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Two travelling waves of equal amplitudes and equal frequencies move in opposite directions along a string. They interfere to produce a standing wave having the equation y = A cos kx sin cot in which A = PO mm, k = P57 cm and co = 78'5 s (a) Find the velocity of the component travelling waves. (b) Find the node closest to the origin in the region x> 0. (c) Find the antinode closest to the origin in the region x> 0. (d) Find the amplitude of the particle at x 2.33 cm.

Two travelling waves of equal amplitudes and equal frequencies move in opposite directions along a string. They interfere to produce a standing wave having the equation y = A cos kx sin cot in which A = PO mm, k = P57 cm and co = 78'5 s (a) Find the velocity of the component travelling waves. (b) Find the node closest to the origin in the region x> 0. (c) Find the antinode closest to the origin in the region x> 0. (d) Find the amplitude of the particle at x 2.33 cm.

Grade:12

3 Answers

Saurabh Koranglekar
askIITians Faculty 10341 Points
one year ago
Dear student

Please write the Question in a standard form or attach an image of the question

Regards
Vikas TU
14149 Points
one year ago
Dear student 
Question is not clear 
Please attach an image, 
We will happy to  help you 
Good Luck
Cheers
Arun
25763 Points
one year ago

Here, the equation of standing wave is  
y=Acoskxsinωt
y=Acoskxsinωt
where  
A=1.0mm=0.1cm,k=1.57cm,
ω=78.5s
As travelling waves are of equal amplitudes equal frequencies and moving in opposite directions, their equations can be written as  
y1=
1=  A/2  sin(ωt−kx)
y2
2=A/2  sin(ωt+kx)
y=y1+y2 =
A/2  [sin(ωt−kx)+sin(ωt+kx)]
=Acoskxsinωt.
(a) Wave velocity of either wave
υ=
ωk=
78.5s/1.57c
=50cm/s
υ=ωk=78.5s-11.57cm-1=50cm/s
(b) For a node,  
y=0,
∴coskx=0,kx=π/2
x=
π/2 whole divided by  k
=3.14/2 whole divided by  1.57
=1cm
(c) For an antinode,  
y=max,
for which  
|coskx|=1
kx=nπ=π  for smallest value of x  
x=
π/k=3.14/1.57=2cm
(d) Teh amplitude of vibration  
=|Acoskx|
r=1.0cos(1.57×2.33radian)
=1.0cos(3.658rad)
=1.0cos(3.658×57degree)
=1.0cos(209∘)= -cos29∘
r=−0.875mm
 

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