First, aphids are often carried onto plants by ants, which like the honeydew they secrete. The ants "farm" the aphids, offering them protection and carrying them to the tenderest new growth on the plants. The ants use their antennae to stroke the aphids, to get them to secrete the honeydew.
Aphids have a somewhat complex life cycle, with females able to produce live offspring -- all wingless females -- without the necessity of breeding. This occurs for several generations, and then some of the female aphids grow wings, fly off to a different plant species, and some of them become males. Breeding then occurs and eggs are laid for the next generation. That's a very simplified description of the process, and each species of aphid has its own variation of that basic pattern.
The point is that not all of the aphids will be removed by just one treatment, whether you use a spray from the garden hose, or a mixture of soap and water. The plain water doesn't kill the aphids, it just washes them away and the ants bring them back. The soapy water
does kill them, but there will always be a few survivors, or new ones that hatch from eggs. To counteract that, the plant(s) must be re-sprayed about every 5 to 7 days. Four sprayings is usually enough to kill all the pests, although ants may again bring a new batch of aphids to the plant later on.
It's important to use a pure soap, not a detergent. Most liquids for dishwashing are actually detergents, because detergents cut grease much better than plain soaps do. However, detergents are harsh and can burn a plant's leaves. Any pure soap will work, regardless of whether it's a solid or a liquid. A bar of Castile soap is fine, as is Ivory soap, but I personally prefer Dr. Bronner's unscented liquid soap.
HTH!
