Question

two travelling waves of equal amplitude and equal frequencies move in opposite directions along a stting. they interfere to produce a standing wave having the equation y=Acos Kx din wt in which A=1.0mm ,k =1.57cm and which=78.5.

find the velocity of the component travelling waves?​

Answer 1

★ See this attachment...

Answer 2

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

Hope it helps!