Question

Two waves traveling in opposite directions on a stretched rope interfere to give the standing wave described by the following wave function:

y(x,t) = 4 sin⁡(2πx) cos⁡(120πt),

where, y is in centimetres, x is in meters, and t is in seconds. The rope is two meters long, L = 2 m, and is fixed at both ends.

The distance between two successive antinodes is:

d_AA = 0.25 m

d_AA = 0.15 m

d_AA = 1 m

d_AA = 0.5m

d_AA = 2 m

Answer 1

we have,

the equation for standing wave as,

________________________________ relation 1

we have in this relation in centimeters ,   in meters and in seconds.

in relation 1,

corresponds to amplitude of vibration of standing wave at position on the rope.

the antinodes are the positions   along the length of the vibrating rope where corresponding  amplitude are maximum.

we know that the trigonometric sine function has range of    ,

that is

that is,

that is,

maximum value of amplitude is 4,

let,

for which the amplitude is maximum that is 4,

that is corresponds to positions of antinodes along the length of the rope.  

that is,

   ________________________________ relation 2

we have,

and

that is,

using this in relation 2 we get,

we have length of the rope

let,

the length of the rope lies between   to  

that is,

must lie between to  

considering this condition we get ,

that is antinodes are located at succesive  positions of

, , and   

so we get the distance   between two succesive antinodes as,