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Question
waves that transmit energy through matter or empty space
Mechanical waves need a medium (matter) to travel. Electromagnetic waves can move through empty space (vacuum) as well as matter. Longitudinal and transverse waves are types of mechanical waves (though electromagnetic waves can also have transverse - like properties in terms of oscillation direction relative to propagation, but the key here is the medium requirement). Since the question is about waves that can transmit energy through matter or empty space, electromagnetic waves fit this description as they don't rely solely on a material medium.
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Electromagnetic Wave