Based on the linear shallow-water equations, analytical solutions for trapped waves over an asymmetric oceanic ridge with an exponential cross-sectional profile are presented. The solutions can become those for a symmetric ridge by setting the same topographic parameter for both sides, in which there are both symmetric and asymmetric trapped waves. The dispersion relationships imply that the wavenumber is related to not only the frequency and water depth at the top of the ridge but also the cross-section profile parameters and the mode number. With increasing frequency, the phase velocity decreases rapidly and approaches the classical shallow-water wave velocity for fixed water depth at the top and profile parameters. Meanwhile, for increasing angular frequency, the energy velocity first increases sharply from less than the shallow-water wave velocity and then decreases gently toward it. For an asymmetric ridge, the amplitude profile is confined to a narrower region on the steeper side and spread in a wider region on the less-steep side. Finally, the analytical solutions are applied to reveal the trapping effect of the Hawaii ridge on the 2011 Tohoku Tsunami in the North Pacific.