- #1

songoku

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- TL;DR Summary
- This is note I got from my teacher.

Let say a horizontal string of length L is tied on one end to a fixed point and the other end of the string is vibrated so that a progressive wave is formed on the string, travels to the fixed point, being reflected back then superposes with incoming wave.

The equation of incoming wave is ##y_1=A \sin(\omega t - kx)## and the equation of the reflected wave will be ##y_2=-A \sin (\omega t+kx)## since the wave travels in opposite direction and undergoes phase change of ##\pi##

The equation of stationary wave will be:

$$y_s=y_1+y_2$$

$$=A \sin(\omega t - kx) -A \sin (\omega t+kx)$$

$$=-2A \cos (\omega t) \sin (kx)$$

My questions

1) Is the value of ##t## in ##y_1## and ##y_2## the same? I think the value will be different because ##t## in ##y_1## is measured from when the string is vibrated and ##t## in ##y_2## is measured when the incoming wave is reflected

2) Same question as question (1) but about ##x##. I also think ##x## in ##y_1## and ##y_2## are different because in ##y_1##, ##x## is horizontal distance of a point on the wave measured from the source of vibration and in ##y_2## the distance is measured from fixed end

3) The equation ##y_1## is for progressive wave traveling to the right but I found from other source (pdf file I downloaded from internet), equation for wave traveling to the right is ##y=A \sin (kx - \omega t)##, so which one is correct? I tried changing the equation a little bit to ##y=-A \sin (\omega t - kx)## but then I got different answer for the question: at the start, in which direction will the wave move, positive or negative vertical direction?

Thanks

The equation of stationary wave will be:

$$y_s=y_1+y_2$$

$$=A \sin(\omega t - kx) -A \sin (\omega t+kx)$$

$$=-2A \cos (\omega t) \sin (kx)$$

My questions

1) Is the value of ##t## in ##y_1## and ##y_2## the same? I think the value will be different because ##t## in ##y_1## is measured from when the string is vibrated and ##t## in ##y_2## is measured when the incoming wave is reflected

2) Same question as question (1) but about ##x##. I also think ##x## in ##y_1## and ##y_2## are different because in ##y_1##, ##x## is horizontal distance of a point on the wave measured from the source of vibration and in ##y_2## the distance is measured from fixed end

3) The equation ##y_1## is for progressive wave traveling to the right but I found from other source (pdf file I downloaded from internet), equation for wave traveling to the right is ##y=A \sin (kx - \omega t)##, so which one is correct? I tried changing the equation a little bit to ##y=-A \sin (\omega t - kx)## but then I got different answer for the question: at the start, in which direction will the wave move, positive or negative vertical direction?

Thanks