In this paper, we investigate the receptivity of a supersonic boundary layer to
impinging acoustic waves. Unlike previous studies of acoustic receptivity, where
the sound waves have phase speeds comparable with or larger than the free-stream
velocity U∞, the acoustic waves here have much slower (O(R
−1/8U∞)) phase velocity,
and their characteristic wavelength and frequency are of O(R
−3/8L) and O(R
1/4U∞/L)
respectively, compatible with the triple-deck structure, where L is the distance to the
leading edge and R the Reynolds number based on L and U∞. A significant feature of
a sound wave on the triple-deck scale is that an O(εs) perturbation in the free stream
generates much stronger (O(εsR
1/8
)) velocity fluctuations in the boundary layer. Two
receptivity mechanisms are considered. The first is new, involving the interaction of
two such acoustic waves and operating in a boundary layer over a smooth wall. The
second involves the interaction between an acoustic wave and the steady perturbation
induced by a wavy wall. The sound–sound, or sound–roughness, interactions generate
a forcing in resonance with a neutral Tollmien–Schlichting (T–S) wave. The latter is
thus excited near the lower branch of the neutral curve, and subsequently undergoes
exponential amplification. The excitation through sound–sound interaction may offer
a possible explanation for the appearance of instability modes downstream of their
neutral locations as was observed in a supersonic boundary layer over a smooth
wall. The triple-deck formalism is adopted to describe impingement and reflection of
the acoustic waves, and ensuing receptivity, allowing the coupling coefficient to be
calculated. The two receptivity processes with the acoustic waves on the triple-deck
scale are much more effective compared with those involving usual sound waves,
with the coupling coefficient being greater by a factor of O(R
1/4
) and O(R
1/8
) in
the sound–sound and sound–roughness interactions, respectively. A parametric study
for both the reflection and coupling coefficients is conducted for representative T–S
waves, to assess the influence of the streamwise and spanwise wavenumbers, and the
phase speed (or frequency) of the acoustic wave.