Welcome to our fourth example video. Considering the basics of traveling waves in this video, we're going to ask ourselves how to find the wave speed V from a function. Let's consider the function D of X T equals 10 centimeters multiplied by the sign of seven per centimeter. Times X plus 12 hurts times t plus pi over three. Okay, and we want to find out wave speed now. The equation you might remember because it's easy is V equals Lambda over t, the wave length divided by period. But because we're looking at the function here where we have Omega and K, it's probably going to be easier for us to use this relationship. V equals omega over K, which you conceive from their conversions is the exact exact same equation. This is nice because we can just plug things in. We have 12 hurts and then we want to convert seven per centimeter to seven per meter. So when we do that, we have seven comes one over centimeters times. There are 100 centimeters in 1 m, so centimeters canceled on left with 700 per meter, so we have divided by 700 per meter that's equal to 12 divided by 700 meters per second. So there's our wave speed that we've found from a function. Now. Another interesting question is, how fast is an individual piece of the material that it's traveling through going up and down Now we know displacement tells us the position on the time, so how far it displaces at a particular time? Um so if we take the derivative of displacement as a function of X and T, we take that direct expected time, it should be the velocity of the displacement. How fast the displacement is changing as a function of time. Okay, so let's take the derivative with respect to time of a sign of K X minus omega T plus Bye. Okay, so remember when we want to take the derivative of the sine function, what we're going to get back is cosine function. We're going to take the derivative of the inside. But the only term that matters is the negative omega t. So we end up with in negative omega a cosine of K X minus omega T plus five. Okay, where the negative comes from the derivative of the inside, not from taking the derivative of sign, which is just positive. Cosine. Okay, so this is a function of how fast particles they're going to move up and down for this particular displacement function.