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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(
Question
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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Physics
5 years
2021-07-27T14:27:45+00:00
2021-07-27T14:27:45+00:00 1 Answers
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Answers ( )
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)