Question

Two waves y₁(x,t) and y₂(x,t) propagate in the air: y₁ = A sin(6x - 12t) y₂ = A sin(5x - 10t)
Find:
a) The equation of the resulting pulse, y₁ + y₂.
b) The distance between 2 consecutive zeros in the elongation.
c) The distance between 2 consecutive absolute maxima of the elongation.

Answer 1

Given:

y1 = Asin(6x - 12t)

y2 = Asin(5x - 12t)

a) By superposition principle, resultant displacement y is given by:

y = y1 + y2

y = Asin(6x - 12t) + Asin(5x - 12t)

Now, using

sinC + sin D = 2sin{(C+D)/2} cos{(C - D)/2}

y = 2Asin{(11x - 24t)/2} cos(x/2)

y = {2Acos(x/2)} sin{(11x/2) - (24t/2)}

y = {2Acos(x/2)} sin{(11x/2) - (12t)} ...........(1)

This is the resultant pulse equation.

In the above equation:

i. resultant amplitude is :

A' = 2Acos(x/2) ..........(2)

ii. wavelength ¥ is given by:

2π/¥ = 11/2

¥ = 4π/11 units

iii. angular frequency w is:

w = 12 units

b.

Now, for minimum amplitude (A' = 0) in eq.(2):

cos(x/2) = 0

x/2 = π/2 , 3π/2 , 5π/2 , 7π/2 , .......

x = π , 3π , 5π , 7π , ......

Now, phase of 2π means one cycle, which is equal to path of one wavelength ¥, so, phase of π is equal to path of ¥/2, so:

x = ¥/2 , 3¥/2 , 5¥/2 , ......

so, distane between two successive minima is:

x_{​​​​​​2}​​​ - x_{​​​​​​1} = (3¥/2) - (¥/2)

∆x = ¥ = 4π/11 units (ans)

c.

Now, for maximum amplitude in eq.(2):

cos(x/2) = 1

x/2 = 0 , π , 2π , 3π , .........

x = 0 , 2π , 4π , 6π , ......

Now, phase of 2π means one cycle, which is equal to path of one wavelength ¥, so:

x = 0 , ¥ , 2¥ , 3¥ , ......

so, distane between two successive maxima is:

x_{​​​​​​2}​​​ - x_{​​​​​​1} = ¥ - 0

∆x = ¥ = 4π/11 units (ans)

(Feel free to ask for any doubts, and please like or dislike the answer as per your experience. Thank you)