Worksheet: Wave Superposition

In this worksheet, we will practice calculating the resultant wave function of multiple wave functions occupying the same space and time as each other.

Q1:

Two sinusoidal waves are moving through a medium in the positive 𝑥-direction, both having an amplitude of 7.00 cm, a wave number of 3.00 m−1, an angular frequency of 2.50 s−1, but one has a phase shift of an angle 𝜋12 rad. What is the displacement of the resultant of these waves at a time 𝑡=2.00s and a position 𝑥=0.530m?

Q2:

Two sine waves travel along a string, modeled as 𝑦(𝑥,𝑡)=0.304.0𝑥+3.0𝑡+𝜋3msin and 𝑦(𝑥,𝑡)=0.60(8.0𝑥6.0𝑡)msin, where 𝑥 is measured in meters and 𝑡 is measured in seconds. What is the height of the resultant wave formed by the interference of the two waves at the position 𝑥=1.0m at the time 𝑡=3.0s?

Q3:

Two sinusoidal waves are moving through a medium in the same direction, both having an amplitude of 3.00 cm, a wavelength of 5.20 m, and a period of 6.52 s, but one of the waves has a phase shift of an angle 𝜙. The resultant of the waves has an amplitude of 5.00 cm. Find 𝜙 by using the fact that sinsinsincos𝑢+𝑣=2𝑢+𝑣2𝑢𝑣2.

Q4:

What is the displacement of the resulting superposition of the three wave functions 𝑦(𝑥,𝑡)=0.040m sin (4.0𝑥2.0𝑡), 𝑦(𝑥,𝑡)=0.040m sin (2.0𝑥+4.0𝑡) , and 𝑦(𝑥,𝑡)=0.040m sin (8.0𝑥1.0𝑡) at the position 𝑥=4.00m at the time 𝑡=8.0s ? For each of the wave functions, 𝑥 is measured in meters and 𝑡 is measured in seconds.

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