Part 4 - Waves | rezarezvan.com

The next chapter in our physics course is about waves, especially mechanical waves.

General about Waves

In this chapter we’ll cover mechanical waves but also electromagnetic waves, in this case only (visible) light.

There are two types of waves we’ll be covering:

Transverse waves
- They have ‘noise’ orthogonal to the wave’s propagation direction.
Longitudinal waves
- They have ‘noise’ parallel with the wave’s propagation direction.

We can describe this noise with the function $y(x, t) = f(x + vt)$ .

Harmonic Waves

Harmonic waves are ‘idealized waves’ where the frequency is a positive integer multiple of the fundamental frequency.

This just means we can describe our waves in terms of sine and cosine functions.

A general formula for harmonic waves is,

y(x, t) = A \sin\left[2\pi\left(\frac{x}{\lambda} - \frac{t}{T}\right)\right].

Let’s cover each term here and see what they mean,

\begin{align*} A & \quad \text{Amplitude } [\mathrm{m}] \newline \lambda & \quad \text{Wavelength } [\mathrm{m}] \newline f & \quad \text{Frequency } [\mathrm{Hz} = \mathrm{1/s}] \newline v & \quad \text{Propagation velocity } [\mathrm{m/s}] \newline \omega & \quad \text{Angular frequency } [\mathrm{rad/s}] \newline T & \quad \text{Period } [\mathrm{s}] \end{align*}

And now let’s write up some relationships between these variables,

\begin{align*} f & = \frac{1}{T}, \newline \omega & = 2\pi f = \frac{2\pi}{T}, \newline v & = \lambda f = \frac{\lambda}{T}, \newline \lambda & = v T. \end{align*}

Now we can rewrite our general formula to something even more general,

y(x, t) = A \sin(kx - \omega t + \phi),

where,

\begin{align*} k & = \frac{2\pi}{\lambda} \ [\mathrm{rad/m}], \newline \phi & \quad \text{Describes our 'starting' position.} \end{align*}

Different Kind of Velocities

There are two different kinds of velocities that we need to differentiate. We can view the entire wave’s velocity as $V_f$ , but at the same time we could view a singular particle’s speed. This is denoted by $V_p$ .

We can write $V_p$ as,

V_p = \frac{\partial y}{\partial t} = -\omega A \cos(kx - \omega t + \phi).

Velocities in a String

Strings, when flicked, will behave like waves - if we fixate a string on each ends, for example in a guitar.

We can calculate the propagation velocity as,

V_f = \sqrt{\frac{T}{\mu}},

where,

\begin{align*} T & \quad \text{Tension force } [\mathrm{N}] \newline \mu & \quad \text{Mass per unit length } [\mathrm{kg/m}] \end{align*}

Intensity

The ‘intensity’ of a wave describes its energy flux. But we can also describe it as,

I \propto A^2.

So the intensity is proportional to the amplitude squared. This will become important once we begin to cover light waves.

Wave interference

Interference is the phenomenon that occurs when two waves collide. It can either result in destructive interference where the waves ‘take each other out’, or in constructive interference, which means we ‘add’ the waves together. It’s quite a natural thing as we can see from the equations,

\begin{align*} y_1 & = A \sin(kx - \omega t), \newline y_2 & = A \sin(kx - \omega t + \phi), \end{align*}

y_{\text{total}} = A [\sin(kx - \omega t) + \sin(kx - \omega t + \phi)].

Using the sine addition formula,

y_{\text{total}} = 2A \cos\left(\frac{\phi}{2}\right) \sin\left(kx - \omega t + \frac{\phi}{2}\right).

The important term is $\frac{\phi}{2}$ , if this is,

\frac{\phi}{2} = m\pi,

where $m$ is a positive integer.

Then we have so-called constructive interference - visually this means we have highs and lows ‘aligned’.

If instead this term is,

\frac{\phi}{2} = (2m + 1) \frac{\pi}{2},

where $m$ is a positive integer.

Then we have destructive interference, or visually, opposites will be aligned (highs with lows, and lows with highs of the other).

More Ways to Write Wave Equation

One way we can rewrite our original wave equation is,

A \sin\left(\omega\left(t - \frac{x}{v}\right)\right).

Which can be useful sometimes!

Reflection and Transmission of Waves

If we have a wave that propagates from one medium to another, we will have so called reflection and transmission of the wave.

If we just visualize what will happen - the reflection of the wave will go in the other direction - whilst the transmission will have the same direction as the wave before.

By doing some math we get,

A_i + A_r = A_t.

Which seems logical, the initial wave added with the reflection wave should have the same amplitude as the new transmission wave.

We can also get the following,

\begin{align*} \frac{\partial}{\partial x} (y_i + y_r) & = \frac{\partial}{\partial x} y_t, \newline \frac{1}{V_1} (A_i - A_r) & = \frac{1}{V_2} A_t. \end{align*}

Which finally gives us the relations,

\begin{align*} A_t & = \frac{2 V_2}{V_2 + V_1} A_i, \newline A_r & = \frac{V_2 - V_1}{V_2 + V_1} A_i. \end{align*}

Standing Waves

Standing waves is the phenomenon when interference occurs but the waves have opposite directions,

\begin{align*} y_1 & = A \sin(kx - \omega t), \newline y_2 & = A \sin(kx + \omega t), \newline y_{\text{total}} & = \dots = 2A \sin(kx) \cos(\omega t). \end{align*}

From this we can see that we will always have some points that are 0. We call these points ‘nodes’. The peaks are therefore called antinodes.

Fixated Wave

Let’s apply our knowledge about standing waves to this example - we have a string which is fixated at both ends. So therefore,

\begin{align*} y(0, t) & = 0, \newline y(L, t) & = 0. \end{align*}

So at $x = 0$ (the start of the string) and $x = L$ (the end of the string).

We get,

y = 2A \sin(0) \cos(\omega t) = 0.

Which is what we wanted, now for $x = L$ ,

y = 2A \sin(kL) \cos(\omega t) = 0.

For this to be equal to 0 - $\sin(kL)$ needs to be 0. So,

\sin(kL) = 0, \quad k = \frac{2\pi}{\lambda} \Rightarrow kL = m\pi \Rightarrow \lambda = \frac{2L}{m}.

This means we only have a subset of all possible Wave-lengths that we can use so that our boundary conditions are fulfilled,

\lambda = 2L, L, \frac{2}{3} L, \dots

Beats

This phenomenon occurs when we have two frequencies which are really close, there isn’t really that much to say, we can look at the formula and derive what happens from there,

y = y_1 + y_2 = 2A \cos\left[2\pi \left(\frac{f_1 - f_2}{2}\right) t\right] \cos\left[2\pi \left(\frac{f_1 + f_2}{2}\right) t\right].

So from this we can see that if $f_1, f_2$ are really close, we will have a really low frequency and a high frequency that is close to $f_1$ .

The Doppler Effect

Is the effect that happens when we have a sound source which moves, or both the listener and sound source move.

If only the listener moves

If the listener moves then we can write the frequency which the listener hears as,

f_L = \frac{V + V_L}{\lambda} = \frac{V + V_L}{V / f_S} = \left(1 + \frac{V_L}{V}\right) f_S.

If both are moving

f_L = \frac{V + V_L}{V + V_S} f_S.

Conclusion

This was it for this part - quite short, waves are quite a natural phenomenon so a lot of things comes from intuition. In the next part we’ll begin to cover light waves!