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Because it is based on an initial seismic analysis and preset criteria, tsunami forecast often fails in assessment of tsunami danger. The level of danger can be determined more or less reliably only when observed sea level data became available. Along with the sea level data, i.e., vertical motions of free water surface, we suggest considering horizontal motions of water that accompany the formation and the propagation of a tsunami. The amplitude of horizontal motions is normally much higher than the amplitude of the vertical motions. Detection of the horizontal motions may provide tsunami warning centers with additional in situ data that can be used for estimation of tsunami strength. In this study, taking the 2011 Tohoku-Oki event as an example, horizontal motions of water in the vicinity of the tsunami source are theoretically examined by means of dynamic and static numerical models developed within the framework of linear shallow-water approximation. It is shown that in the vicinity of the tsunami source within a wide area of about 0.5 million square kilometers, the amplitude of horizontal motions exceeded 10 m, whereas in some shallow-water areas the amplitude amounted to hundreds of meters. Possible methods of in situ detection of the horizontal motions are discussed.