Is it permissible for a wife to masturbate to achieve orgasm, whether in the presence or absence of her husband, with his knowledge, and is ghusl required in this case, and is this act considered masturbation, and what is the ruling on this act in the event of the husband's absence due to his being with another wife?

Using the hand to satisfy lust is a form of forbidden masturbation, whether in the presence or absence of the husband, and whether he knows about it or not. It is obligatory to repent to Allah from this act, and to increase in supplication and fasting. As for ghusl (definition) (ritual bath), it is only obligatory if semen is discharged.