they felt excited when finding the ___abandoned ____ knife by the lake.

when finding the abandoned knife by the lake-------这是分词结构作时间状语.可以改写成以下句子而不改变句意：
They felt excited when they found the abandoned knife by the lake.---------时间状语从句
而整个句子可以简单地成为：
They felt excited to find the abandoned knife by the lake.--------简单句