Is there any Ayah in the Quran which allows husbands punish their wives because of disobedience?
Yes, there is. The ayah in the Quran says:
Men are in charge of women by [right of] what Allah has given one over the other and what they spend [for maintenance] from their wealth. So righteous women are devoutly obedient, guarding in [the husband's] absence what Allah would have them guard. But those [wives] from whom you fear arrogance - [first] advise them; [then if they persist], forsake them in bed; and [finally], strike them. But if they obey you [once more], seek no means against them. Indeed, Allah is ever Exalted and Grand. Quran 4/34
Also, a hadith in "Sahih Muslim" backs up the ayah above:
Prophet Muhammad (PBUH) said: "Fear Allah concerning women! Verily you have taken them on the security of Allah, and intercourse with them has been made lawful unto you by words of Allah. You too have rights over them, and that they should not allow anyone to sit on your bed [i.e., not let them into the house] whom you do not like. But if they do that, you can chastise them but not severely. Their rights upon you are that you should provide them with food and clothing in a fitting manner"
Before jumping to any conclusion. You need to read the rest of the answer...
Q1/ When are we allowed to beat them ?
But those [wives] from whom you fear arrogance.
Q2/ Is there another solution ?
Actually, yes. There are 2 other solutions in the same verse before reaching this step and other hadiths by the prophet telling us not to us force with our wives.
The 1st solution in the verse says:
[first] advise them;
The 2nd solution,
[then if they persist], forsake them in bed"
You must note that you are not allowed to jump to the 3rd solution(beating your wife) directly. You use the 3rd solution only when you have tried the 2 solutions above.
These solutions are for sincere people. In other words: Don't just give her a 1 second advice or forsake her in bed for 1 day and you think now you are allowed to beat her)
If the problem persists and instead of divorce, the husband may resort to another measure that may save the marriage, at least in some cases. Then he would use the last solution.
Although it is permisible to beat our wives, it is worth mentioning that Allah has asked us in many verses to be patient with our wives:
In the Quran:
O you who have believed, it is not lawful for you to inherit women by compulsion. And do not make difficulties for them in order to take [back] part of what you gave them unless they commit a clear immorality. And live with them in kindness. For if you dislike them - perhaps you dislike a thing and Allah makes therein much good.
Another solution in the Quran is:
And if you fear dissension between the two, send an arbitrator from his people and an arbitrator from her people. If they both desire reconciliation, Allah will cause it between them. Indeed, Allah is ever Knowing and Acquainted [with all things]. Quran 4/35
Not to mention dozens of verses which ask muslim to be patient. We should not ignore all these verses and beat our wives even if it permissible for us to do so.
Furthermore, the prophet never hit his wives or nor a slave. When you read his seerah(bio) he would always solve any family issue without resorting to force/violence. He even praised those who don't beat their wives:
In the Sunnah:
Do not beat the female servants of Allaah;" "Some (women) visited my family complaining about their husbands (beating them). These (husbands) are not the best of you;" and also, "[It is a shame that] one of you beats his wife like [an unscrupulous person] beats a slave and maybe he sleeps with her at the end of the day." [Riyadh Al-Saaliheen]
and in another hadith,
Abu Bakr came seeking permission to enter upon the Prophet, peace and blessings be upon him, and he heard Aisha raising her voice over the Messenger of Allah. He entered and said, “O daughter of Umm Ruman! Are you raising your voice over the Messenger of Allah?” So the Prophet stood between them and when Abu Bakr left, he said to her, “Do you see how I saved you from him?” Then Abu Bakr again sought permission to enter upon the Prophet and he heard them both laughing. He entered and said, “O Messenger of Allah, allow me to share in your peace just as you have allowed me to share in your conflict.” Source: Musnad Ahmad 17927
And as Muslims when we hear such hadiths, we should imitate and follow the teachings of the prophet Mohammed (PBUH) as the quran commands:
Say, [O Muhammad], "If you should love Allah , then follow me, [so] Allah will love you and forgive you your sins. And Allah is Forgiving and Merciful." http://quran.com/3/31
What kind of beating ?
Islam has specific guidelines that a Muslim must follow if he wants to beat his wife.
- It must be a light striking.
Ibn Abbas, The most famous interpreter of the Quran, when asked about this verse and the way you beat the wife, he answered:
You beat them with a Miswak
Miswak is a thin rod [google images here]. By choosing the thin rod as a tool,
Ibn Abbas is informing us that beating should not cause any bodily harm, or even be harsh. Instead, It is more like a way to chastise someone's wife.
- Muslims are not allowed to beat the face of an animal. Then Imagine how unpermisable it is to hit the face of a human ? In the hadith of the prophet:
Messenger (PBUH) forbade (the animals to be beaten) on the face or cauterisation on the face. Muslim
But why does the Quran say we should beat them while the prophet didn't raise his hands over any women ?
To answer this question you should know that Islamic teachings are universal in nature. They respond to the needs and circumstances of diverse times, cultures and circumstances. Some measures may work in some cases and cultures or with certain persons but may not be effective in others. By definition, a "permissible" act is neither required, encouraged nor forbidden. So may be beating the wife would in some cultures would actually help the relationship between the husband and the wife.
Last Note: Did you notice how the 1st verse at the beginning of my answer ended with a warning to those who would cause severe damage, abuse this verse, or beat their wives without any reason ? Allah said that he is the exalted and the Grand. Thus, who ever disobey allah must know that he is capable of punishing that person or even worse just as he punished his wife.