<< Chapter < Page
  Waves and optics   Page 1 / 1
Chapter >> Page >
The wave equation and some basic properties of its solution are given

Waves

The wave equation

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

2 y x 2 = 1 v 2 2 y 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.

Waves add

Say you have two waves governed by two equations Since they are traveling in the same medium, v is the same 2 f 1 x 2 = 1 v 2 2 f 1 t 2 2 f 2 x 2 = 1 v 2 2 f 2 t 2 add these 2 f 1 x 2 + 2 f 2 x 2 = 1 v 2 2 f 1 t 2 + 1 v 2 2 f 2 t 2 2 x 2 ( f 1 + f 2 ) = 1 v 2 2 t 2 ( f 1 + f 2 ) Thus f 1 + f 2 is a solution to the wave equation

Lets say we have two functions, f 1 ( x v t ) and f 2 ( x + v t ) . Each of these functions individually satisfy the wave equation. note that y = f 1 ( x v t ) + f 2 ( x + v t ) will also satisfy the wave equation. In fact any number of functions of the form f ( x v t ) or f ( x + v t ) 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 2 f x 2 = 1 v 2 2 f 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 ( x v t ) is a solution to the wave equation. Lets define f ( a ) = d f d a and f ( a ) = d 2 f d a 2 . Then using the chain rule y x = f ( x v t ) ( x v t ) x = f ( x v t ) = f ( x v t ) , and 2 y x 2 = f ( x v t ) . Also y t = f ( x v t ) ( x v t ) t = v f ( x v t ) = v f ( x v t ) 2 y t 2 = v 2 f ( x v t ) . We see that this satisfies the wave equation.

Lets take the example of a Gaussian pulse. f ( x v t ) = A e ( x v t ) 2 / 2 σ 2

Then f x = 2 ( x v t ) 2 σ 2 A e ( x v t ) 2 / 2 σ 2

and f t = 2 ( x v t ) ( v ) 2 σ 2 A e ( x v t ) 2 / 2 σ 2 or 2 f ( x v t ) t 2 = v 2 2 f ( x v t ) 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 ( x , t ) = f ( x v t ) Then can see that if you increase time and x by Δ t and Δ x for a point on the traveling wave of constant amplitude f ( x v t ) = f ( ( x + Δ x ) v ( t + Δ t ) ) . Which is true if Δ x v Δ t = 0 or v = Δ x Δ t Thus f ( x v t ) describes a wave that is moving in the positive x direction. Likewise f ( x + v t ) 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 = f ( x ) , fixed to a coordinate system O ( x , y )

Now let the O system, together with the pulse, move to the right along the x-axis at uniform speed v relative to a fixed coordinate system O ( x , y ) .
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 , where x = x 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 = f ( x ) = f ( x v t ) If the pulse moves to the left, the sign of v must be reversed, so that we maywrite y = f ( x ± v t ) as the general form of a traveling wave. Notice that we have assumed x = x 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 sin ( x v t ) since the part in brackets has dimensions of length. Instead we use y = A sin 2 π λ ( x v t ) . Notice that y ( x = 0 , t ) = y ( x = λ , t ) which gives us the definition of the wavelength λ .

Also note that the frequency is ν = v λ . The angular frequency is defined to be ω 2 π ν = 2 π v λ . Finally the wave number is k 2 π λ . So we could have written our wave as y = A sin ( k x ω t ) Note that some books say k = 1 λ

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 ( x , t ) = A n sin ( 2 π x λ n ) cos ω n t y n ( x , t ) = A n sin ( 2 π x λ n ) cos ( 2 π λ n v t ) . Now lets do the following: make use of sin ( θ + φ ) + sin ( θ φ ) = 2 sin θ cos φ Also lets just take the first normal mode and drop the n's Finally, define A A 1 / 2 Then y ( x , t ) = 2 A sin ( 2 π x λ ) cos ( 2 π λ v t ) becomes y ( x , t ) = A sin [ 2 π λ ( x v t ) ] + A sin [ 2 π λ ( x + v t ) ] 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

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Waves and optics. OpenStax CNX. Nov 17, 2005 Download for free at http://cnx.org/content/col10279/1.33
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Waves and optics' conversation and receive update notifications?

Ask