| Popis: |
The exact second‐order transient solution to the interaction of an arbitrary wave with a plane wave is given by Westervelt [P. J. Westervelt, 3320 (1994)]. Let the arbitrary wave be χ(x0−m⋅r), a plane wave traveling in the m direction. In this case σ=(1−n⋅m)−1χ and the solution to Eckart’s equation becomes ρsc20=E12 +(1−n⋅m)−1(Λ+2n⋅m)=cos θ ∇2ψ2, which is identical to Eq. (10) of Westervelt [P. J. Westervelt, 934 (1957)] provided the substitutions ψ2=1/2(w1w2)−1W12 and n⋅m=cos θ are made. It is asserted that this exact solution serves as an approximate solution to the far‐field interaction of arbitrary sources. This is done by allowing m and n to be space dependent. As an example, the exact solution for the cardioid wave χ=(4πr)−1[n⋅m(G,0+Gr−1)−G ,0] interactions with a plane wave is obtained from σ=−(4πr)−1G,0, where G=G(x0−r) and m=rr−1. In the far field of the cardioid source, σ=(1−n⋅m)−1χ, as in the plane wave–plane wave interaction, thus demonstrating the assertion. |
