ABSTRACT
[a] Consider a cavity resonator with interior region D ⊂ ℝ 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/ieq0114.tif"/> and boundary surface ∂D. The wave field within the cavity ψ(ω,r) satisfies the Helmholtz equation ( ∇ 2 + k 2 ) ψ ( ω , r ) = 0 , r ∈ D https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0953.tif"/> with boundary condition ψ ( r ) = ψ 0 ( r ) , r ∈ ∂ D https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0954.tif"/> Explain how you would solve this problem using the Green’s function, ie, in terms of a function G ( r ∣ r ′ ) , r , r ′ ∈ D https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/ieq0115.tif"/> satisfying ( ∇ r 2 + k 2 ) G ( r ∣ r ′ ) = δ 3 ( r − r ′ ) , r , r ′ ∈ D https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0955.tif"/> and G ( r ∣ r ′ ) = 0 , r ∈ ∂ D , r ′ ∈ D https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0956.tif"/> How would the solution be modified if the above Dirichlet boundary condition on ψ is replaced by the Neumann boundary condition ∂ ψ ( r ) / ∂ n ^ = ψ 0 ( r ) , r ∈ ∂ D https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0957.tif"/> hint: Use Green’s identity ∫ D [ G ( r ∣ r ′ ) ∇ 2 ψ ( r ) − ψ ( r ) ∇ 2 G ( r ∣ r ′ ) ] = ∫ ∂ D [ G ( r ∣ r ′ ) ∂ ψ ( r ) / ∂ n ^ − ψ ( r ) ∂ G ( r ∣ r ′ ) / ∂ n ^ ] d S ( r ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0958.tif"/> 108Explain how you would obtain corrections to ψ if the gravitational field in the form of a time independent g μν (r) is taken into account and ψ satisfies the Laplace-Beltrami-Helmholtz equation: ( g k m − g ψ , m ) , k + j ω ( g k 0 − g ψ , k ) + j ω ( g k 0 − g ψ ) , k − ω 2 g 00 − g ψ = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0959.tif"/>
