# 16.2 Mathematics of waves  (Page 7/11)

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If you shake the end of a stretched spring up and down with a frequency f , you can produce a sinusoidal, transverse wave propagating down the spring. Does the wave number depend on the frequency you are shaking the spring?

The wavelength is equal to the velocity of the wave times the frequency and the wave number is equal to $k=\frac{2\pi }{\lambda },$ so yes, the wave number will depend on the frequency and also depend on the velocity of the wave propagating through the spring.

Does the vertical speed of a segment of a horizontal taut string through which a sinusoidal, transverse wave is propagating depend on the wave speed of the transverse wave?

In this section, we have considered waves that move at a constant wave speed. Does the medium accelerate?

The medium moves in simple harmonic motion as the wave propagates through the medium, continuously changing speed, therefore it accelerates. The acceleration of the medium is due to the restoring force of the medium, which acts in the opposite direction of the displacement.

If you drop a pebble in a pond you may notice that several concentric ripples are produced, not just a single ripple. Why do you think that is?

## Problems

A pulse can be described as a single wave disturbance that moves through a medium. Consider a pulse that is defined at time $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s}$ by the equation $y\left(x\right)=\frac{6.00\phantom{\rule{0.2em}{0ex}}{\text{m}}^{3}}{{x}^{2}+2.00\phantom{\rule{0.2em}{0ex}}{\text{m}}^{2}}$ centered around $x=0.00\phantom{\rule{0.2em}{0ex}}\text{m}.$ The pulse moves with a velocity of $v=3.00\phantom{\rule{0.2em}{0ex}}\text{m/s}$ in the positive x -direction. (a) What is the amplitude of the pulse? (b) What is the equation of the pulse as a function of position and time? (c) Where is the pulse centered at time $t=5.00\phantom{\rule{0.2em}{0ex}}\text{s}$ ?

A transverse wave on a string is modeled with the wave function $y\left(x,t\right)=\left(0.20\phantom{\rule{0.2em}{0ex}}\text{cm}\right)\text{sin}\left(2.00\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x-3.00\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}t+\frac{\pi }{16}\right).$ What is the height of the string with respect to the equilibrium position at a position $x=4.00\phantom{\rule{0.2em}{0ex}}\text{m}$ and a time $t=10.00\phantom{\rule{0.2em}{0ex}}\text{s}?$

$y\left(x,t\right)=-0.037\phantom{\rule{0.2em}{0ex}}\text{cm}$

Consider the wave function $y\left(x,t\right)=\left(3.00\phantom{\rule{0.2em}{0ex}}\text{cm}\right)\text{sin}\left(0.4\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x+2.00\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}t+\frac{\pi }{10}\right).$ What are the period, wavelength, speed, and initial phase shift of the wave modeled by the wave function?

A pulse is defined as $y\left(x,t\right)={e}^{-2.77\phantom{\rule{0.2em}{0ex}}{\left(\frac{2.00\left(x-2.00\phantom{\rule{0.2em}{0ex}}\text{m/s}\left(t\right)\right)}{5.00\phantom{\rule{0.2em}{0ex}}\text{m}}\right)}^{2}}.$ Use a spreadsheet, or other computer program, to plot the pulse as the height of medium y as a function of position x . Plot the pulse at times $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s}$ and $t=3.00\phantom{\rule{0.2em}{0ex}}\text{s}$ on the same graph. Where is the pulse centered at time $t=3.00\phantom{\rule{0.2em}{0ex}}\text{s}$ ? Use your spreadsheet to check your answer. The pulse will move $\text{Δ}x=6.00\phantom{\rule{0.2em}{0ex}}\text{m}$ .

A wave is modeled at time $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s}$ with a wave function that depends on position. The equation is $y\left(x\right)=\left(0.30\phantom{\rule{0.2em}{0ex}}\text{m}\right)\text{sin}\left(6.28\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x\right)$ . The wave travels a distance of 4.00 meters in 0.50 s in the positive x -direction. Write an equation for the wave as a function of position and time.

A wave is modeled with the function $y\left(x,t\right)=\left(0.25\phantom{\rule{0.2em}{0ex}}\text{m}\right)\text{cos}\left(0.30\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x-0.90\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}t+\frac{\pi }{3}\right).$ Find the (a) amplitude, (b) wave number, (c) angular frequency, (d) wave speed, (e) phase shift, (f) wavelength, and (g) period of the wave.

a. $A=0.25\phantom{\rule{0.2em}{0ex}}\text{m};$ b. $k=0.30\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1};$ c. $\omega =0.90\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1};$ d. $v=3.0\phantom{\rule{0.2em}{0ex}}\text{m/s};$ e. $\varphi =\pi \text{/}3\phantom{\rule{0.2em}{0ex}}\text{rad};$ f. $\lambda =20.93\phantom{\rule{0.2em}{0ex}}\text{m}$ ; g. $T=6.98\phantom{\rule{0.2em}{0ex}}\text{s}$

A surface ocean wave has an amplitude of 0.60 m and the distance from trough to trough is 8.00 m. It moves at a constant wave speed of 1.50 m/s propagating in the positive x -direction. At $t=0,$ the water displacement at $x=0$ is zero, and ${v}_{y}$ is positive. (a) Assuming the wave can be modeled as a sine wave, write a wave function to model the wave. (b) Use a spreadsheet to plot the wave function at times $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s}$ and $t=2.00\phantom{\rule{0.2em}{0ex}}\text{s}$ on the same graph. Verify that the wave moves 3.00 m in those 2.00 s.

A wave is modeled by the wave function $y\left(x,t\right)=\left(0.30\phantom{\rule{0.2em}{0ex}}\text{m}\right)\text{sin}\left[\frac{2\pi }{4.50\phantom{\rule{0.2em}{0ex}}\text{m}}\left(x-18.00\frac{\text{m}}{\text{s}}t\right)\right].$ What are the amplitude, wavelength, wave speed, period, and frequency of the wave?

$A=0.30\phantom{\rule{0.2em}{0ex}}\text{m},\phantom{\rule{0.2em}{0ex}}\lambda =4.50\phantom{\rule{0.2em}{0ex}}\text{m},\phantom{\rule{0.2em}{0ex}}v=18.00\phantom{\rule{0.2em}{0ex}}\text{m/s},\phantom{\rule{0.2em}{0ex}}f=4.00\phantom{\rule{0.2em}{0ex}}\text{Hz},\phantom{\rule{0.2em}{0ex}}T=0.25\phantom{\rule{0.2em}{0ex}}\text{s}$

A transverse wave on a string is described with the wave function $y\left(x,t\right)=\left(0.50\phantom{\rule{0.2em}{0ex}}\text{cm}\right)\text{sin}\left(1.57\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x-6.28\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}t\right)$ . (a) What is the wave velocity of the wave? (b) What is the magnitude of the maximum velocity of the string perpendicular to the direction of the motion?

A swimmer in the ocean observes one day that the ocean surface waves are periodic and resemble a sine wave. The swimmer estimates that the vertical distance between the crest and the trough of each wave is approximately 0.45 m, and the distance between each crest is approximately 1.8 m. The swimmer counts that 12 waves pass every two minutes. Determine the simple harmonic wave function that would describes these waves.

$y\left(x,t\right)=0.23\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(3.49\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x-0.63\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}t\right)$

Consider a wave described by the wave function $y\left(x,t\right)=0.3\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(2.00\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x-628.00\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}t\right).$ (a) How many crests pass by an observer at a fixed location in 2.00 minutes? (b) How far has the wave traveled in that time?

Consider two waves defined by the wave functions ${y}_{1}\left(x,t\right)=0.50\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\frac{2\pi }{3.00\phantom{\rule{0.2em}{0ex}}\text{m}}x+\frac{2\pi }{4.00\phantom{\rule{0.2em}{0ex}}\text{s}}t\right)$ and ${y}_{2}\left(x,t\right)=0.50\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\frac{2\pi }{6.00\phantom{\rule{0.2em}{0ex}}\text{m}}x-\frac{2\pi }{4.00\phantom{\rule{0.2em}{0ex}}\text{s}}t\right).$ What are the similarities and differences between the two waves?

They have the same angular frequency, frequency, and period. They are traveling in opposite directions and ${y}_{2}\left(x,t\right)$ has twice the wavelength as ${y}_{1}\left(x,t\right)$ and is moving at half the wave speed.

Consider two waves defined by the wave functions ${y}_{1}\left(x,t\right)=0.20\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\frac{2\pi }{6.00\phantom{\rule{0.2em}{0ex}}\text{m}}x-\frac{2\pi }{4.00\phantom{\rule{0.2em}{0ex}}\text{s}}t\right)$ and ${y}_{2}\left(x,t\right)=0.20\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{cos}\left(\frac{2\pi }{6.00\phantom{\rule{0.2em}{0ex}}\text{m}}x-\frac{2\pi }{4.00\phantom{\rule{0.2em}{0ex}}\text{s}}t\right).$ What are the similarities and differences between the two waves?

The speed of a transverse wave on a string is 300.00 m/s, its wavelength is 0.50 m, and the amplitude is 20.00 cm. How much time is required for a particle on the string to move through a distance of 5.00 km?

Each particle of the medium moves a distance of 4 A each period. The period can be found by dividing the velocity by the wavelength: $t=10.42\phantom{\rule{0.2em}{0ex}}\text{s}$

In Example, we calculated the final speed of a roller coaster that descended 20 m in height and had an initial speed of 5 m/s downhill. Suppose the roller coaster had had an initial speed of 5 m/s uphill instead, and it coasted uphill, stopped, and then rolled back down to a final point 20 m bel
A steel lift column in a service station is 4 meter long and .2 meter in diameter. Young's modulus for steel is 20 X 1010N/m2.  By how much does the column shrink when a 5000- kg truck is on it?
what exactly is a transverse wave
does newton's first law mean that we don't need gravity to be attracted
no, it just means that a brick isn't gonna move unless something makes it move. if in the air, moves down because of gravity. if on floor, doesn't move unless something has it move, like a hand pushing the brick. first law is that an object will stay at rest or motion unless another force acts upon
Grant
yeah but once gravity has already been exerted .. i am saying that it need not be constantly exerted now according to newtons first law
Dharmee
gravity is constantly being exerted. gravity is the force of attractiveness between two objects. you and another person exert a force on each other but the reason you two don't come together is because earth's effect on both of you is much greater
Grant
maybe the reason we dont come together is our inertia only and not gravity
Dharmee
this is the definition of inertia: a property of matter by which it continues in its existing state of rest or uniform motion in a straight line, unless that state is changed by an external force.
Grant
the earth has a much higher affect on us force wise that me and you together on each other, that's why we don't attract, relatively speaking of course
Grant
quite clear explanation but i just want my mind to be open to any theory at all .. its possible that maybe gravity does not exist at all or even the opposite can be true .. i dont want a fixed state of mind thats all
Dharmee
why wouldn't gravity exist? gravity is just the attractive force between two objects, at least to my understanding.
Grant
earth moves in a circular motion so yes it does need a constant force for a circular motion but incase of objects on earth i feel maybe there is no force of attraction towards the centre and its our inertia forcing us to stay at a point as once gravity had acted on the object
Dharmee
why should it exist .. i mean its all an assumption and the evidences are empirical
Dharmee
We have equations to prove it and lies of evidence to support. we orbit because we have a velocity and the sun is pulling us. Gravity is a law, we know it exists.
Grant
yeah sure there are equations but they are based on observations and assumptions
Dharmee
g is obtained by a simple pendulum experiment ...
Dharmee
gravity is tested by dropping a rock...
Grant
and also there were so many newtonian laws proved wrong by einstein . jus saying that its a law doesnt mean it cant be wrong
Dharmee
pendulum is good for showing energy transfer, here is an article on the detection of gravitational waves: ***ligo.org/detections.php
Grant
yeah but g is calculated by pendulum oscillations ..
Dharmee
thats what .. einstein s fabric model explains that force of attraction by sun on earth but i am talking about force of attraction by earth on objects on earth
Dharmee
no... this is how gravity is calculated:F = G*((m sub 1*m sub 2)/r^2)
Grant
gravitational constant is obtained EXPERIMENTALLY
Dharmee
the G part
Dharmee
Calculate the time of one oscillation or the period (T) by dividing the total time by the number of oscillations you counted. Use your calculated (T) along with the exact length of the pendulum (L) in the above formula to find "g." This is your measured value for "g."
Dharmee
G is the universal gravitational constant. F is the gravity
Grant
search up the gravity equation
Grant
yeahh G is obtained experimentally
Dharmee
sure yes
Grant
thats what .. after all its EXPERIMENTALLY calculated so its empirical
Dharmee
yes... so where do we disagree?
Grant
its empirical whixh means it can be proved wrong
Dharmee
so cant just say why wouldnt gravity exists
Dharmee
the constant, sure but extremely unlikely it is wrong. gravity however exists, there are equations and loads of support surrounding the concept. unfortunately I don't have a high enough background in physics but have this discussion with a physicist
Grant
can u suggest a platform where i can?
Dharmee
stack overflow
Grant
stack exchange, physics section***
Grant
its an app?
Dharmee
there is! it is also a website as well
Grant
okayy
Dharmee
nice talking to you
Dharmee
***physics.stackexchange.com/
Grant
likewise :)
Grant
What is the percentage by massof oxygen in Al2(so4)3
A spring with 50g mass suspended from it,has its length extended by 7.8cm 1.1 determine the spring constant? 1.2 it is observed that the length of the spring decreases by 4.7cm,from its original length, when a toy is place on top of it. what is the mass of the toy?
solution mass = 50g= 0.05kg force= 50 x 10= 500N extension= 7.8cm = 0.078m using the formula Force= Ke K = force/extension 500/.078 = 6410.25N/m
Sampson
1.2 Decrease in length= -4.7cm =-0.047m mass=? acceleration due to gravity= 10 force = K x e force= mass x acceleration m x a = K x e mass = K x e/acceleration = 6410.25 x 0.047/10 = 30.13kg
Sampson
1.1 6.28Nm-¹
Anita
1.2 0.03kg or 30g
Anita
I used g=9.8ms-²
Anita
you should explain how yoy got the answer Anita
Grant
ok
Anita
with the fomular F=mg I got the value for force because now the force acting on the spring is the weight of the object and also you have to convert from grams to kilograms and cm to meter
Anita
so the spring constant K=F/e where F is force and e is extension
Anita
In this first example why didn't we use P=P° + ¶hg where ¶ is density
Density = force applied x area p=fA =p = mga, then a=h therefore substitute =p =mgh
Hlehle
Hlehle
sorry I had a little typo in my question
Anita
Density = m/v (mass/volume) simple as that
Augustine
Hlehle vilakazi how density is equal to force * area and you also wrote p= mgh which is machenical potential energy ? how ?
Manorama
what is wave
who can state the third equation of motion
Alfred
wave is a distrubance that travelled in medium from one point to another with carry energy .
Manorama
wave is a periodic disturbance that carries energy from one medium to another..
Augustine
what exactly is a transverse wave then?
Dharmee
two particles rotate in a rigid body then acceleration will be ?
same acceleration for all particles because all prticles will be moving with same angular velocity.so at any time interval u find same acceleration of all the prticles
Zaheer
what is electromagnetism
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
What is th formular for force
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 rate of change of momentum of a body is directly proportional to the force exerted on that body.
Rani
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?
Consider a wave produced on a stretched spring by holding one end and shaking it up and down. Does the wavelength depend on the distance you move your hand up and down?
no, only the frequency and the material of the spring
Chun
Tech
beat line read important. line under line
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