Here is the scene where the cop is shown removing sound cables while Forrest Gump was giving a speech. As a result, the crowd couldn’t hear him:
Why did the cop remove sound cables while Forrest Gump was giving a speech?
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At the time the movie describes, there were already huge demonstrations and Vietnam War Protests. Many people couldn't understand why there were such mass protest. Both sides were acting and supporting their ideas, and there were marches and riots:
The Deputies and soldiers were taunted and assaulted with vegetables, rocks, and bottles. The troops inside the Pentagon rushed outside as the violence escalated. A full-scale riot erupted.
Forrest was wearing his uniform, and acting as if he was supporting the protest. Which led police officer to see him as a traitor to the cause. He then wanted to shut him down. Without being able to talk about the war and its atrocities, Forrest isn't a support to the protesters anymore. That's also why a protester is really pissed off and comes to fight the policeman.
When you (or after you did) serve in the US Army, you often hear people saying: "Thank you for your service". How do you thank someone who is willing to risk their life for yours? Actually, not by saying their duty is worthless. The Vietnam war being legitimate or not (this is not the point here), many soldiers were just doing what they thought was their job: protect their country. During the rallies, people were fighting against the war, thus many people in the Forces saw their action criticized and belittled. When Forrest was about to speak, the man who unplugged the mics wanted to prevent that: a possible rant against war and soldiers.
In the context of the movie, it's because the event is an anti-war rally. It's attended by people opposed to the US presence in Vietnam. The veterans giving speeches are talking about their own experiences in Vietnam and how bad it is for American soldiers there.
The cop is politically opposed to the anti-war movement, and doesn't want the people being stirred by the first hand experiences of the veterans waiting to speak to them.
In the context of making the movie, I believe that they wanted Gump's statement to be incredibly profound, but they couldn't have it go out to the general public because it would effectively change history. So having the PA system be sabotaged means that Gump can give an incredibly moving first hand account of the war without having to ignore the real-world consequences it would bring.