Standing Waves

If you have a solid understanding of what Travelling waves are (see previous sub-section if you need a refresher) then when you add up a sine wave moving to the right with a wave moving to the left, you get a standing wave.

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

Using \sin(a) + \sin(b) = 2 \sin((a+b)/2) \cos((a-b)/2) and simplifying the above equation we get :

y(x,t) = 2\sin(kx)\cos(\omega t)

A plot of this looks like the following:

Red circles denote the nodes where the amplitude of vibration is 0. Anti nodes on the other hand are the regions of maximum amplitude oscillations.

For the most part when one is referring to standing waves, it is customary to just talk about the resultant wave that you see above.

But one should understand that the way you form this is by taking a right moving wave and adding it up with a left-moving wave:

Blue – Traveling wave moving to the right. Red- Traveling wave moving to the left

Beats + Doppler effect : How do radar guns work?

Concepts involved:

Doppler Effect and Beats

Radar stands for RAdio Detection And Ranging. The way radar guns work to find out the speed of an object is that a high frequency radio wave is transmitted from the gun onto the object whose speed you would like to measure.

This wave bounces off the moving object and then returns back to the radar gun. A circuit in the gun amplifies the signals and adds them together.

Case – I

Let’s consider a stationary object. Any wave that hits the object reflects back and get back to you in the same frequency. If you add these signals up you would get a wave with the following amplitude:

y(t)  =\sin(2 \pi f t) + \sin(2 \pi f t + \phi)

Case-II

Now if that object is moving with some velocity, your reflected wave would be doppler shifted. This means the frequency of the received wave is different than the transmitted wave. And when you add those two signals up you get:

y = sin(2 \pi f t) + sin(2 \pi f_1 t) \rightarrow \text{Beats Phenomenon}

f_{beats} = | f - f_1 |

Since each velocity corresponds to a particular beat frequency, radar guns use the beat frequency as a measure to find out how fast an object is moving.

If you would like to learn more about the circuit underlying radar guns, a useful resource would be this video.

The travelling wave: An intuition

The aim of this section is to understand the traveling wave solution.

We all know about our favorite function – ‘The sinusoid’.

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y becomes 0 whenever sin(x) = 0 i.e x = n \pi  (n=1,2,3 ...)

Now the form of the traveling sine wave is as follows:

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When does the value for y become 0 ? Well, it is when

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As you can see this value of x is dependent on the value of time t ,which means that the points where y=0 moves to the right as time progresses.

\text{(t=0)} x = n \pi

\text{(t=1)} x = n \pi  + \omega

\text{(t=2)} x = n \pi + 2\omega

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Here is a slowly moving forward sine wave for reference.

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You can now try to convince yourself that a sine wave moving to the left is given by :

y= A sin(x + \omega t )

** The general form of a traveling wave is y = sin(kx - \omega t) . For the sake of simplicity we have considered the case where k=1 but understand that you can apply the same argument to any value of k as well.

Vibrating string fixed at both ends

Consider a string that is clamped at x = 0 and x= L (i.e y(0) = 0; y(L) = 0 ) undergoing traverse vibrations. And you would like to know the motion of the string.

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Maybe you know a priori that the solutions are sinusoids but you have no information on its wave number.

So you start trying out every single possibility of the wave number.

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Few more solutions that satisfy the boundary condition

The important thing to understand here is that If there weren’t any boundary conditions that was imposed on the string then all possible sinusoidal wave would be a solution to the problem.

But the existence of a boundary condition acts as a constraint and restricts the total number of possibilities.

Fun Sidenote:

Perhaps you have stumbled upon the word ‘quantization’, ‘quantized’, ‘discrete energy states’ when people talk about atoms. If not, you will most probably hear it somewhere when you are taking an undergraduate class in physics.

If you have an electron in a hydrogen atom, there are only specific energy levels it can be observed to occupy when its energy is measured because the electron is trapped in the atom which is a type of Boundary condition -> restricts the total number of states possible -> identical to the string scenario -> leads to quantization

But if the electron is unbound or there are no boundary conditions, the electron can in theory take any energy state it wants. You cannot have quantized states if you do not have boundary conditions.

Simple Harmonic Oscillator: An intuition

You have a mass suspended on a spring. We want to know where the mass will be at any instant of time.

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The physical solution

Now before we get on to the math, let us first visualize the motion by attaching a spray paint bottle as the mass.

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Oh, wait that seems like a function that we are familiar with – The sinusoid.

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Without even having to write down a single equation, we have found out the solution to our problem. The motion that is traced  by the mass is a sinusoid.

But what do I mean by a sinusoid ?

If you took the plotted paper and tried to create that function with the help of sum of polynomials i.e x, x2, x3 … Now you this what it would like :

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By taking an infinite of these polynomial sums you get the function Since this series of polynomial occurs a lot, its given the name – sine.

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A note on the cosine

Since cosine is merely the sine function pushed to the side, the analogy works the same way.

Therefore we can conclude that the motion described a mass attached to a spring is a sine or a cosine function:

y(t) = A \sin( \omega t + \phi) \ \text{or} \  A \cos(\omega t + \phi)

Hopefully this gives you an intuition on why the solution to a simple harmonic oscillator is given by the sine or the cosine function. Following runs through how you would go about getting it from solving a differential equation.

\text{solution} \ x = A \cos(\omega t + \phi) \  \text{or} A \sin(\omega t + \phi)