O see how two traveling waves of the same frequency create a standing wave. Consider a traveling wave described by the formula y1(x, t)=Asin(kx−ωt). This function might represent the lateral displacement of a string, a local electric field, the position of the surface of a body of water, or any of a number of other physical manifestations of waves.

Find ye(x) and yt(t). Keep in mind that yt(t) should be a trigonometric function of unit amplitude.
Express your answers in terms of A, k, x, ?, and t. Separate the two functions with a comma.?

yt(t) = cos(wt)

ye(t) = 2*A*sin(kx)

Explanation:

Given:-

Consider a traveling wave described by the formula

                                  y(x,t)=Asin(kx−ωt)

Find:-

Find ye(x) and yt(t). Keep in mind that yt(t) should be a trigonometric function of unit amplitude.

Express your answers in terms of A, k, x, ?, and t.

Solution:-

- We are to express the given y1 ( x, t ) in the form of sum of two waves y1 and y2:

                             y ( x, t ) = y1(x,t) + y2(x,t)

Where,

                             y1(x,t) = A*sin(kx−ωt)   ... wave travelling in +x direction

                             y2(x,t) = A*sin(kx+wt)  ... wave travelling in -x direction

- Such that ye(x) is a single variable function of x and yt(t) is a single variable unit amplitude function of (t).

                             y ( x, t ) = y1(x,t) + y2(x,t) = ye(x)*yt(t)

                            A*sin(kx−ωt) + A*sin(kx+wt) = ye(x)*yt(t)

- Apply sum to product formula on the left hand side:

                        = 2A* [ sin ( (kx -wt + kx +wt) / 2 ) * cos ( ((kx -wt - kx - wt) / 2 )]

                        = 2*A*[sin (kx) * cos (-wt) ]

Where,

                        cos (-wt ) = cos (wt)

                        = 2*A*sin (kx) * cos (wt)

- The function yt(t) must be a unit amplitude:

                       yt(t) = cos(wt)

                       ye(t) = 2*A*sin(kx)

carbon is found in all living things. carbon atoms move constantly through

living organisms, the oceans, the atmosphere, and the earth’s crust in what is

known as the carbon cycle. the directions taken by carbon atoms through this

cycle are very complicated and can take millions of years to make a full circle.

all animals, from humans to the dinosaurs are part of the carbon cycle. when

animals eat food, they get carbon in the form of carbohydrates and proteins.

in animals, oxygen combines with food in the cells to produce energy for daily

activity and then gives off carbon. the carbon combines with oxygen to form

carbon dioxide (co2) and is released back into the atmosphere as a waste

product when animals breathe and exhale.

from 145-65 million years ago, earth was much

hotter than today and covered with dense,

tropical swamp forests. the trees and other

plants were immense and

provided an endless supply of

food for the giant animals that

roamed the land. somewhere in

the air above one of these

forests, a lone carbon atom has joined

up with two oxygen atoms to form a molecule of co2.

the co2 molecule was sucked into the tiny holes (stomata) on the leaf of a

fern plant and joined with sunlight, chlorophyll and water to make food and

energy in the plant’s cells through photosynthesis. the oxygen (o2) from the

co2 molecule was sent back into the atmosphere; the carbon atom (c) was

detached and used to make a molecule of sugar.

if that carbon atom had been eaten instead, a totally

different situation would have developed. let’s say

that a giant plant-eating dinosaur (herbivore) ate the

fern for breakfast, and swallowed the carbon in the

fern in the form of carbohydrates and proteins. in

the process of respiration, oxygen combined with

the carbohydrates and proteins in the dinosaur’s cells

to provide energy for its daily activity.  

the skeletons of most mammals reach a certain size and then stop growing. however, many animals, including some mammals, keep growing throughout their lives. kangaroos, for example, just keep growing and growing until they die. also, the longer an animal lives the greater the chance it might catch some deadly disease.