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.?

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shahnoorazhar3 shahnoorazhar3 · Answer 1 · 2020-04-09T05:40:43+02:00

Answer:

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)