# Wave equations

So, consider a tranverse travelling wave. Let's say that it is moving in the x-direction, and the displacement of its medium as it travels is perpendicular to that motion: in the y-direction. We can describe the position of particles in the medium in the following manner:

As time passes, the position of particles will change. They will oscillate up-and-down as the wave passes. Is there some way to figure out how fast the particles will move?

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Q:  How fast will the particles move in the tranverse direction?

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Yes, there is a way. Since we know the y-position as a function of time, we can simply take the derivative with respect to time to find the y-velocity as a function of time:

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Q:  Write an equation for vy as a function of space and time.

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Fine.

But we don't have to stop there. We can take things one step further.

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Q:  Is there a way to figure out the tranverse acceleration
of particles in the medium?

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Yup. Since we know the y-velocity as a function of time, we can simply take the derivative with respect to time to find the y-acceleration as a function of time:

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Q:  Write an equation for ay as a function of space and time.

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