It may be the wastes that the plugs attach to which are leaking. Also look around the perimeter of the sink is it well siliconed and no holes? It can seep through there.
The kitchen sink pipes are leaking and require immediate repair.
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Result is broken pipes, so symptoms are: 1) no Water at faucet 2) leaking water elsewhere.
Plumbers are people who earn a living dealing with pipes.
Are you talking about pipes under a slab here? If so you probably have a hot water line under the slab leaking that is heating the water in the cold line.
There may be a blockage in the pipes specifically connected to the kitchen sink, preventing water from flowing through. This can be caused by debris or a clog in the pipes.
It is the water rushing through the pipes creating a pumping sound
To fix a leaking kitchen sink, first identify the source of the leak. It could be a loose connection, a damaged pipe, or a faulty seal. Tighten any loose connections, replace damaged pipes, and reseal any areas where water may be escaping. If you are unsure how to fix the leak, consider hiring a professional plumber for assistance.
The kitchen is probably closest to the water heater and the shower is the farthest. Water cools down the farther it travels through the pipes.
If it runs continually, as you say, then it can only be the inlet water pipe or fitting, leaking.
Clogged drains are caused by the accumulation of sticky material in the water pipes. Many times stuff like hair, food, plastic pouches.. gets stuck in the pipes and starts accumulating. Because of this the water from the kitchen or shower sink takes time to drain. Many times the drain pipes emit foul odor. Once the pipes are blocked, either the water stops draining from the sink or the debris starts coming out of the sink and toilets. The clogged water may also start leaking and also rust the pipes.[Source:A-General Plumbing and Sewer Service]