# 16.2 Mathematics of waves  (Page 7/11)

 Page 7 / 11

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: $t=10.42\phantom{\rule{0.2em}{0ex}}\text{s}$

A central force is given as F vector (r),where a=2NM².Assuming the potential energy at infinity to be zero,calculate the potential energy of a particle located at the point (3,4)
what is a vector
A vector is any physical quantity which has a magnitude as well as a direction associated to it. Which means a vector is some physical quantity which has magnitude and direction.
malayala
what is matter
Seth
nice
Faith
What is the equation illustrating Williamsons ether synthesis
What is the equation illustrating Williamsons ether synthesis
Kingdom
what is simple harmonic motion
examples: vibrating prongs of a tuning fork and a guittar string.
Salman
It is a repetitive periodic motion of a system about an equilibrium position
Felix
SHM is the repitition process of to and fro motion.
Younus
SHM is the motion in which the restoring force is directly proportional to the displacement of body from its mean position and is opposite in direction to the displacement. From Hooke's law F=-kx
Kushal
SHM is the motion in which the restoring force is directly proportional to the displacement of body from its mean position and is opposite in direction to the displacement. From Hooke's law F=-kx
Kushal
what is a wave?
show that coefficient of friction of solid block inclined at an angle is equivalent to trignometric tangent of angle
DAVID
Wave is the transfer of energy due to the periodic vibration of the particles in the medium.
Kushal
wave is the transfer of energy
Vindora
Wave is the transfer of particles in a fluid or any way.
Younus
thanks for that definition.
Hi everyone please can dere be motion without force?
Lafon
no...
Enyia
Thanks
Lafon
hi
Omomaro
whats is schrodinger equation
Omomaro
l went spiral spring
Xalat
what is position?
position is simply where you are or where you were
Shii
position is the location of an object with respect to a two or three dimensional axes or space.
Bamidele
Can dere be motion without force?
Lafon
what is the law of homogeinity?
two electric lines of force never interested each other. why?
if two electric lines of force intersect eachother then their will be two tangent at a point which represent the two forces which is impossible.
Amar
proof that for BBC lattice structure 4r\root 5 and find Apf for the BBC structure
what is physics?
physics is deine as the specific measrument of of volume, area,nd distances...
Olakojo
if a string of 2m is suspended an an extended 3m elasticity is been applied.... is hooks law obeyed?
Enyia
if a string of 2m is suspended an an extended 3m elasticity is been applied.... is hooks law obeyed?
Enyia
yes
Alex
proof that for a BBC lattice structure a= 4r/ root 5 find the APF for the BBC structure
Eric
if a string of 2m is suspended an an extended 3m elasticity is been applied.... is hooks law obeyed?
tell me conceptual quetions of mechanics
I want to solve a physical question
ahmed
ok
PUBG
a displacement vector has a magnitude of 1.62km and point due north . another displacement vector B has a magnitude of 2.48 km and points due east.determine the magnitude and direction of (a) a+ b and (b) a_ b
quantum
George
a+b=2.9
SUNJO
a+b
Yekeen
use Pythogorous
Dhritwan
A student opens a 12kgs door by applying a constant force of 40N at a perpendicular distance of 0.9m from the hinges. if the door is 2.0m high and 1.0m wide determine the magnitude of the angular acceleration of the door. ( assume that the door rotates freely on its hinges.) please assist me to d
Mike
what is conditions met to produce shm
what is shm
Manzoor
shm?
Grant
Why is Maxwell saying that light is an electromagnetic wave?
Bong
1st condition; It(th e BBC's system) must have some inertia which will enable it to possess Kinetic energy 2. must be able to store potential energy
Calleb
I meant "the system" not the BBC'S....."
Calleb
Manzoor
kindly tell us the name of your university
Manzoor
GUlam Ishaq Khan INSTITUTE of engineering science
ali
Department of Environment Ionian University Zante Greece