L and \(\Lambda\) here must not be confused with these symbols in → Sect. B.11 .
The wave equation is a second-order linear differential equation, with p,q possibly functions of r: \(L\left({f(r)}\right)={f}^{{\prime\prime}}(r)+p\cdot{f}^{{\prime}}(r)+q\cdot f(r)=0\). (1)
The adjoint wave equation is \(\Lambda\left({g(r)}\right)={g}^{{\prime\prime}}(r)-p\cdot{g}^{{\prime}}(r)+(q-{p}^{{\prime}})\cdot g(r)=0\). (2)
Both satisfy the identity \(g\cdot L(f)-f\cdot\Lambda(g)=\displaystyle\frac{d\, P(g,f)}{dr}.\hfill(3)\)
P(g,f) is the bilinear concomitant . If \(\Lambda\)(g)=0 can be solved, then solutions of L(f)=0 are \(\begin{array}[]{@{}rcl}f_{1}(r)&=&g(r)\cdot e^{{-\int{p\, dr}}},\\ f_{2}(r)&=&f_{1}(r)\cdot\displaystyle\int{\displaystyle\frac{e^{{-\int{p(s)\, ds}}}}{g^{2}}dr}.\\ \end{array}\) (4)
The general solution is \(f(r)=a\cdot f_{{1}}\cdot\) \((r)+b\cdot f_{{2}}(r)\).
In the special case q(r)=dp(r)/dr, \(\begin{array}[]{@{}l}g(r)=\displaystyle\int{e^{{-\int{p(s)\, ds}}}}\, dr,\\...
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References
Cremer, L., Heckl, M.: Koerperschall, 2nd edn. Springer, Berlin (1995)
Cummings, A.: Sound Generation in a Duct with a Bulk-Reacting Liner. Proc. Inst. Acoust. 11 part 5, 643–650 (1989)
Felsen, L.B., Marcuvitz, N.: Radiation and Scattering of Waves, p. 89 Prentice Hall, London (1973)
Gottlieb: J. Sound Vibr. 40, 521–533 (1975)
Kleinstein, Gunzburger: J. Sound Vibr. 48, 169–178 (1976)
Mechel, F.P.: Schallabsorber, Vol. I, Ch. 3: Sound fields: Fundamentals. Hirzel, Stuttgart (1989)
Mechel, F.P.: Schallabsorber, Vol. I, Ch. 11: Surface Waves. Hirzel, Stuttgart (1989)
Mechel, F.P.: Schallabsorber, Vol. I, Ch. 12: Periodic Structures. Hirzel, Stuttgart (1989)
Mechel, F.P.: Schallabsorber, Vol. II, Ch. 3: Field equations for viscous and heat conducting media. Hirzel, Stuttgart (1995)
Mechel, F.P.: Schallabsorber, Vol. III, Ch. 18: Multi-layer finite walls. Hirzel, Stuttgart (1998)
Mechel, F.P.: Schallabsorber, Vol. III, Ch. 26: Rectangular duct with local lining. Hirzel, Stuttgart (1998)
Mechel, F.P.: Schallabsorber, Vol. III, Ch. 27: Rectangular duct with lateral lining Hirzel, Stuttgart (1998)
Mechel, F.P.: A Principle of Superposition. Acta Acustica (2000)
Moon, P., Spencer, D.E.: Field Theory Handbook, 2nd edn. Springer, Heidelberg (1971)
Morse, P.M., Feshbach, H.: Metheods of Theoretical Physics, part I McGraw-Hill, New York (1953)
Ochmann, M., Donner, U.: Investigation of silencers with asymmetrical lining; I: Theory. Acta Acustica 2, 247–255 (1994)
Pierce, A.D.: Acoustics, Ch. 4: McGraw-Hill, New York (1981)
Skudrzyk, E.: The Foundations of Acoustics, Springer, New York (1971)
VDI-Waermeatlas, 4th edition, VDI, Duesseldorf (1984)
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(2008). Adjoint Wave Equation . In: Mechel, F.P. (eds) Formulas of Acoustics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-76833-3_30
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