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)