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.

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:

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