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Traveling-wave optomechanical interactions, known as Brillouin interactions, have now been established as a powerful and versatile resource for photonic sources, sensors, and radio-frequency processors. However, established Brillouin-based interactions with sufficient interaction strengths involve short phonon lifetimes, which critically limit their performance for applications, including radio-frequency filtering and optomechanical storage devices. Here, we investigate a new paradigm of optomechanical interactions with tightly confined fundamental acoustic modes, which enables the unique and desirable combination of high optomechanical coupling, long phonon lifetimes, tunable phonon frequencies, and single-sideband amplification. Using sensitive four-wave mixing spectroscopy controlling for noise and spatial mode coupling, optomechanical interactions with long $><\#comment/>2\text{µ<\#comment/>}\mathrm{s}$phonon lifetimes and strong $><\#comment/>400{\mathrm{W}}^{-<\#comment/>1}{\mathrm{m}}^{-<\#comment/>1}$coupling are observed in a tapered fiber. In addition, we demonstrate novel phonon self-interference effects resulting from the unique combination of an axially varying device geometry with long phonon lifetimes. A generalized theoretical model, in excellent agreement with experiments, is developed with broad applicability to inhomogeneous optomechanical systems.