Question 7.2: Determine the energy flux densities of the incident, reflect...
Determine the energy flux densities of the incident, reflected and transmitted waves. Verify the conservation of energy at the interface.
The vector energy flux density of an acoustic wave has the components S_i=-B\sum\limits_{j}\dot{u} _j(\partial u_j/\partial x_i) . Taking the real parts of the waves [7.28] and using the expressions of the reflection and transmission coefficients, we obtain:
\underline{u} =e\underline{u} _me^{i(ωt-ke.r)} \ , \ \ \ \ \underline{u^\prime} =e^\prime \underline{u} _m^\prime e^{i(ωt-k^\prime e^\prime .r)} \ , \ \ \ \ \ \underline{u}^{\prime \prime} =e^{\prime \prime} \underline{u} _m^{\prime \prime} e^{i(ωt-k^{\prime \prime}e^{\prime \prime}.r)} [7.28]
S_i=e_ikB_1ωu_m^2 \sin^2(ωt-ke.r) \ \ \ \Rightarrow \ \ \ S=eZ_1ω^2u_m^2 \sin^2(ωt-ke.r) \\ S^\prime _i=e^\prime _ikB_1ω{u^\prime _m}^2 \sin^2(ωt-k^\prime e^\prime .r) \ \ \ \Rightarrow \ \ \ S^\prime =e^\prime Z_1ω^2R_u^2 u_m^2 \sin^2(ωt-ke^\prime .r) \\ S^{\prime \prime}_i=e^{\prime \prime}_ik^{\prime \prime}B_1ω{u^{\prime \prime}_m}^2 \sin^2(ωt-k^{\prime \prime}e^{\prime \prime}.r) \ \ \ \Rightarrow \ \ \ S^{\prime \prime}=e^{\prime \prime}Z_2ω^2 7_u^2u_m^2 \sin^2(ωt-k^{\prime \prime}e^{\prime \prime}.r) .
To verify the conservation of energy, let us consider an element of area dS of the interface. The power that dS receives from medium (1) toward the positive z is
dS(S+S^\prime).e_z=dS\left[(e.e_z)+(e^\prime.e_z)R_u^2\right] Z_1ω^2u_m^2 \sin^2(ωt-ke.r) \\ =dS( \cos \theta -\cos \theta ^\prime R_u^2)Z_1ω^2u_m^2 \sin^2(ωt-ke.r).The transmitted power across dS toward the positive z in medium (2) is
dS(S^{\prime \prime}.e_z)=dS(e^{\prime \prime}.e_z) 7_u^2Z_2ω^2u_m^2 \sin^2(ωt-k^{\prime \prime}e^{\prime \prime}.r) \\ =dS \cos\theta ^{\prime \prime} 7_u^2Z_2ω^2u_m^2 \sin^2(ωt-k^{\prime \prime}e^{\prime \prime}.r)As ωt-k^{\prime \prime}e^{\prime \prime}.r=ωt-ke.r on the interface, the conservation of energy may be written in the form Z_1(1-R_u^2) \cos \theta =Z_2 7_u^2 \cos\theta ^{\prime \prime}, which is verified by the expressions of [7.30] for R_u and 7_u .
R_u\equiv \frac{\underline{u^\prime}_m }{\underline{u}_m } =\frac{Z_2 \cos\theta -Z_1 \cos\theta ^{\prime \prime}}{Z_2 \cos\theta +Z_1 \cos\theta ^{\prime \prime}} , \ \ \ \ \ \ 7_u\equiv \frac{\underline{u^{\prime \prime}}_m }{\underline{u}_m } =\frac{2Z_1 \cos\theta }{Z_2 \cos\theta +Z_1 \cos\theta ^{\prime \prime}} [7.30]