# 2.1 Waves

 Page 1 / 1
The wave equation and some basic properties of its solution are given

## The wave equation

In deriving the motion of a string under tension we came up with an equation:

$\frac{{\partial }^{2}y}{\partial {x}^{2}}=\frac{1}{{{\text{v}}}^{2}}\frac{{\partial }^{2}y}{\partial {t}^{2}}$ which is known as the wave equation. We will show that this leads to waves below, but first, let us note the fact that solutions of this equation can beadded to give additional solutions.

Say you have two waves governed by two equations Since they are traveling in the same medium, ${\text{v}}$ is the same $\frac{{\partial }^{2}{f}_{1}}{\partial {x}^{2}}=\frac{1}{{{\text{v}}}^{2}}\frac{{\partial }^{2}{f}_{1}}{\partial {t}^{2}}$ $\frac{{\partial }^{2}{f}_{2}}{\partial {x}^{2}}=\frac{1}{{{\text{v}}}^{2}}\frac{{\partial }^{2}{f}_{2}}{\partial {t}^{2}}$ add these $\frac{{\partial }^{2}{f}_{1}}{\partial {x}^{2}}+\frac{{\partial }^{2}{f}_{2}}{\partial {x}^{2}}=\frac{1}{{{\text{v}}}^{2}}\frac{{\partial }^{2}{f}_{1}}{\partial {t}^{2}}+\frac{1}{{{\text{v}}}^{2}}\frac{{\partial }^{2}{f}_{2}}{\partial {t}^{2}}$ $\frac{{\partial }^{2}}{\partial {x}^{2}}\left({f}_{1}+{f}_{2}\right)=\frac{1}{{{\text{v}}}^{2}}\frac{{\partial }^{2}}{\partial {t}^{2}}\left({f}_{1}+{f}_{2}\right)$ Thus ${f}_{1}+{f}_{2}$ is a solution to the wave equation

Lets say we have two functions, ${f}_{1}\left(x-{\text{v}}t\right)$ and ${f}_{2}\left(x+{\text{v}}t\right)$ . Each of these functions individually satisfy the wave equation. note that $y={f}_{1}\left(x-{\text{v}}t\right)+{f}_{2}\left(x+{\text{v}}t\right)$ will also satisfy the wave equation. In fact any number of functions of the form $f\left(x-{\text{v}}t\right)$ or $f\left(x+{\text{v}}t\right)$ can be added together and will satisfy the wave equation. This is a very profound property of waves. For example it will allow us to describe a verycomplex wave form, as the summation of simpler wave forms. The fact that waves add is a consequence of the fact that the wave equation $\frac{{\partial }^{2}f}{\partial {x}^{2}}=\frac{1}{{{\text{v}}}^{2}}\frac{{\partial }^{2}f}{\partial {t}^{2}}$ is linear, that is $f$ and its derivatives only appear to first order. Thus any linear combination of solutions of the equation is itself a solution to the equation.

## General form

Any well behaved (ie. no discontinuities, differentiable) function of the form $y=f\left(x-{\text{v}}t\right)$ is a solution to the wave equation. Lets define ${f}^{\prime }\left(a\right)=\frac{df}{da}$ and ${f}^{\prime \prime }\left(a\right)=\frac{{d}^{2}f}{d{a}^{2}}\text{.}$ Then using the chain rule $\begin{array}{c}\frac{\partial y}{\partial x}=\frac{\partial f}{\partial \left(x-{\text{v}}t\right)}\frac{\partial \left(x-{\text{v}}t\right)}{\partial x}\\ =\frac{\partial f}{\partial \left(x-{\text{v}}t\right)}={f}^{\prime }\left(x-{\text{v}}t\right)\text{,}\end{array}$ and $\frac{{\partial }^{2}y}{\partial {x}^{2}}={f}^{\prime \prime }\left(x-{\text{v}}t\right)\text{.}$ Also $\begin{array}{c}\frac{\partial y}{\partial t}=\frac{\partial f}{\partial \left(x-{\text{v}}t\right)}\frac{\partial \left(x-{\text{v}}t\right)}{\partial t}\\ =-{\text{v}}\frac{\partial f}{\partial \left(x-{\text{v}}t\right)}\\ =-{\text{v}}{f}^{\prime }\left(x-{\text{v}}t\right)\end{array}$ $\frac{{\partial }^{2}y}{\partial {t}^{2}}={{\text{v}}}^{2}{f}^{\prime \prime }\left(x-{\text{v}}t\right)\text{.}$ We see that this satisfies the wave equation.

Lets take the example of a Gaussian pulse. $f\left(x-{\text{v}}t\right)=A{e}^{-{\left(x-{\text{v}}t\right)}^{2}/2{\sigma }^{2}}$

Then $\frac{\partial f}{\partial x}=\frac{-2\left(x-{\text{v}}t\right)}{2{\sigma }^{2}}A{e}^{-{\left(x-{\text{v}}t\right)}^{2}/2{\sigma }^{2}}$ 

and $\frac{\partial f}{\partial t}=\frac{-2\left(x-{\text{v}}t\right)\left(-{\text{v}}\right)}{2{\sigma }^{2}}A{e}^{-{\left(x-{\text{v}}t\right)}^{2}/2{\sigma }^{2}}$  or $\frac{{\partial }^{2}f\left(x-{\text{v}}t\right)}{\partial {t}^{2}}={{\text{v}}}^{2}\frac{{\partial }^{2}f\left(x-{\text{v}}t\right)}{\partial {x}^{2}}$ That is it satisfies the wave equation.

## The velocity of a wave

To find the velocity of a wave, consider the wave: $y\left(x,t\right)=f\left(x-{\text{v}}t\right)$ Then can see that if you increase time and x by $\Delta t$ and $\Delta x$ for a point on the traveling wave of constant amplitude $f\left(x-{\text{v}}t\right)=f\left(\left(x+\Delta x\right)-{\text{v}}\left(t+\Delta t\right)\right)\text{.}$ Which is true if $\Delta x-{\text{v}}\Delta t=0$ or ${\text{v}}=\frac{\Delta x}{\Delta t}$ Thus $f\left(x-{\text{v}}t\right)$ describes a wave that is moving in the positive $x$ direction. Likewise $f\left(x+{\text{v}}t\right)$ describes a wave moving in the negative x direction.

Lots of students get this backwards so watch out!

Another way to picture this is to consider a one dimensional wave pulse of arbitrary shape, described by ${y}^{\prime }=f\left({x}^{\prime }\right)$ , fixed to a coordinate system ${O}^{\prime }\left({x}^{\prime },{y}^{\prime }\right)$

Now let the ${O}^{\prime }$ system, together with the pulse, move to the right along the x-axis at uniform speed v relative to a fixed coordinate system $O\left(x,y\right)$ . As it moves, the pulse is assumed to maintain its shape. Any point P on the pulsecan be described by either of two coordinates $x$ or ${x}^{\prime }$ , where ${x}^{\prime }=x-{\text{v}}t$ . The $y$ coordinate is identical in either system. In the stationary coordinate system's frame of reference, the moving pulse has the mathematical form $y={y}^{\prime }=f\left({x}^{\prime }\right)=f\left(x-{\text{v}}t\right)$ If the pulse moves to the left, the sign of v must be reversed, so that we maywrite $y=f\left(x±{\text{v}}t\right)$ as the general form of a traveling wave. Notice that we have assumed $x={x}^{\prime }$ at $t=0$ .

Waves carry momentum, energy (possibly angular momentum) but not matter

## Wavelength, wavenumber etc.

We will often use a sinusoidal form for the wave. However we can't use $y=A{\mathrm{sin}}\left(x-{\text{v}}t\right)$ since the part in brackets has dimensions of length. Instead we use $y=A{\mathrm{sin}}\frac{2\pi }{\lambda }\left(x-{\text{v}}t\right)\text{.}$ Notice that $y\left(x=0,t\right)=y\left(x=\lambda ,t\right)$ which gives us the definition of the wavelength $\lambda$ .

Also note that the frequency is $\nu =\frac{{\text{v}}}{\lambda }\text{.}$ The angular frequency is defined to be $\omega \equiv 2\pi \nu =\frac{2\pi {\text{v}}}{\lambda }\text{.}$ Finally the wave number is $k\equiv \frac{2\pi }{\lambda }\text{.}$ So we could have written our wave as $y=A{\mathrm{sin}}\left(kx-\omega t\right)$ Note that some books say $k=\frac{1}{\lambda }$

## Normal modes on a string as an example of wave addition

Lets go back to our solution for normal modes on a string: ${y}_{n}\left(x,t\right)={A}_{n}{\mathrm{sin}}\left(\frac{2\pi x}{{\lambda }_{n}}\right){\mathrm{cos}}{\omega }_{n}t$ ${y}_{n}\left(x,t\right)={A}_{n}{\mathrm{sin}}\left(\frac{2\pi x}{{\lambda }_{n}}\right){\mathrm{cos}}\left(\frac{2\pi }{{\lambda }_{n}}{\text{v}}t\right)\text{.}$ Now lets do the following: make use of ${\mathrm{sin}}\left(\theta +\phi \right)+{\mathrm{sin}}\left(\theta -\phi \right)=2{\mathrm{sin}}\theta {\mathrm{cos}}\phi$ Also lets just take the first normal mode and drop the n's Finally, define $A\equiv {A}_{1}/2$ Then $y\left(x,t\right)=2A{\mathrm{sin}}\left(\frac{2\pi x}{\lambda }\right){\mathrm{cos}}\left(\frac{2\pi }{\lambda }{\text{v}}t\right)$ becomes $y\left(x,t\right)=A{\mathrm{sin}}\left[\frac{2\pi }{\lambda }\left(x-{\text{v}}t\right)\right]+A{\mathrm{sin}}\left[\frac{2\pi }{\lambda }\left(x+{\text{v}}t\right)\right]$ These are two waves of equal amplitude and speed traveling in opposite directions.We can plot what happens when we do this. The following animation was made with Mathematica using the command

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Got questions? Join the online conversation and get instant answers!