Four sound waves are to be sent through the same tube of air, in the same direction: s 1 ( x , t ) = ( 9.00 n m ) cos ⁡ ( 2 π x − 700 π t ) s 2 ( x , t ) = ( 9.00 n m ) cos ⁡ ( 2 π x − 700 π t + 0.7 π ) s 3 ( x , t ) = ( 9.00 n m ) cos ⁡ ( 2 π x 700 π t + π ) s 4 ( x , t ) = ( 9.00 n m ) cos ⁡ ( 2 π x − 700 π t + 1.7 π )What is the amplitude of the resultant wave?

Question

Four sound waves are to be sent through the same tube of air, in the same direction:      s          1        (  x  ,  t  )  =  (  9.00  n  m  )  cos  ⁡  (  2  π  x  −  700  π  t  )      s          2        (  x  ,  t  )  =  (  9.00  n  m  )  cos  ⁡  (  2  π  x  −  700  π  t  +  0.7  π  )      s          3        (  x  ,  t  )  =  (  9.00  n  m  )  cos  ⁡  (  2  π  x  700  π  t  +  π  )      s          4        (  x  ,  t  )  =  (  9.00  n  m  )  cos  ⁡  (  2  π  x  −  700  π  t  +  1.7  π  )What is the amplitude of the resultant wave?

cercimw · Accepted Answer

The sound waves are      s          1        (  x  ,  t  )  =  (  9.00  n  m  )  cos  ⁡  (  2  π  x  −  700  π  t  )      s          2        (  x  ,  t  )  =  (  9.00  n  m  )  cos  ⁡  (  2  π  x  −  700  π  t  +  0.7  π  )      s          3        (  x  ,  t  )  =  (  9.00  n  m  )  cos  ⁡  (  2  π  x  700  π  t  +  π  )      s          4        (  x  ,  t  )  =  (  9.00  n  m  )  cos  ⁡  (  2  π  x  −  700  π  t  +  1.7  π  )From super position The net wave is  S  (  x  ,  t  )  =      s          1        (  x  ,  t  )  +      s          2        (  x  ,  t  )  +      s          3        (  x  ,  t  )  +      s          4        (  x  ,  t  )By using the sum identities  cos  ⁡  x  +  cos  ⁡  y  =  2  cos  ⁡  (            x      +      y        2    )  cos  ⁡  (            x      −      y        2    )  S  (  x  ,  t  )  =  (      s          1        (  x  ,  t  )  +      s          3        (  x  ,  t  )  )  +  (      s          2        (  x  ,  t  )  +      s          4        (  x  ,  t  )  )  =  (  18  n  m  )  cos  ⁡  (  2  π  x  −  700  π  t  +  π      /    2  )  cos  ⁡  (  π      /    2  )  +  (  18  n  m  )  cos  ⁡  (  2  π  x  −  700  π  t  +  1.2  π  )  cos  ⁡  (  π      /    2  )  =  0Result:S=0

Four sound waves are to be sent through the same tube of air, in the same direction: s_1(x,t)=(9.00nm)cos(2pix−700pit) s_2(x,t)=(9.00nm)cos(2pix−700pit+0.7pi) s_3(x,t)=(9.00nm)cos(2pix700pit+pi) s_4(x,t)=(9.00nm)cos(2pix−700pit+1.7pi) What is the amplitude of the resultant wave?

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