Final answer:
Seismic waves and sound waves can be transmitted through the core of a material like Earth, as they are mechanical waves that require a medium to travel. Electromagnetic waves, such as light waves, do not require a medium and so do not travel through the core of materials.
Step-by-step explanation:
Among the types of waves listed—sound waves, electromagnetic waves, seismic waves, and light waves—the ones that can be transmitted through the core of a material such as Earth are seismic waves and sound waves. Seismic waves are akin to sound waves as they require a medium to travel through and can be understood as mechanical waves. Sound waves are categorized as mechanical waves and can travel through solids, liquids, and gases. Seismic waves, similar to sound waves in mediums, can travel through Earth's layers and are used to study its interior structure. As established in seismic studies, there are different types of seismic waves, with some capable of traveling through Earth's liquid core, namely the compressional or longitudinal waves. On the other hand, shear or transverse waves cannot travel through liquid and thus do not pass through the Earth's core.
Light waves and electromagnetic waves, which include visible light and X-rays, are examples of electromagnetic waves which are types of transverse waves that do not require a medium to propagate and thus are not the type that can travel through the core of a material. They can propagate through a vacuum, which differentiates them from mechanical waves like sound and seismic waves.