Spherical Polar Coordinates (SPC) r θ Φ X Y Z O x y z r: radial distance θ: polar angle Φ: azimuthal angle Conversion from CC * to SPC: * CC: Cartesian Coordinates Conversion from SPC to CC:

Spherical Polar Coordinates (SPC) r θ Φ X Y Z O x y z r: radial distance θ: polar angle Φ: azimuthal angle Unit vectors in SPC: r

Vector Differential Operators in SPC 1) Gradient in SPC 2) Divergence in SPC 3) Curl in SPC

4) Laplacian in SPC Vector Differential Operators in SPC

Plane Waves

Suppose (x 0, y 0, z 0 ) is a particular point on this plane. X Y Z k (x 0, y 0, z 0 ) r0r0 (x, y, z) r r – r 0 O Consider some vector k One can always find a plane to which k is perpendicular. Point (x 0, y 0, z 0 ) would, then, have a position vector r 0 We can then consider any arbitrary point (x, y, z) lying on the same plane. This point (x, y, z) would also have its position vector r Both these points lie on the same plane. Therefore, the vector r – r 0 would always lie on the plane, and be perpendicular to the vector k. Hence,

X Y Z k (x 0, y 0, z 0 ) r0r0 (x, y, z) r r – r 0 O We have, Thus, Since, Because k is a fixed vector and (x 0, y 0, z 0 ) is a fixed point, Therefore, necessary condition for the plane to be perpendicular to the vector k is:

X Y Z k O In the equation Taking different values of 'a' would result in a number of planes that are parallel to each other, and all perpendicular to the given vector k. a1a1 a2a2 a3a3 a4a4 If the dimensions of the vector k is such that kr gives phase, then kr can be used as the argument of sinusoid. We can then define a suitable function ψ that varies sinusoidally as per kr

Let ψ be a function that varies sinusoidally over these parallel planes. ψ becomes the requisite profile function. Then we can express ψ as: or as or, in general, as Since ψ is a periodic function, it repeats itself over some displacement, say λ, along the direction of k. This may be expressed as:

This implies: Thus, Hence, Therefore, the vector k represents the propagation of the wave, and is known as the propagation vector.

Thus, represents the profile of a plane wave moving along or opposite to k. Therefore, the travelling harmonic plane wave is: where '+' sign implies a wave travelling along k and '-' sign implies a wave opposite to k Suppose the wavefront moves a distance dr k along in time dt. Then we must have, Thus,Hence,

Again, we have, In terms of the directional cosines: Thus, The wave function becomes: (1) Differentiating (1) with respect to x: (2) Differentiating (1) with respect to t: (3)

Differentiating (2) with respect to x again: (4) Differentiating (3) with respect to t again: (5) Proceeding in a similar manner, we finally obtain (A) (B) (C) (D)

(A) (B) (C) (D) Adding (A), (B) and (C), we get: (6) Using (D) in (6), we finally get:

Since, This yields the 3D differential wave equation:

Spherical Waves

 Emitted by a point isotropic source.  We use SPC because of the spherical symmetry. In SPC, the Laplacian operator becomes: (1) Spherical wavefronts Since the wave is assumed to be spherically symmetric, it only depends on r, i.e., Thus, we have

(1) Therefore, (2)

Hence, the differential wave equation in SPC becomes: (2) The differential wave equation in 3D is: (3) Using (2) in (3): (4) Therefore,

(4) Compare (4) with the 1D differential wave equation: Thus, (4) is equivalent to the 1D differential wave equation with rψ being the wave function. Therefore, This represents any spherically symmetric wave propagating radially outward from the source (at the origin) at a constant speed v.

This represents any spherically symmetric wave propagating radially inward towards the source (at the origin) at a constant speed v.

X Y Z O

X Y Z O

The general solution is: If the spherical wave is harmonic: or, In general, where A 0 : source strength This represents a travelling harmonic spherical wave.

Assam Don Bosco University Thank You!