### Abstract

The effect of a linear shear flow of a Newtonian fluid in the region 0<*z*<∞ on the fluctuations at the surface of an elastic medium of thickness *H* in the region −*H*<*z*<0 is analyzed in the regime Re≫1 and Λ∼1, where Re=ργ*H* ^{2}/η is the Reynolds number and Λ=(ργ^{2} *H* ^{2}/*E*)^{1/2} is the ratio of the inertial stresses in the fluid and the elastic stresses in the solid. Here ρ and η are the fluid density and viscosity,*E* is the coefficient of elasticity of the solid, and γ is the mean strain rate in the fluid. A linear analysis is used to determine the effect of the flow on the fluctuations in the surface displacement, and an asymptotic expansion in the small parameter ε=(Λ/Re) is employed. The dynamics in the bulk of the fluid is inviscid in the leading approximation, and the leading order growth rate is imaginary because energy is conserved in the absence of viscous dissipation. There are multiple frequencies of oscillation, all of which satisfy the equations of motion. An increase in the fluid velocity increases the frequency of the downstream traveling waves, and decreases the frequency of the upstream traveling waves. The structure factor for the surface modes of the upstream traveling waves increases with an increase in the fluid velocity because the kinetic energy of the fluctuations decreases due to the lower frequency.

An opposite effect is observed for the downstream traveling waves; in addition, it is observed that the structure factor has a double‐peaked structure and reaches zero at an intermediate value at sufficiently high velocities. This is due to a divergence in the ratio of the tangential and normal displacements, and a consequent divergence in the energy required for the normal fluctuations at the surface. There is an *O*(ε^{1/2}) correction to the growth rate due to the presence of a viscous boundary layer of thickness *H*ε^{1/2} in the fluid at the interface. The *O*(ε^{1/2}) calculation shows that the real part of the growth rate is negative for all values of Λ and wave number *k*, except along certain lines in the Λ−*k* parameter space where the real part of the growth rate is zero, because the amplitude of the boundary layer velocity becomes zero along these lines. The real part of the *O*(ε) correction to the growth rate at these points is negative, indicating the presence of a small stabilizing effect due to the dissipation in the bulk of the fluid and the elastic medium.

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