Two transverse waveforms (A and B) travel through the same medium with a phase difference of 180°. If the amplitude of the waveforms is ±100 m, what is the combined displacement of both waveforms as waveform A reaches maximum displacement?

Question

Two transverse waveforms (A and B) travel through the same medium with a phase difference of 180°.  If the amplitude of the waveforms is ±100 m, what is the combined displacement of both waveforms as waveform A reaches maximum displacement?

Quizplus · Accepted Answer

11ecba2d_1312_6539_a3bf_e3ad8d87cbfe_MD0008_00 Mechanical waves transmit energy by displacing matter through temporary, periodic oscillations (ie, vibrations). Transverse mechanical waves are modeled by sinusoidal functions that express the momentary displacement of the propagation medium (y-axis) through a cyclical interval (x-axis). Sinusoidal functions are used to model mechanical waves because each function contains terms analogous to the fundamental properties of waves (eg, frequency, amplitude, periodicity). In particular, sinusoidal functions used to model mechanical waves repeat every 360° (or 2π radians), which is one complete cycle. For example, a wave function yielding a maximum displacement at 90° will once again yield maximum displacement at 450° following another complete cycle (90° + 360° = 450°). Phase difference (or phase shift) refers to the degree to which the cyclical nature of two equal-frequency waveforms are aligned. For example, a phase difference of 0° indicates that both wave functions are fully overlaid and reach minimum or maximum displacement together at the same point in the waveform cycle. Conversely, a phase difference of 180° (half a cycle) indicates that the maximum of one wave function occurs concurrent to the minimum of the other waveform. As a result, phase difference expresses the degree of constructive and destructive interference between two waveforms. In this question, waveforms A and B are out of phase by 180°. Therefore, when waveform A reaches its maximum displacement (+100 m) waveform B reaches its minimum displacement (-100 m). The expression for total displacement caused by two interfering waveforms is the sum of the individual relative displacements (y-axis values) of each waveform: Total displacement = Waveform A displacement + Waveform B displacement Total displacement = (100 m + -100 m) = 0 m Therefore, the combined displacement of both waveforms as waveform A reaches maximum is 0 m. Educational objective: Phase difference relates the alignment of two equal-frequency waveforms to one another. Phase differences can result in constructive interference, destructive interference, or no interference.

Two Transverse Waveforms (A and B) Travel Through the Same