Question

The equation of a stationary wave is given by y(x,t) = 2sin πx cos100πt, where x and y
are measured in metre and t in second. Calculate the amplitude, wavelength and
frequency of component waves whose superposition generated this stationary wave. Also
write the equations of component waves.

Answer 1

y  (x , t ) =  2  Sin πx    Cos  100π t 
     x and y are in meters  and t is in seconds.

we use the formula in trigonometry  2 Sin A Cos B = Sin (A+B)  + Sin (A-B)

  y (x, t)  =  2  Sin (π x)     Cos (100 π t)
                 =  Sin (πx + 100πt)    +    Sin  (πx - 100 πt)
                 =  Sin (πx + 100πt)    -    Sin  (100πt  -  πx )
                 =  Sin (100 πt + πx)   +  Sin  (100 πt  -  πx + π)

general formula  for a standing wave :  y (x, t) = A Sin (ω t - k x + Ф)

  these are the component waves which are part of the stationary wave.

component wave 1:    y1 (x,t)  =  sin (π  x  +  100 π t)
       angular frequency = ω    = 100 π  radians/sec
      frequency = f  = 50 Hz = ω/2π              Time period: 1/f = 0.02 Sec.
       Amplitude = A = 1 m
       wave number  k =  - π  rad/meter                       we have formula   ω = k v
           velocity  v =  ω / k  = - 100 meters/sec
         wavelength λ =  v / f  =  100/50  meters =  2 meters

  this component of the wave is traveling in the negative x direction. so its velocity is negative.
 
==========================

component wave 2:    y1 (x,t)  =  sin ( 100 π t - π x +  π )
       angular frequency = ω    = 100 π  radians/sec
      frequency = f  = 50 Hz = ω/2π              Time period: 1/f = 0.02 Sec.
       Amplitude = A = 1 m
       wave number  k =  π  rad/meter                       we have formula   ω = k v
           velocity  v =  ω / k  = 100 meters/sec
         wavelength λ =  v / f  =  100/50   meters =  2 meters
   initial phase angle = π radians

 this wave is traveling in the positive x direction.  so  k and v are positive.

But the waves are have a phase difference also.

Answer 2

Answer:y  (x , t ) =  2  Sin πx    Cos  100π t 

    x and y are in meters  and t is in seconds.

we use the formula in trigonometry  2 Sin A Cos B = Sin (A+B)  + Sin (A-B)

 y (x, t)  =  2  Sin (π x)     Cos (100 π t)

                =  Sin (πx + 100πt)    +    Sin  (πx - 100 πt)

                =  Sin (πx + 100πt)    -    Sin  (100πt  -  πx )

                =  Sin (100 πt + πx)   +  Sin  (100 πt  -  πx + π)

general formula  for a standing wave :  y (x, t) = A Sin (ω t - k x + Ф)

 these are the component waves which are part of the stationary wave.

component wave 1:    y1 (x,t)  =  sin (π  x  +  100 π t)

      angular frequency = ω    = 100 π  radians/sec

     frequency = f  = 50 Hz = ω/2π              Time period: 1/f = 0.02 Sec.

      Amplitude = A = 1 m

      wave number  k =  - π  rad/meter                       we have formula   ω = k v

          velocity  v =  ω / k  = - 100 meters/sec

        wavelength λ =  v / f  =  100/50  meters =  2 meters

 this component of the wave is traveling in the negative x direction. so its velocity is negative.

 

==========================

component wave 2:    y1 (x,t)  =  sin ( 100 π t - π x +  π )

      angular frequency = ω    = 100 π  radians/sec

     frequency = f  = 50 Hz = ω/2π              Time period: 1/f = 0.02 Sec.

      Amplitude = A = 1 m

      wave number  k =  π  rad/meter                       we have formula   ω = k v

          velocity  v =  ω / k  = 100 meters/sec

        wavelength λ =  v / f  =  100/50   meters =  2 meters

  initial phase angle = π radians

 this wave is traveling in the positive x direction.  so  k and v are positive.

But the waves are have a phase difference also.