To understand the concept of nodes of a standing wave. the nodes of a standing wave are points where the displacement of the wave is zero at all times. nodes are important for matching boundary conditions, for example that the point at which a string is tied to a support has zero displacement at all times (i. e., the point of attachment does not move). consider a standing wave, where y represents the transverse displacement of a string that extends along the x direction. here is a common mathematical form for such a wave:
y(x, t) = acos(x, t)sin(wt)

where a is the maximum transverse displacement of the string (the amplitude of the wave), which is assumed to be nonzero, k is the wavenumber, w is the angular frequency of the wave, and t is time.
at which three points x1, x2 and x3 closest to x=0 but with x> 0, will the displacement of the string be zero for all times? these are the first three nodal points. express the first three nonzero nodal points as multiples of the wavelength lambda, using constants like pie . list the factors that multiply in increasing order, separated by commas.

scattering of light by tiny particles in air to produce the colors we see on sky, is dependent on the following factors:

(i) wavelength of light

(ii) size of the scattering particle  

(iii) distance traveled by light through the atmosphere.

according to rayleigh's law of scattering, the amount of scattering is inversely proportional to the fourth power of its wavelength. shorter the wavelength, greater is its scattering. hence violet/blue light is scattered more than red.

sun's light is a composite light ( visible light) with wavelengths in the range 400 nm (violet) to 700 nm (red). as sunlight passes through our atmosphere, the molecules of air scatter light according to their wavelengths. during sunrise and sunset, the sun's position is below the horizon. hence light from the sun travels a greater distance in the atmosphere when compared to its position a noon.

as sunlight enters the earth's atmosphere, blue light is scattered away and at the end of its journey in the sky, only red and orange wavelengths remain in the beam.

hence at sunrise and sunset, the sky appears reddish orange.

54 degrees incident angle = reflected angle

angle is from normal turned to line of ray