Physics with Calculus/Mechanics/Harmonic Motion, Waves, and Sounds

Simple harmonic motion
A body executes simple harmonic motion if its acceleration is proportional to its displacement from a fixed point, and is always in the direction of that point.

To explore simple harmonic motion (SHM) let's take the example of a spring with a mass in the absence of gravity (interestingly, you get SHM even with gravity present). If this is our ideal spring, the force is $$~kx$$ where $$~k$$ is a measure of the stiffness of the spring and $$~x$$ is the displacement. The force is toward the origin if that is the equilibrium position of the spring, so we write $$~-kx$$ to remind ourselves of that. Now, Newton's second law becomes

$$m \frac{d^2 x}{dt^2} = -kx$$.

This differential equation is easy enough to solve, and the answer is $$~A cos( \omega_0 t + \phi)$$ where $$~A$$ and $$~\phi$$ are arbitrary constants and $$~\omega_0 = \sqrt{k/m}$$. It does not really matter how we got the solution, because we are physicists, not mathematicians. This is the answer we are expecting, so we try it, and lo and behold, it works. If you do not believe me, substitute it in. Moreover, this is the complete solution, and you will just have to believe me on that because it is slightly more difficult to prove.

Without loss of generality, we will take $$~\phi$$, also called the phase shift, to be zero (if you are concerned about this, we are just defining where $$~t=0$$ is).

Now, a remarkable thing we recognize about the solution is that the frequency (radians per second), $$~\omega_0$$ is independent of $$~A$$. That is, no matter how big the oscillations are, the frequency is the same. A pendulum approximately undergoes SHM, so this is why they are used in clocks, the amplitude doesn't affect the period! By the way, we have added the subscript zero to omega because we are going to have some other omegas soon.

Some terms to remember are frequency, f (cycles per second) = $$~ \omega_0 / (2 \pi) $$ and the period, T = $$~ 1/f $$. These are not so important, but often people will specify the frequency or the period instead of the angular frequency, so they can be helpful.

Now, to get the velocity, differentiate the position, and to get the acceleration, differentiate the velocity. We have,

$$~v = A \omega_0 sin ( \omega_0 t)$$ and $$~a = -A \omega_0^2 cos( \omega_0 t) = - \omega_0^2 x$$.

Now, we have avoided saying what $$~A$$ is. It turns out, it depends on the problem, or the initial conditions. We can say the velocity or position of the oscillator at some t is something and then use the expression for $$~v$$ or $$~a$$ to find $$~A$$. You can do the same thing with the phase if you want, but it is a little tedious and doesn't tell us much.

Notice the greatest velocity is at the equilibrium position $$~(x = 0)$$ of the oscillation. We can go on making such statements, but they are all extremely obvious if you simply plot out the position, velocity, and acceleration on the same graph.

Damping
An object that oscillates freely oscillates at its natural frequency. If it loses no energy, it will continue to oscillate forever. Damping is when an oscillating mass loses energy. There are 3 types of damping:

1) Light - The amplitude gradually decreases over time

2) Heavy - The mass would overshoot 0 displacement

3) Critical - The displacement decreases to 0 without any oscillation.

The cause of damping is frictional forces, e.g. Car suspension

Let's try to quantify this a bit. Say there is a friction force which is proportional to the velocity (this is a pretty good approximation in many cases) with constant of proportionality c. Then, by Newton's second law,

$$~m \frac{d^2 x}{dt^2} + c \frac{dx}{dt} + kx = 0$$.

This equation is a little trickier to solve than without the friction. We going to use a very nice trick which you will find throughout physics, and whenever you have similar equations. Notice that if $$~x$$ is a solution and $$~y$$ is a solution, then $$~ax + by$$ is also a solution, where $$~a$$ and $$~b$$ are constants (real or complex). This property means the equation is called "linear." We know that $$~e^{ix} = cos(x) + i sin(x)$$. Assume $$~x$$ is $$~Ae^{i \omega t}$$. Then we just take the real part of $$~x$$ and we get our answer because the equation is linear, but exponentials are so much easier to work with than sines and cosines. The equation of motion becomes

$$~(-m \omega^2 + i \omega c + k) e^{i \omega t} = 0$$

So,

$$~(m \omega^2 - i \omega c - k) = 0$$ or $$~\omega_0 = ic/2m \plusmn \sqrt{(c/m)^2 /4 - k/m}$$.

Defining $$ ~\gamma = c/m$$, and remembering $$\omega_0^2 = k/m$$

$$~\omega = i \gamma /2 \plusmn \sqrt{-\gamma ^2 /4 + \omega_0^2}$$.

Defining $$ ~\omega_\gamma = \sqrt{-\gamma ^2 /4 + \omega_0^2} $$, we have the general solution

$$ ~Ae^{-\gamma/2 + i\omega_\gamma} + Be^{-\gamma/2 - i\omega_\gamma} = e^{-\gamma/2} (A e^{i\omega_\gamma} + B e^{-i\omega_\gamma} ) $$.

All we do is take the real part of this with Euler's identity, and we have,

$$ ~C e^{-\gamma/2} cos( \omega_\gamma t + \phi) $$,

where $$~C$$ and $$~\phi$$ are just $$~A$$ and $$~B$$ written a different way. You can find them if you want, but they won't be very helpful. Notice that the oscillator oscillates with ever decreasing amplitude, but not at it's "natural" frequency, but at a different frequency.

It is conceivable that $$~\omega_\gamma$$ is imaginary, in which case, the entire solution is just a negative exponential! This is called critical damping, when it just turns into being an exponential instead of oscillatory motion.

Waves
We all have seen water waves, such as the ripples in a pond or puddle. Although the wave moves from one point to another, the water itself does not go with it. So, what are waves? They are somewhat difficult to define, yet they are present throughout physics -- water waves, seismic waves, sound waves, light waves, and many others. Waves are generally characterized by propagating themselves; if you leave a wave alone, it keeps going. But already we have exceptions because some materials absorb waves, so it won't necessarily keep going. Furthermore, waves usually travel through a medium, which is a fancy word for substance. Water waves travel in water, sound waves travel in air (or just about anything else). However, light does not seem to travel in a medium, but rather through empty space. So, a wave does not even need something to travel in, it seems. In any case, instead of asking around trying to figure out precisely what waves are, let's try to say some useful things about waves.

First, we will introduce a common wave equation (the equation which describes how waves move), which holds true for electromagnetic waves in vacuum, for small waves in a tight string, and for small sound waves. Then we will take a look at it's solutions and try to understand some of the principles of waves.

$$ ~\nabla^2 E = \frac{1}{c^2} \frac{\partial^2 E}{\partial^2 t}$$

$$ ~\nabla^2 E \equiv \frac{\partial^2 E}{\partial^2 x} + \frac{\partial^2 E}{\partial^2 y} + \frac{\partial^2 E}{\partial^2 z}$$

$$~E$$ can be pressure, displacement, or a component of the electric field, or something else, but in any case, this is the equation you get for many things.

First, let's look at the one dimensional equation.

$$~ \frac{\partial^2 E}{\partial^2 z} = \frac{1}{c^2} \frac{\partial^2 E}{\partial^2 t}$$.

I will tell you the solutions to this equations, because that is what we care about. It is a little more difficult to prove that they are the only solutions than it is to verify they are solutions, but you will have to take my word for it. We don't really care how we get the solutions because we are physicists not mathematicians.

$$ ~E = A f(x + ct) + B g(x - ct) $$

where $$~A$$ and $$~B$$ are arbitrary constants, and $$~f$$ and $$~g$$ are arbitrary functions. Interestingly enough, it does not matter what $$~f$$ and $$~g$$ are! Physically speaking, the two terms correspond waves moving to the left and to the right at the speed c. Notice that if I have two solutions, say, $$~E$$ and $$~E'$$, then $$~ aE + bE' $$ is also a solution where $$~a$$ and $$~b$$ are arbitrary constants. This is one of the most useful properties of the solutions to this wave equation. This property is known as linearity.

So, we know everything about waves since we have the solution to the wave equation, right? Wrong. Let's take some particularly special solutions to this equation and study them a bit.

$$~A cos(-\omega t + k x + \phi) $$

is a solution provided that

$$~\frac{\omega}{k} = c $$.

$$~k$$ is called the wave number or wave vector (it will be a vector when we look at the three dimensional solutions), $$~\omega$$ is called the frequency (measured in radians per second, not cycles per second), $$~\phi$$ is called the phase shift, and $$~A$$ is called the amplitude. Physically this corresponds to a sine wave of frequency $$~k$$ moving to the right, with each point going up and down with frequency $$~\omega$$. That is, $$~k$$ is the frequency in space and $$~\omega$$ is the frequency in time. $$~c$$ is called the phase velocity in this case (we will see that there is another kind of velocity, but it happens to be the same as this velocity for this particular equation).

Some people like to talk about frequency in cycles per second and give it a special name, $$~f$$. Some people like to talk about wavelength, $$~\lambda$$, which is just $$~\frac{2 \pi}{k}$$ and is the distance from crest to crest. Fairly obviously, $$~\lambda f = c $$ because $$~\frac{\omega}{k} = c $$. Some people call this one of the most important or beautiful relationships of physics, but I would disagree. It really is not that profound, it is just telling us that if $$~f$$ crests pass us per second, and the distance from crest to crest is $$ ~\lambda $$, the speed of the wave is just $$ ~f \lambda $$ which is just using conversion factors. It is actually pretty mundane.

So, what can we do with these solutions we have found? Why, make new solutions! We already know that if we have two solutions we can add them together to get another solution. We use Fourier analysis and you will find it just about anywhere you find linear systems. It turns out that we can write any periodic function, discontinuous or not, as the sum of many sine waves and cosine waves of different frequencies. So, we usually construct a solution based on a boundary condition by finding the coefficients for the different components. This is the big deal about sine waves -- they add up to make any function you want.

Three Dimensional Solutions
To generalize from one to three dimensions, we convert k and x into vectors: $$k\rightarrow\vec k$$ and $$x\rightarrow\vec r$$. To create a scalar phase, we need to take the dot product between these vectors:

$$cos(-\omega t + k x + \phi) \rightarrow cos(-\omega t + \vec k\cdot\vec r + \phi)$$