There are some onomatopoeia that have entered Japanese through Chinese. They are typically rare, but they feature the mouth radical. The reason why they deserve special attention is that you may not realize at first hand that they're onomatopoeic.
Furthermore, we must understand that things Sino-Japanese tend to be literary. Because native onomatopoeia is so natural of spoken speech, we can expect Chinese onomatopoeic expressions to enter Japanese as rather technical words and may not quite be used or viewed as onomatopoeia. Some readily are viewed as such as seen in Ex. 1 and 2.
Nevertheless, it is certain that you will hardly find most of these spoken. These words are important, though, in higher class literature in which even today new Sino-Japanese expressions may be imported. For you as the advanced learner, running into new characters in this section may very well be the most beneficial and important thing to this lesson.
To make one's first cry.
To continue to laugh loudly.
Clear and resounding tone/sound
The resounding of high reputation
To play cheerfully.
A reverberating echo
To babble and speak rubbish.
The groaning of a dead spirit
Next are examples that really seem to have lost any sense of being onomatopoeic in Japanese. But if you think about some of them, you can still picture how they may relate to sound. For instance, 丁寧 (polite), which you will see has another spelling resembling the rest of these words, itself emulates the 'sound' of polite speech.
To let out a roar.
To declare defiantly.
The harmonizing, mentally and physically, of two parties engaged in an activity
Etymology Note: 阿吽 in the last example actually ultimately derives from the interjection Ohm from Sanskrit.