Lecture 7.4: The Laplacian in polar coordinates Matthew Macauley - PowerPoint PPT Presentation

Lecture 7.4: The Laplacian in polar coordinates Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4340, Advanced Engineering Mathematics M. Macauley (Clemson) Lecture 7.4: The Laplacian in polar coordinates Advanced Engineering Mathematics 1 / 11

Goal To solve the heat equation over a circular plate, or the wave equation over a circular drum, we need translate the Laplacian ∆ = ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 = ∂ 2 x + ∂ 2 y into polar coordinates ( r , θ ), where x = r cos θ and y = r sin θ . First, let’s write ∂ u ∂ x and ∂ u ∂ y in polar coordinates. M. Macauley (Clemson) Lecture 7.4: The Laplacian in polar coordinates Advanced Engineering Mathematics 2 / 11

Some messy calculations The Laplacian is the sum of the following two differential operators: � ∂ � ∂ � 2 � � 2 � 2 � � 2 cos θ ∂ ∂ r − sin θ ∂ sin θ ∂ ∂ r + cos θ ∂ = , = . ∂ x r ∂θ ∂ y r ∂θ M. Macauley (Clemson) Lecture 7.4: The Laplacian in polar coordinates Advanced Engineering Mathematics 3 / 11

Next goal The Laplacian operator in polar coordinates is ∂ r + ∂ 2 ∂ 2 ∆ = 1 ∂ r 2 + 1 ∂θ 2 = 1 r + 1 ∂ r ∂ r + ∂ 2 r 2 ∂ 2 θ . r 2 r Find the eigenvalues λ nm (fundamental frequencies) and the eigenfunctions f nm ( r , θ ) (fundamental nodes). Naturally, this depends on the boundary conditions. Clearly, in θ , the BCs have to be periodic: f ( r , θ + 2 π ) = f ( r , θ ). In r , the BCs can be: Dirichlet: f ( a , θ ) = 0 Neumann: f r ( a , θ ) = 0 Mixed: α 1 f ( a , θ ) + α 2 f r ( a , θ ) = 0. We will only consider Dirichlet BCs conditions in this lecture . M. Macauley (Clemson) Lecture 7.4: The Laplacian in polar coordinates Advanced Engineering Mathematics 4 / 11

Dirichlet boundary conditions Example Solve the following BVP for the Helmholtz equation in polar coordinates ∆ f = f rr + 1 r f r + 1 r 2 f θθ = − λ f , f (1 , θ ) = 0 , f ( r , θ + 2 π ) = f ( r , θ ) . M. Macauley (Clemson) Lecture 7.4: The Laplacian in polar coordinates Advanced Engineering Mathematics 5 / 11

Summary so far To solve the heat equation over a circular plate, or the wave equation over a circular drum, we need to translate the Laplacian ∆ = ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 = ∂ 2 x + ∂ 2 y into polar coordinates ( r , θ ), where x = r cos θ and y = r sin θ . This becomes the operator ∂ r + ∂ 2 ∂ 2 ∆ = 1 ∂ r 2 + 1 ∂θ 2 = 1 r + 1 ∂ r ∂ r + ∂ 2 r 2 ∂ 2 θ . r 2 r Its eigenvalues and eigenfunctions are λ nm = ω 2 nm , f nm ( r , θ ) = cos( n θ ) J n ( ω nm r ) , g nm ( r , θ ) = sin( n θ ) J n ( ω nm r ) , where ω nm is the m th positive root of J n ( r ), the Bessel function of the first kind of order n . These functions form a basis for the solution space of Helmholtz’s equation, ∆ u = − λ u . As such, every solution h ( r , θ ) under Dirichlet BCs can be written as f nm ( r ,θ ) g nm ( r ,θ ) ∞ ∞ � �� � � �� � � � h ( r , θ ) = cos( n θ ) J n ( ω nm r ) + b nm sin( n θ ) J n ( ω nm r ) a nm n =0 m =1 ∞ ∞ � � � � = J n ( ω nm r ) a nm cos( n θ ) + b nm sin( n θ ) . n =0 m =1 M. Macauley (Clemson) Lecture 7.4: The Laplacian in polar coordinates Advanced Engineering Mathematics 6 / 11

Fourier-Bessel series, revisited Every solution h ( r , θ ) to ∆ u = − λ u , u (1 , θ ) = 0 , u ( r , θ + 2 π ) = u ( r , θ ) can be written uniquely as f nm ( r ,θ ) g nm ( r ,θ ) ∞ ∞ � �� � � �� � � � h ( r , θ ) = a nm cos( n θ ) J n ( ω nm r ) + b nm sin( n θ ) J n ( ω nm r ) n =0 m =1 ∞ ∞ � � � � = J n ( ω nm r ) a nm cos( n θ ) + b nm sin( n θ ) . n =0 m =1 This is called a Fourier-Bessel series. By orthogonality, and the identity ˆ 1 n ( ω x ) x dx = 1 � 2 = � �� � �� � � � � 2 , J 2 � J n ( ω x ) J n ( ω x ) , J n ( ω x ) = J n +1 ( ω ) 2 0 ˆ 1 ˆ π ˜ � h , f nm � D h · f nm dA 2 a nm = � f nm , f nm � = = h ( r , θ ) J n ( ω nm r ) cos( n θ ) r dr d θ || f nm || 2 J n +1 ( ω nm ) 2 − π 0 ˆ π ˆ 1 ˜ D h · g nm dA � h , g nm � 2 b nm = � g nm , g nm � = = h ( r , θ ) J n ( ω nm r ) sin( n θ ) r dr d θ. || g nm || 2 J n +1 ( ω nm ) 2 − π 0 M. Macauley (Clemson) Lecture 7.4: The Laplacian in polar coordinates Advanced Engineering Mathematics 7 / 11

Bessel functions of the first kind � x ∞ ( − 1) m � 2 m + ν � J ν ( x ) = . m !( ν + m )! 2 m =0 M. Macauley (Clemson) Lecture 7.4: The Laplacian in polar coordinates Advanced Engineering Mathematics 8 / 11

Fourier-Bessel series from J 0 ( x ) � x ∞ ∞ 1 � 2 m � � ( − 1) m f ( x ) = c n J 0 ( ω n x ) , J 0 ( x ) = ( m !) 2 2 n =0 m =0 Figure: First 5 solutions to ( xy ′ ) ′ = − λ x 2 . M. Macauley (Clemson) Lecture 7.4: The Laplacian in polar coordinates Advanced Engineering Mathematics 9 / 11

Eigenfunctions of the Laplacian in the unit square λ nm = n 2 + m 2 , f nm ( x , y ) = sin nx sin my M. Macauley (Clemson) Lecture 7.4: The Laplacian in polar coordinates Advanced Engineering Mathematics 10 / 11

Eigenfunctions of the Laplacian in the unit disk λ nm = ω 2 nm , f nm ( r , θ ) = cos( n θ ) J n ( ω nm r ) , g nm ( r , θ ) = sin( n θ ) J n ( ω nm r ) M. Macauley (Clemson) Lecture 7.4: The Laplacian in polar coordinates Advanced Engineering Mathematics 11 / 11