Three sinusoidal waves of the same frequency travel along a...

Three sinusoidal waves of the same frequency travel along a string in the positive direction of an $x$ axis. Their amplitudes are $y_{1}, y_{1} / 2,$ and $y_{1} / 3,$ and their phase constants are $0, \pi / 2,$ and $\pi,$ respectively. What are the (a) amplitude and (b) phase constant of the resultant wave? (c) Plot the wave form of the resultant wave at $t=0,$ and discuss its behavior as $t$ increases.

Three sinusoidal waves of the same frequency travel along a string in the positive direction of an $x$ axis. Their
amplitudes are $y_{1}, y_{1} / 2,$ and $y_{1} / 3,$ and their phase constants
are $0, \pi / 2,$ and $\pi,$ respectively. What are the (a) amplitude and
(b) phase constant of the resultant wave? (c) Plot the wave
form of the resultant wave at $t=0,$ and discuss its behavior as $t$
increases.

Key Concepts

Superposition Principle

The superposition principle states that when two or more waves meet, their displacements add together point by point. In the context of sinusoidal waves, this means that the total wave is the sum of the individual sinusoidal functions, which can result in constructive or destructive interference depending on the relative phases and amplitudes.

Phasor Representation

Phasor representation involves expressing sinusoidal functions as rotating vectors (phasors) in the complex plane. Each wave is represented by a vector whose length is the amplitude and whose angle corresponds to the phase constant. This method simplifies the process of adding sinusoidal functions of the same frequency by converting the problem into one of vector addition.

Vector Addition

Vector addition is the process of summing the individual phasors corresponding to the component waves. By adding their horizontal (real) and vertical (imaginary) components, one can determine the overall (resultant) amplitude and phase constant of the combined wave. This technique provides a clear visualization of how different phases and amplitudes interact.

Interference

Interference occurs when waves of the same frequency combine, leading to regions of increased (constructive interference) or decreased (destructive interference) amplitude. The specific interference pattern results from the phase differences between the waves, which influence the overall shape and amplitude of the resulting wave.

Time Evolution of Waves

The analysis of wave behavior as time progresses involves understanding that a wave's phase continuously shifts with time. For sinusoidal waves of the same frequency, the combined waveform will also oscillate sinusoidally. The time evolution maintains the shape determined by the initial phase constants and amplitudes but shifts in position along the propagation direction according to the wave’s speed.