Determine whether the traveling wave v(x, t) = E_m cos(βx − ωt) represents a backward wave or not. Statement: The wave E_m cos(βx − ωt) is a backward wave.
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ATrue
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BFalse
Answer
Correct Answer: False
Explanation
Introduction:The direction of travel of a sinusoidal wave is inferred from the sign convention in its phase. Recognizing forward versus backward waves is fundamental in electromagnetics and transmission-line analysis.
Given Data / Assumptions:
- Waveform: v(x,t) = E_m cos(βx − ωt)
- β > 0, ω > 0
- Medium is uniform
Concept / Approach:For a phasor with phase ψ = βx − ωt, setting ψ = constant gives βx − ωt = constant → x = (ω/β) t + constant/β, which increases with time. Hence the phase front moves in +x direction: a forward wave. A backward wave traveling toward −x would have phase βx + ωt (or equivalently cos(ωt + φ + βx)).
Step-by-Step Reasoning:
1) Take βx − ωt = C → x(t) = (ω/β) t + C/β.2) Since ω/β > 0, x increases with t → motion toward +x.3) Therefore, v(x,t) represents a forward wave, not a backward wave.Verification / Alternative check:Equivalently, write v(x,t) = Re{E_m e^{j(βx − ωt)}}; the sign convention with −ωt and +βx indicates +x propagation.
Why Other Options Are Wrong:
- True: would require βx + ωt for backward travel in this convention.
Common Pitfalls:Mixing time-domain and phasor sign conventions; assuming any “−ωt” implies backward propagation.
Final Answer:False