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Worksheet: Wave Superposition

Q1:

Two sinusoidal waves are moving through a medium in the positive π‘₯ -direction, both having an amplitude of 7.00 cm, a wave number of 3.00 mβˆ’1, an angular frequency of 2.50 sβˆ’1, and a period of 6.00 s, but one has a phase shift of an angle πœ‹ 1 2 rad. What is the displacement of the resultant of these waves at a time 𝑑 = 2 . 0 0 s and a position π‘₯ = 0 . 5 3 0 m ?

Q2:

Two sine waves travel along a string, modeled as 𝑦 ( π‘₯ , 𝑑 ) = 0 . 3 0 ο€» 4 . 0 π‘₯ + 3 . 0 𝑑 + πœ‹ 3  1 m s i n and 𝑦 ( π‘₯ , 𝑑 ) = 0 . 6 0 ( 8 . 0 π‘₯ βˆ’ 6 . 0 𝑑 ) 2 m s i n , where π‘₯ is measured in meters and 𝑑 is measured in seconds. What is the height of the resultant wave formed by the interference of the two waves at the position π‘₯ = 1 . 0 m at the time 𝑑 = 3 . 0 s ?

Q3:

What is the displacement of the resulting superposition of the three wave functions 𝑦 ( π‘₯ , 𝑑 ) = 0 . 0 4 0 m sin ( 4 . 0 π‘₯ βˆ’ 2 . 0 𝑑 ) , 𝑦 ( π‘₯ , 𝑑 ) = 0 . 0 4 0 m sin ( 2 . 0 π‘₯ + 4 . 0 𝑑 ) , and 𝑦 ( π‘₯ , 𝑑 ) = 0 . 0 4 0 m sin ( 8 . 0 π‘₯ βˆ’ 1 . 0 𝑑 ) at the position π‘₯ = 4 . 0 0 m at the time 𝑑 = 8 . 0 s ? For each of the wave functions, π‘₯ is measured in meters and 𝑑 is measured in seconds.

Q4:

Two sinusoidal waves are moving through a medium in the same direction, both having an amplitude of 3.00 cm, wavelength of 5.20 m, and a period of 6.52 s, but one of the waves has a phase shift of an angle πœ™ . The resultant of the waves has an amplitude of 5.00 cm. Find πœ™ by using the fact that s i n s i n s i n c o s 𝑒 + 𝑣 = 2 ο€Ό 𝑒 + 𝑣 2  ο€» 𝑒 βˆ’ 𝑣 2  .