Question

To 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.

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

Answer 1

The spatial function ye(x) is A sin(kx) and the temporal function yt(t) for a traveling wave described as y1(x, t) = A sin(kx - ωt) is sin(ωt) after normalizing to unit amplitude.

Step-by-step explanation:

To find the functions ye(x) and yt(t), we need to consider the wave equation y1(x, t) = A sin(kx − ωt). Since ye(x) should describe the spatial part of the wave and yt(t) should describe the temporal part, we separate the variables. The spatial part is obtained by fixing the time, which makes ye(x) = A sin(kx). Similarly, the temporal part can be obtained by observing the wave at a fixed position, resulting in yt(t) = sin(−ωt), which can be rewritten to a unit amplitude as yt(t) = sin(ωt), assuming that the wave amplitude A is absorbed in the equation's constants.

Answer 2

y_e = 2A sin kx

y_t = cos wt

Step-by-step explanation:

A standing wave is produced by the reflection of a traveling wave in an obstacle, let's write the initial traveling wave

y₁ = A sin (kx -wt)

let's write the reflected wave

y₂ = A sin (kx + wt)

we find the sum of these two waves

y = y₁ + y₂

y = A sin (kx -wt) + A sin (kx + wt)

let's develop the double angles

y = A [sin kx cos wt - cos kx sin wt + sin kx cos wt + cos kx sin wt]

y = A [2 sin kx cox wt]

we can write this resulting wave

y_e = 2A sin kx

y_t = cos wt