# 16.2 Mathematics of waves  (Page 5/11)

 Page 5 / 11
$\frac{{\partial }^{2}y\left(x,t\right)}{\partial {x}^{2}}=\frac{1}{{v}^{2}}\phantom{\rule{0.2em}{0ex}}\frac{{\partial }^{2}y\left(x,t\right)}{\partial {t}^{2}}.$

[link] is the linear wave equation, which is one of the most important equations in physics and engineering. We derived it here for a transverse wave, but it is equally important when investigating longitudinal waves. This relationship was also derived using a sinusoidal wave, but it successfully describes any wave or pulse that has the form $y\left(x,t\right)=f\left(x\mp vt\right).$ These waves result due to a linear restoring force of the medium—thus, the name linear wave equation. Any wave function that satisfies this equation is a linear wave function.

An interesting aspect of the linear wave equation is that if two wave functions are individually solutions to the linear wave equation, then the sum of the two linear wave functions is also a solution to the wave equation. Consider two transverse waves that propagate along the x -axis, occupying the same medium. Assume that the individual waves can be modeled with the wave functions ${y}_{1}\left(x,t\right)=f\left(x\mp vt\right)$ and ${y}_{2}\left(x,t\right)=g\left(x\mp vt\right),$ which are solutions to the linear wave equations and are therefore linear wave functions. The sum of the wave functions is the wave function

${y}_{1}\left(x,t\right)+{y}_{2}\left(x,t\right)=f\left(x\mp vt\right)+g\left(x\mp vt\right).$

Consider the linear wave equation:

$\begin{array}{ccc}\hfill \frac{{\partial }^{2}\left(f+g\right)}{\partial {x}^{2}}& =\hfill & \frac{1}{{v}^{2}}\phantom{\rule{0.2em}{0ex}}\frac{{\partial }^{2}\left(f+g\right)}{\partial {t}^{2}}\hfill \\ \hfill \frac{{\partial }^{2}f}{\partial {x}^{2}}+\frac{{\partial }^{2}g}{\partial {x}^{2}}& =\hfill & \frac{1}{{v}^{2}}\left[\frac{{\partial }^{2}f}{\partial {t}^{2}}+\frac{{\partial }^{2}g}{\partial {t}^{2}}\right].\hfill \end{array}$

This has shown that if two linear wave functions are added algebraically, the resulting wave function is also linear. This wave function models the displacement of the medium of the resulting wave at each position along the x -axis. If two linear waves occupy the same medium, they are said to interfere. If these waves can be modeled with a linear wave function, these wave functions add to form the wave equation of the wave resulting from the interference of the individual waves. The displacement of the medium at every point of the resulting wave is the algebraic sum of the displacements due to the individual waves.

Taking this analysis a step further, if wave functions ${y}_{1}\left(x,t\right)=f\left(x\mp vt\right)$ and ${y}_{2}\left(x,t\right)=g\left(x\mp vt\right)$ are solutions to the linear wave equation, then $A{y}_{1}\left(x,t\right)+B{y}_{2}\left(x,y\right),$ where A and B are constants, is also a solution to the linear wave equation. This property is known as the principle of superposition. Interference and superposition are covered in more detail in Interference of Waves .

## Interference of waves on a string

Consider a very long string held taut by two students, one on each end. Student A oscillates the end of the string producing a wave modeled with the wave function ${y}_{1}\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{sin}\left(kx-\omega t\right)$ and student B oscillates the string producing at twice the frequency, moving in the opposite direction. Both waves move at the same speed $v=\frac{\omega }{k}.$ The two waves interfere to form a resulting wave whose wave function is ${y}_{R}\left(x,t\right)={y}_{1}\left(x,t\right)+{y}_{2}\left(x,t\right).$ Find the velocity of the resulting wave using the linear wave equation $\frac{{\partial }^{2}y\left(x,t\right)}{\partial {x}^{2}}=\frac{1}{{v}^{2}}\phantom{\rule{0.2em}{0ex}}\frac{{\partial }^{2}y\left(x,t\right)}{\partial {t}^{2}}.$

## Strategy

First, write the wave function for the wave created by the second student. Note that the angular frequency of the second wave is twice the frequency of the first wave $\left(2\omega \right)$ , and since the velocity of the two waves are the same, the wave number of the second wave is twice that of the first wave $\left(2k\right).$ Next, write the wave equation for the resulting wave function, which is the sum of the two individual wave functions. Then find the second partial derivative with respect to position and the second partial derivative with respect to time. Use the linear wave equation to find the velocity of the resulting wave.

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It is the study of the electromagnetic force, one of the four fundamental forces of nature. ... It includes the electric force, which pushes all charged particles, and the magnetic force, which only pushes moving charges.
Energy
what is units?
units as in how
praise
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F = m x a
Santos
State newton's second law of motion
can u tell me I cant remember
Indigo
force is equal to mass times acceleration
Santos
The acceleration of a system is directly proportional to the and in the same direction as the external force acting on the system and inversely proportional to its mass that is f=ma
David
The uniform seesaw shown below is balanced on a fulcrum located 3.0 m from the left end. The smaller boy on the right has a mass of 40 kg and the bigger boy on the left has a mass 80 kg. What is the mass of the board?
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how can one calculate the value of a given quantity
means?
Manorama
To determine the exact value of a percent of a given quantity we need to express the given percent as fraction and multiply it by the given number.
AMIT
meaning
Winford
briefly discuss rocket in physics
ok let's discuss
Jay
What is physics
physics is the study of natural phenomena with concern with matter and energy and relationships between them
Ibrahim
a potential difference of 10.0v is connected across a 1.0AuF in an LC circuit. calculate the inductance of the inductor that should be connected to the capacitor for the circuit to oscillate at 1125Hza potential difference of 10.0v is connected across a 1.0AuF in an LC circuit. calculate the inducta
L= 0.002H
NNAEMEKA
how did you get it?
Favour
is the magnetic field of earth changing
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tibebeab
can you explain it
Guhan
Energy can not either created nor destroyed .therefore who created? and how did it come to existence?
this greatly depend on the kind of energy. for gravitational energy, it is result of the shattering effect violent collision of two black holes on the space-time which caused space time to be disturbed. this is according to recent study on gravitons and gravitational ripple. and many other studies
tibebeab
and not every thing have to pop into existence. and it could have always been there . and some scientists think that energy might have been the only entity in the euclidean(imaginary time T=it) which is time undergone wick rotation.
tibebeab
What is projectile?
An object that is launched from a device
Grant
2 dimensional motion under constant acceleration due to gravity
Awais
Not always 2D Awais
Grant
Awais
why not? a bullet is a projectile, so is a rock I throw
Grant
bullet travel in x and y comment same as rock which is 2 dimensional
Awais
components
Awais
no all pf you are wrong. projectile is any object propelled through space by excretion of a force which cease after launch
tibebeab
for awais, there is no such thing as constant acceleration due to gravity, because gravity change from place to place and from different height
tibebeab
it is the object not the motion or its components
tibebeab
where are body center of mass on present.
on the mid point
Suzana
is the magnetic field of the earth changing?
tibebeab
does shock waves come to effect when in earth's inner atmosphere or can it have an effect on the thermosphere or ionosphere?
tibebeab
and for the question from bal want do you mean human body or just any object in space
tibebeab
A stone is dropped into a well of 19.6m deep and the impact of sound heared after 2.056 second ,find the velocity of sound in air.
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Kyla
In this case, the velocity of sound is 350 m/s.
Zahangir
why?
Kyla
some calculations is need. then you will get exact result.
Zahangir
i mean how? isn't it just a d over t?
Kyla
calculate the time it takes the stone to hit the ground then minus the stone's time to the total time... then divide the total distance by the difference of the time
Snuggly
awit lenard. Hahahah ari ga to!
Kyla