Generalization of Brittingham’s localized solutions to the wave equation

A family of localized solutions of Brittingham’s type is constructed for different cylindric coordinates. We use method of incomplete separation of variables with zero separation constant and, then, the Bateman transformation, which enables us to obtain solutions in the form of relatively undistorted progressing waves containing two arbitrary functions, each of which depends on a specific phase function. PACS. 43.20.Bi Mathematical theory of wave propagation – 41.20.Jb Electromagnetic wave propagation; radiowave propagation The purpose of this Rapid Note is generalization of the family of Brittingham’s localized solutions to the wave equation known as the focus wave modes. In fact, they are presently represented by three specific solutions: Gauss [1], Hermite-Gauss [2], and Bessel-Gauss [3] modes. Most of reported Brittingham’s type solutions [4–6] reduces to Ψ (ρ, φ, z, τ) = ρe (z − τ) v ( z + τ + ρ z − τ ) , (1) where m is an integer, ρ, φ, z are the circular-cylinder coordinates, τ = ct is the time variable, c is the wavefront velocity, and v is an arbitrary function. Our investigation is connected with generalization of solutions of this particular type. In point of Courant and Hilbert’s terminology [7], localized solutions (1) represent relatively undistorted progressing waves Ψ (r, τ) = g (r, τ) f (Φ (r, τ)) (2) where r defines an observation point in some coordinate system, f (Φ) is an arbitrary function with continuous partial derivatives while Φ and g are fixed functions, called the phase function and the distortion (or attenuation) factor. For Ψ being a solution of the wave equation, the phase function must satisfy the Hamilton-Jacobi equation (∇Φ) − (∂Φ/∂τ) = 0. (3) The undistorted progressing waves are of great importance for telecommunications, launching directional scalar and electromagnetic waves (missiles), and other applications. a On leave from Research Institute for Laser Physics, St. Petersburg 199034, Russia e-mail: [email protected] Following [8], we construct the explicit solutions of the homogeneous wave equation [ 1 h1h2 ( ∂ ∂x1 ( h2 h1 ∂ ∂x1 ) + ∂ ∂x2 ( h1 h2 ∂ ∂x2 )) + ∂ ∂z2 − ∂ 2 ∂τ2 ] Ψ = 0 (4) (h1 and h2 are the metric coefficients) in different orthogonal cylindric coordinate systems x1, x2, z in the form of a family of wavefunctions with invariable profiles Ψ = w (x1, x2) v (τ, z), that corresponds to incomplete separation of variables [9]. Putting the separation constant equal to zero we get the solution of the wave equation (4) as Ψ = w (x1 ± ix2) v (τ ± z) (5) where w and v are arbitrary differentiable functions. Here v satisfies the 1D wave equation ( ∂/∂z − ∂/∂τ ) v = 0 while w (x1 ± ix2) meets ( ∂/∂x1 + ∂ /∂x2 ) w = 0, which for the rectangular (Cartesian), elliptic-cylinder, parabolic-cylinder, and bipolar coordinate systems is sufficient for satisfying the wave equation because h1 = h2. For the remaining circular-cylinder coordinates, x1 = ρ, x2 = φ, equation (4) leads to another representation, w (ρ, φ) = w ( ρe±iφ ) , which is completely equivalent to Cartesian-coordinate form w (x± iy). Finally, the whole wavefunction is subjected to one of the Bateman transformations [10] whose R a p id e N o te R a p id N o te 478 The European Physical Journal B Cartesian-coordinate representation is Ψ (x0, y0, z0, τ0)→ Ψ̃ (x0, y0, z0, τ0) = 1 z0 − τ0 Ψ ( x0 z0 − τ0 , y0 z0 − τ0 , r 0 − τ 0 − 1 2 (z0 − τ0) , r 0 − τ 0 + 1 2 (z0 − τ0) ) , x0 = x/λ, y0 = y/λ, z0 = z/λ, τ0 = τ/λ, r0 = √ x0 + y 2 0 + z 2 0, (6) λ is a real constant parameter. The function Ψ̃ due to transformation (6) is as well a solution of the wave equation, and here we use the method applied in [5,12] for construction of new solutions from known wavefunctions. Expressing the transversal coordinates through their dimensionless Cartesian counterparts x1 = x1 (x0, y0) , x2 = x2 (x0, y0) and fixing the phase sign, v = v (τ + z), one gets the general result for new wavefunctions Ψ̃ = 1 z0 − τ0 w (φ) v (Φ) (7) where φ is the transformed argument x1 ( x0 z0−τ0 , y0 z0−τ0 ) ± ix2 ( x0 z0−τ0 , y0 z0−τ0 ) while Φ takes the form z0 + τ0 + x0+y 2 0 z0−τ0 , which is easily seen to be of type (2) with g = w(φ) z0−τ0 and f (Φ) = v (Φ). Thus, in the case of incomplete separation of variables with zero separation constant, g is defined within a factor w (φ), an arbitrary function of one prescribed argument. Examining different cylindric coordinate systems, we get the following new families of localized wave structures: (i) Rectangular coordinates, x1 = x, x2 = y, h1 = h2 = 1, Ψ̃ = λ z − τ w (φr) v (Φr) , φr = λ x± iy z − τ , Φr = 1 λ ( z + τ + x + y z − τ ) . (8) In the circular-cylinder coordinates, x1 = ρ, x2 = φ, h1 = 1, h2 = ρ, −∞ < ρ < ∞, 0 ≤ φ < 2π, x = ρ cosφ, y = ρ sinφ, one has equivalent representation written as Ψ̃ = λ z − τ w (φc) v (Φc) , φc = λ ρe±iφ z − τ , Φc = 1 λ ( z + τ + ρ z − τ ) . (9) Localized waves (1) are easily seen to be a special case of (9) for λ = 1 and w ( ρe ) = ρe. (ii) Elliptic-cylinder coordinates, x1 = μ, x2 = θ, h1 = h2 = a ( sinh μ+ cos θ ) , −∞ < μ <∞, 0 ≤ θ < 2π, x = a coshμ cos θ, y = a sinhμ sin θ, 0 < a <∞ Ψ̃ = λ z − τ w (φe) v (Φe) , φe = a z − τ cosh (μ± iθ) , Φe = 1 λ ( z + τ + a z − τ ( sinh μ+ cos θ )) .