How Do You Calculate the Combined Phase Angle Shift of Multiple Waves?

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SUMMARY

The combined phase angle shift of multiple waves can be calculated using vector addition of their respective amplitudes and phase angles. In this discussion, three waves with amplitudes and phase angles were analyzed: wave 1 (A=5, φ=0), wave 2 (A=5, φ=π/4), and wave 3 (A=9, φ=π/2). The correct approach involves converting the polar coordinates (A, φ) into Cartesian coordinates (x, y) for each wave, followed by standard vector addition to find the resultant amplitude and phase angle. The formula φ' = arctan(a2sin(φ)/(a1+a2cos(φ)) is essential for calculating the phase shift between two waves.

PREREQUISITES
  • Understanding of wave properties, including amplitude and phase angle
  • Familiarity with polar and Cartesian coordinate systems
  • Knowledge of vector addition techniques
  • Basic proficiency in trigonometric functions and their applications
NEXT STEPS
  • Learn how to convert polar coordinates to Cartesian coordinates
  • Study vector addition in the context of wave interference
  • Explore the law of cosines as it applies to wave phase shifts
  • Investigate the implications of phase differences in wave mechanics
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Students studying physics, particularly those focusing on wave mechanics, as well as educators and anyone interested in understanding the mathematical principles behind wave interactions.

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Homework Statement



If I have three different waves each with difference amplitude and phase angle how would i find the combined phase angle shift?

wave 1: A=5 phi=0
wave 2: A=5 phi=pi/4
wave 3: A=9 phi=pi/2

Homework Equations



phi'=arctan(a2sin(phi)/(a1+a2cos(phi)), where phi in this case is the difference between the phase angles of two waves.

The Attempt at a Solution



My first strategy was to add wave 1 and 2 and then add their sum to wave three. this gave me the correct combined amplitude of 15.29. I then applied the same thinking to the phase angles but the answer i got was incorrect so i tried vector addition and using the law of cosines and I got a nonsensical answer - arcos(-1.41). Needless to say i need a nudge in the right direction. Thanks for the help
 
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Think of the (A,Φ) values as polar coordinates of vectors. Convert them to (x,y) coordinates, and then it's standard vector addition from there.
 

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