The Physics Philes, lesson 98: The Most Important Equation in All of Physics
After a week hiatus for finals, I’m back with more physic-y goodness. Now…let’s see. Where did we leave off? Ah, here we go. It’s time to talk about particle velocity and acceleration, and, in doing so, we’ll find one of the most important equations in physics. So let’s get started.
In a way, determining the transverse particle velocity and acceleration isn’t all that different from determining linear velocity and linear acceleration. We already know to express a sinusoidal wave as a function:
When we have a regular linear position function, all we need to do to find the velocity function is take the derivative. We do something similar to find the particle velocity. However, instead of the function being dependent on one variable, as is the case with linear motion, the function is dependent on two variables – x and t. So, if we want to find the particle velocity with respect to time, we need to take the partial derivative. That is, we only allow one of the variables to change. In this case, t. So, when we take the partial derivative, we get:
Because we only allow t to vary, this equation allows us to look at a particular point on a string. This equation shows that the transverse velocity varies with time, which is what we would expect with simple harmonic motion. The particle could be moving with a maximum velocity of ωA, which can be greater than, less than, or equal to the wave speed.
We find the acceleration of the particle in a similar way. In linear motion, to find the acceleration all we have to do is take the derivative of the velocity function. To find the transverse acceleration of a particle in a wave, we need to take the partial derivative of the particle velocity function:
These partial derivatives have been taken with respect to time, but we can also take the partial derivatives with respect to the position x. This will tell us the shape of the wave at any time, like you’ve taken a picture of it. The first partial derivative is the slope of the string at any point. The second partial derivative is the curvature of the string, expressed like this:
We can use the angular frequency relationship ω=vk of a periodic wave to see that the ratio of the second derivatives with respect to t and x gives us the wave speed squared. Or, in math:
Furthermore, we find that:
This last equation is called a the wave equation, and it’s one of the most important equations in physics. Whenever we find this equation, we know that the disturbance will propagate as a wave along the x-axis at a wave speed v. We’ve been talking about sinusoidal waves, but this holds for any wave. Electric and magnetic fields satisfy this equation with a wave speed that turns out to be the speed of light. This lead us to determine that light is an electromagnetic wave. So, you know, it’s a pretty important equation.
So, now that we have the equations, how are we supposed to interpret it? When the wave have an upward curvature (the second derivative of y with respect to x greater than 0) the acceleration of that point is positive. When the wave has a downward curvature (the second derivative of y with respect to x less than 0) the acceleration is negative. The acceleration is zero when the second derivative of y with respect to x equals zero. It’s important to remember that this is these equations have to do with the transverse velocity and acceleration, so it’s movement is perpendicular to the propagation direction of the wave.
Even though we’ve been talking about transverse waves, this all still works with longitudinal waves. We still take y to represent the displacement of the particle from its equilibrium position, but in longitudinal waves that will be parallel to wave propagation rather than perpendicular to it.
OK! Now you know how to find the velocity and acceleration of a particle a wave is traveling through. Kind of makes you feel powerful, doesn’t it? We’re not done with mechanical waves, though. Next week we’ll look at the speed of a transverse wave. Fun times ahead!
Featured image credit: Peter Kaminski via Flickr