Two sinusoidal waves with the same amplitude and wavelength travel through each

Question

Two sinusoidal waves with the same amplitude and wavelength travel through each other along a string that is stretched along an x axis. Their resultant wave is shown twice in the figure, as the antinode A travels from an extreme upward displacement to an extreme downward displacement in 5.91 ms. The tick marks along the axis are separated by 12.0 cm; height H is 1.80 cm. Let the equations for one of the two waves be of the form

y(x, t) = ym sin(kx + t).

In the equation for the other wave, what are (a) ym, (b) k, (c) , and (d) the sign in front of ?

Explanation / Answer

This is a classical example of travelling wave. Please some theory to get acquainted with the topic.

a) The term ym is called the Amplitude of the wave. Lets see how we can determine this.

ym is followed by sin(something). Now we know that sin(something) can vary from -1 to +1 only as we keep on changing the "something" inside it.

So the maximum value of the wave can be +ym and minimum can be -ym.

from figure it can be seen that H = ym - (-ym) = 2*ym

ym = H/2 = 1.8/2 cm = 0.9 cm

b) Now, k is given by 2*pi/lambda, where lambda is the wavelength. It is the distance between to peaks)

From the picture, the two consecutive peaks(choose either the dark ones or the dotted ones) are separated by two tick marks on the axis. hence lambda = 2*12 = 24 cm.

k = 2*3.14/24 = 0.261

c) w is called the period. It is the time taken by any point on the wave to complete one cycle. It is given that from extreme top to extreme bottom is covered in 5.91 ms. The return will also take the same time.

hence, w=2*5.91ms = 11.82 ms

d) the sign depends on the direction of travel of the wave. If travelling towards right then its negative else positive.

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